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"
Y
A X
t h e branch has a member i n each i n t e r v a l o f
(remembering K6nig
order,
Y i s an i n f i n i t e branch i n
..Am]
Then f o r some t r u e l y f i n i t e
)
,
€aN.Suppose
t r u e l y f i n i t e m, q
m(*)
J,
"says" t h a t
(whose o r d e r i s i n c l u d e d i n t h e n a t u r a l o r d e r o f & N ) .
By t h e d e f i n i t i o n o f
A
j,
€aR a,, f i r s t
<
y
A
(X < 2 <
y)]
s a t i s f i e s " f o r every
e(,,p,q,A,
we can r e p l a c e
lemma) t h e s e t
Y)
Hence
m(*)+i'
i n the interval
Am(*)+i t* ( i n t h e t r e e
( 3 2 E
t
t* such t h a t f o r e v e r y
,
(F2)
i t i s easy t o see t h a t
m(*)+l
x < y < a (x,y)
there i s
,...,Am)". by
m+l,
hence c l e a r l y
153
On Logical Sentences in PA
ip
€ f i N:
{p
E
a k ( 3 Y ) *(Y,
aN:
f o r every a
P, ill = E
y
...,Am )
BR a s
which belongs t o 2.7.
t* ( i n t h e t r e e
(x,y)
B(,,p,q,
l.
required.
m: For every n a t u r a l number
3 n CPQh ( n ,
in t h e i n t e r v a l
Am+1
E
t h e r e i s an element below Al,
t* such t h a t f o r every
there i s
A,+,
x < y < a , X E Am + l ,
k we can prove i n PA(Q
MM
) t h e statement
k).
Proof: F i r s t we prove t h a t t h e conclusion i s t r u e , i . e . t r u e i n t h e universe
6?)if
i t i s a model of say second o r d e r Peano a r i t h m e t i c .
Then f o r every n , t h e r e i s a p a i r
Suppose t h a t the conclusion f a i l s f o r k.
(Fn, H ) which forms acounterexample t o CP'
of functions
eh
n
We now d e f i n e by induction on i For
5
k i n f i n i t e sets
so t h a t R i + ,
Ai,
5 Ai.
i = 0 t h e r e i s no problem. Let A, = B, = t h e set of a l l natural numbers.
So suppose we have defined
A
for j s i
j
and we s h a l l define
By t h e i n f i n i t e Ramsey theorem we can get an i n f i n i t e ( a ) f o r every y Ai n y )
Now
(n,k).
z
<
in
A:,
f
j
for j = i+l.
Af 5 Ai such t h a t : F ( A , n y , A, n y , ...,
i s t h e function
Y¶Z
A
Z
restricted t o y.
f
does not depend on
z, i . e . i f
y
z,
< z,,
E
A
then
YSZ
SO l e t
f
= f
Y
for y
Y,Z
( b ) For
Y,
<
<
A' i '
z
Y, < y 2 i n
f
= YlZ,
f
YJ,
.
1
fy,ry, = fy,ry0 [Simply apply Ramsey theorem t o t h e natural t h r e e place c o l o u r i n g , f i r s t f o r ( a ) then f o r ( b ) l So
f
=
U If
Y
rz
are i n A i l
: z < y
i s a kcolouring ( o f t h e natural num
b e r s ) . By t h e i n f i n i t e Ramsey theorem t h e r e i s i n
&3,
an i n f i n i t e s e t A:
A:,
which i s fhomogenous. Now we s h a l l deal with
H.
Again t h e r e is an i n f i n i t e
A: 5 A f
(A;
of course) such t h a t :
(c) i f H Z ( p , A, n y ,
y
<
z
. .., Ai
are i n
A:,
ny,
9)
< z
a r e in
95 Ai
n
y,
p
does not depend on
<
y
t h e n t h e value of
z.
Moreover (d) i f
x
<
y
A;,
9 5 Ai
nx,
p
<
x then t h e value of
BR,
S. SHELAH
154
..., Ai
HZ(p, A, n y.
For every p
9)
n y,
63,
E
and i n f i n i t e s e t
zero i f f f o r a r b i t r a r i l y large q HZ(p, A,
..., Ai
y,
n
<
. 9):
z in A i / q
i t is
the value of
i s zero. Clearly a f i n i t e change i n
HZ, and "for a r b i t r a r i l y large q
"for every large enough q "
z )
we define T ( p ,
5 A:
f o r every y
E ( % ~
n y, D n y  q )
n o t change the value of
(and on
does not depend on y
3
does
can be replaced by
"
(see 2 . 2 ) .
Now we want t o apply the Galvin Prikry theorem t o T, more exactly t o a parametrized version of i t
( p as the parameter). Simply i t e r a t e the usual Galvin
Prikry theorem on the natural numbers, and take the diagonal intersection. What we get we call
.
A;
Again remembering 2.2, and the conclusion of the GalvinPrikry theorem, we can
so t h a t f o r every p,
find Aitl 5 A:, Ai/q
the value of
i f f o r some q > p ,
..., A 1.
HZ(p,A, n y ,
n Y, Ai+,nYq)
f o r every y
i s zero,
<
z in
then 9 = P
will serve. If
..., Ak
A,, n
A,,
9
Aktl
are defined, choose y
y > contradicts thechoice of
Fz, HZ
<
z in Aktl
and < A , n y,
.
However we want t o prove t h i s in PA(QMM) (and not in ZFC o r i n second order number theory). The proof i s the same, replacing "a s e t of natural numbers" by "a definable s e t of natural numbers". Why i s < < F , H > : n < W > definable? We canchoose for each n , a minimal n n pair < F n , H n > (by a simple enough coding). We can apply the i n f i n i t e Ramsey theoMM
rem as Macintyre [MI proves theparallel statement holds (from PA(Q ), of course the proof depends on the formula defining the colouring and the i n f i n i t e s e t ) . Wore exactly he proves that i f
,...,xn,
~(x,

p, z)
A
i s i n f i n i t e and definable,
i s such that
then there i s a definable i n f i n i t e
V x1
B c A and co
<
,...,xn
( 3 2 < c ) ~ ( x ,..., , xn,p,z)
c such that
We are l e f t with the "parametrized GalvinPrikry". Let a < * b mean:
<
a , b code f i n i t e increasing sequences, ,
e ( a ) > respectively, and ca(m) : m < a ( a ) > i s a proper i n i t i a l seg
ment of < b ( m ) : m
<
e(b)>.
155
On Logical Sentences in PA We d e f i n e
A:
A:
(after
has been d e f i n e d )
by d e f i n i n g by i n d u c t i o n on
I, kk, 6, such t h a t k,
(a)
<
... <
k
2 1
L o , C1,
and
..., sIl
{0,11
E
> < * a < b and f o r e v e r y p < k f o r some q, (QMMa,b) C < k o , ..., k f i  l y, z i f q < y < z, y E A?, z E A? t h e n HZ(p, A, n y, A . n y.
(b) f o r every
...,
1
Ia(m) : m < z ( a ) > / q ) =
1
cPl.
c ~ = 0~ . Now we can c a r r y t h e d e f i n i t i o n (as i n [ M I n o t i n g t h e f o r m u l a s we use have (c) i f compatible w i t h
(a) t (b),
bounded complexity).So we f i n i s h t h e p r o o f o f 2 . 7 .
5 3. A t r u e
nysentence o f P A n o t p r o v a b l e i n PA
I n summer '80 Friedman and H a r r i n g t o n o f f e r e d h o t l y t h e i r view t h a t i t i s one o f t h e main problems o f contemporary l o g i c t o f i n d mathematical sentences as ment i o n e d i n t h e t y t l e , as w e l l as t o f i n d n a t u r a l t h e o r i e s w i t h incomparable c o n s i stency s t r e n g t h . The " t e c h n i c a l d i f f i c u l t y " i s i m m a t e r i a l ; i n f a c t t h e easiness o f t h e p r o o f may i n d i c a t e t h e profoundness and n a t u r a l i t y o f t h e sentence. Now an answer t o such q u e s t i o n i s n a t u r a l l y more open t o debate t h a n t h e usual mathematic a l problem.
A s t h e a u t o r d i d n o t want t o go i n t o such d i s c u s s i o n , and H a r r i n g t o n wanted a s o l u t i o n , an agreement was reached: i f t h e a u t h o r c o u l d f i n d a s o l u t i o n which H a r r i n g t o n would t h i n k i s O.K.,
he would w r i t e i t up, d i s c u s s i t and p u b l i s h i t .
The c o n t e n t o f t h i s s e c t i o n was done i n s p r i n g '81, H a r r i n g t o n O.K.ed i t , as w e l l as I 4 ( w h i c h was done i n summer '80) b u t was t o o l a z y t o f u l f i l l h i s promise. 3.1. C o n t e x t :
N be a n a t u r a l number
1) Let
r
= < r : I < I(;)> 9"
a finitesequence
o f n a t u r a l numbers.
2) L e t
K = K i = {(A,
<,
R)
: A
6 R
a subset o f
N,
<
a l i n e a r order o f
A
a sequence o f r e l a t i o n s o v e r an
A
,
,
r place
Ik e ( R ) = I(;)}.
3 ) Members o f
K
a r e denoted by
A, B
,
letting
power o f t h e s e t o f t h e s e t o f elements o f 4) I n (4a)
K
A =
/ ~ ,l
so
i s the
llAll
A.
we d e f i n e
A <en B
( B an end e x t e n s i o n o f
A) i f A
i s a submodel o f
B
and
S. SHELAH
156 x
IBI
E
,
 1.4
(4b) A < B
y
satisfying
implies
y < x.
of
A'
into G
IIb/l 5 IlAll
A',
A <
en B over
i n t o @(N)
Let
+
1,
A < B and f o r any en t h e r e i s an embedding
.
A
f r o m @(N)
3.2. E ( N , r , k , n )  p r i n c i p l e : F
PI
( B an u n i v e r s a l end e x t e n s i o n o f A) i f
A'
5) A function
1 ) Domain:
E
F
is
be a k  p l a c e f u n c t i o n s a t i s f y i n g
i s d e f i n e d on i n c r e a s i n g sequences o f l e n g t h
2) Choice F u n c t i o n :
..., A k )
F(A,,
...
3 ) Isomorphism I n v a r i a n c y : i f A 1 < i s an isomorphism
g
from A k
< Ak
onto
k
K
from
.
IA,[
E
if / G ( A ) I z f ( 1 A I ) .
fsmall
B
k
E
K,
B1 <
mapping
... < B k
At
onto
E
Bil
K , and F
then
there and
g
commute, i .e. F(B~.
..., B k ) =
4 ) Weak H e r e d i t a r i t y : F o r e v e r y such t h a t : i f
f
B
1
=
0)
then
Then  there
A,
+)
A2 <
<
i s a submodel o f
B~
F(A,
...,
g(F(Al,
,...,A k )
= F(Bl
...
. k t h e r e i s an x small f u n c t i o n
< Ak,
f(Bill)
Ai,l
n Ail
5 Bil
(stipulating
,..., B k ) .
i s an i n c r e a s i n g sequence < A :~ il < n > on which
F
depends on
the f i r s t structure only. 3.3.
Fact
F to
F'
if
N > 22 k+n+'(r(i)+l),
F a s i n 3.2 and
N' >
a k  p l a c e f u n c t i o n s a t i s f y i n g 1 )  4 ) o f 3.2 f o r
N,
t h e n we can e x t e n d
N ' , i n one and o n l y one
way. __ P r o o f . By 3.2 ( 4 ) ( 3 ) (as i n t h e p r o o f o f 1.3 ( 1 ) ( b ) ) . 3.4.
Claim: I n
PA+PH
$* = ( V
r,
we can p r o v e (the
k, il)
r,
(N,
k , n )  p r i n c i p l e h o l d s f o r e.g.
,k+n+ f C r ( i ) + l l
).
N = 2 __ P r o o f : D e f i n e by i n d u c t i o n on
m
e,
i s s m a l l e r enough t h a n
putation). Applying
PH
and i n d i s c e r n i b l e f o r F ' ( A ~, 1
e,
a model
Am<*A
e
we g e t a s e t (F'
as i n 3.2
..., A .
'k
) = F'(A. Ji
.
e
c
K
w i t h universe
il,
so t h a t i f
( j u s t t a k e c a r e o f 3.1 ( 4 ) ( b ) , easy com
C
o f n a t u r a l numbers
for
,
A
N'
..., Ajk)
(V x < y ~ C ) C 2 2 ~ < y l ,
l a r g e enough)
,
and
ICI
>
4 Min C
157
On Logical Sentences in PA (we use an e q u i v a l e n t v a r i a n t o f P.H.).
: i
F
C>,
E
As i n 5 1
depends on t h e f i r s t v a r i a b l e o n l y . Now as i n t h e p r o o f o f 1.3
( 1 ) ( b ) we c o l l a p s e t h e s o l u t i o n below 3.5.
Claim: I n
PA+$*
N
.
we can p r o v e t h e c o n s i s t e n c y o f PA (hence
Proof. We can b u i l d a nonstandard model o f dard,
N
l a r g e enough, and
..., Ak)
F(A,,
mal code f o r which
1
( v y)
e a s i l y f o r subsequences o f
r
PA,+$*,
<4> d e f i n e
F :
i s d e f i n e d as f o l l o w s :
let
{3Xcp(X.y)
=
..., A k > / = +
M, choose
cp(x,y)
3X[cp(X,y)
A
(v
z
< X)
and t h e n
( i f t h e r e i s one).
A,,
cp(x,y)
non s t a n 
be a f o r m u l a w i t h m i n i 
ic p ( Z , j ) l
(by the lexicographic order o f
E
k, n
"induction fails,i.e.
and t h e n t a k e minimal x
PAP$*).
1''
<Max
y.
y o , yl,
The r e s t i s as i n I 1
...>
)
the only
a d d i t i o n a l p o i n t i s why a r e a d d i t i o n and m u l t i p l i c a t i o n d e f i n a b l e ? T h i s i s by 3.1 ( 4 ) ( b ) .
5
4. On c o n s i s t e n c y s t r e n g t h
Let extending CON(T)
T PA
denote h e r e a ( r e c u r s i v e ) t h e o r y ( w i t h f i n i t e  s o r t , f i n i t e  l a n g u a g e ) b u t i t may a l s o "speak" on r e a l s and even a r b i t r a r y s e t s .
be t h e sentence ( i n
D e f i n i t i o n : We say
PA
T, scs T,
language) s a y i n g
T
i s consistent.
( t h e consistency strength o f
equal) than t h e consistency strength o f
T2)
if
PA
Let
t CON(T,)
T,
i s smaller (or f
CON(T,):
I t was observed t h a t e s s e n t i a l l y " l a r g e c a r d i n a l axioms a r e l i n e a r l y ordered"
(though i n some cases t h i s "has n o t y e t been proved").More e x a c t l y i t seems t h a t a l l s e t t h e o r i e s which has been c o n s i d e r e d so f a r , a r e l i n e a r l y o r d e r e d by Solovay
( I t h i n k ) has
found
T's
which a r e
5
incomparable,
s
cs b u t t h e y were
.
cs " p a r a d o x i a l " (i .e. have s e l f  r e f e r e n t i a l sentences). We s h a l l t r y t o g e t more r e a sonable ones.
* * * * * * * Let
PA+
be
PA t C O N ( P A ) .
We work i n s i d e
P A . A model w i l l mean one which
i s definable. Let
T,,
T,
be c o n s i s t e n t t h e o r i e s ( i n o u r " u n i v e r s e " which s a t i s f i e s PA).
s. SHELAH
158
T,
" s a y i n g " t h e r e i s a model o f
(think o f
PA+,
PA
+
CFMSI, o r o f course As T,
+
ZFC,
T, " s a y i n g " t h e r e i s a model o f
ATR, see Friedman, McAloon and Simpson
T,tCON(TI)
hence t h e r e i s MI
( 1 ) q Q ( n ) says
n
i s c o n s i s t e n t , hence ( b y Godel incompleteness has a model M,
M,
E
f ''$2(n)''
I=
such t h a t
(n
. By
t h e r e q u i r e m e n t on T,
i s a nonstandard i n t e g e r )
where
i s a n a t u r a l number and t h e r e i s a p r o o f o f s i z e
n
PA
ZFC+large c a r d i n a l s ) .
T, +CON(T,) + iCON(T, +CONTI)
iCON(T,),
n
of
t 1 CON ( Te)
PA
f ( 2 ) +,(n)
says
$
Q
(n)
but
f ( 3 ) T~ = PA + ( 3 n ) r$,(n) i s consistent with
i s t h e f i r s t such number.
n
As we have assumed t h a t
with
TI,
o r use
or
i s consistent,
theorem) M,
I rrlCAo;
T, says t h a t
has a model, c l e a r l y
+,~,,(2~")1 f
+ (Vm) l$,(m)
PA + 3 n$,(n)
( a s even
PA+
PA
i s consistent
.
PA+)
By a theorem o f Friedman (based on a n a l y z i n g Godel incompletness theorem) f
( 4 ) Tb = PA + ( 3 n ) C$,(n)
'
( 5 ) PA+
+
Ta, Tb
are
s
cs
incomparable.
L e t us p r o v e e.q.
*
Ta 1 CON(Ta)
rBecause f o r any model nition
11
PA
We s h a l l p r o v e t h a t csTb
217
+.
i s consistent with
Ta
$,(2
f
o f a mode:
phism from
No
Clearly
N,
of
N o of PA
(i.e.
PA++Ta, No
b e i n g a model o f
1: "N,
i n t o a p r o p e r i n i t i a l segment o f N,
satisfies
of bounded f o r m u l a s and
Ta PA,
has a d e f i 
i s a model o f P A " ) and an isomorN,.
as end e x t e n s i o n s p r e s e r v e t h e s a t i s f a c t i o n
f
$,(x),
PA+
$,(x)
a r e such f o r m u l a s . 1
Note a l s o t h a t (6)
PA
+
.
Ta I 7CON(Tb)
[Otherwise t h e r e i s a model a d e f i n i t i o n o f a model segment o f No I= T
a' formulas,
N, No
(i.e. f "$,(n)
and
f
N, o f No
No
of
PA+Ta+CON(Tb)
Tb a n d a n i s o m o r p h i s m g of
hence i n
No
No
there i s
onto a proper i n i t i a l
s a t i s f i e s t h e sentences s a y i n g t h o s e t h i n g s ) . AS 211 f and I $,(2 ) " f o r some n, b u t as $, $1 a r e bounded f commute w i t h e x p o n e n t i a t i o n , c l e a r l y N, I. "$,(g(n)) and
159
On Logical Sentences in PA 1 $,(22g(n))".
So
f 2m N, I= Tb hence f o r some m, N , I= "$,(m) and $ , ( 2 ) " . f f and $,(g(n))",hence by q 2 ' s definition g ( n ) , m have t o be
But
f
N , k "$,(m)
equal. B u t
N, k "$,(2
2m
)
and 7 $ 1 ( 2 ' g ( m ) ) " hence
g(n),
rn
should be unequal,
contraddictionl. By ( 5 ) and ( 6 ) clearly
PA+
+
C O N ( T a ) I$
(as PA+ + T~
CON(Tb)
is
consistent (by ( 3 ) ) ; t h i s implies
PA I f CON(Ta)
(7)
+
So Ta $ csTb.
CON(Tb)
. Tb $ csTa
i s t o t a l l y analogous. n f NOW (Woodin suggests) wecan replace the sentences ( 3n ) r $ , ( n ) +  1 $ ~ ( 2 ' ) I n f ( 3 n ) r $ , ( n ) + $ J 2 ' ) I by inequalities of the indicator functions corresponding t o
and
The proof of
f , , f,
T I and T, ( i . e . the function T , , T,).
t o the consistency of
exhibiting the
E.g. ( V n ) [ i f
f,(n)
ITsentences :
corresponding
i s defined then
so i s
f , ( f , ( f , ( n ) ) 1 and i t s negation. So i f we accept those functions as "mathematical"
( n o t j u s t reasonable methamathematical ) we get mathematical theories of incomparable consistency strength. (Originally we have used three t h e o r i e s ) . We a r r i v e t o the dangerous question o f which PA
function: on
see Paris and Harrington [ P H I ,
T's
have matheratical indicator
on many theories ( l i k e ZFC+large
cardinal) see Friedman rFl1 on ATR, see Friedman McAloon and Simpson rFMSl on 1
IT,CA,
see 5 2. Alternatively f o r an indicator function
f ( f * ( n ) + l ) , and use
$,
$3
where
$,
f
f*
define
= "the f i r s t
by
f*(O) = 0
n f o r which f ( n ) i s
:Q
f*(n+l)=
mod 4".
+ * * * * * * * Notice the following two phenomena
( A ) For any two natural s e t theories, not only they are
5
cs
comparable, b u t one
i s interpretable in the other ( o r expected t o be s o ) . ( 6 ) Similarly, f o r any theories, e.q. undecidab l i t y r e s u l t s are gotten by i n t e r 
pretation. Friedman rFr21 proved a theorem saying ( A ) i s really true. Concerning ( B ) however, in CSh1, (under C H )
the monadic theory of the re 1 order i s proven undecidable
without the usual interpretation. I n Gurevich and Shelah
CGSl1 t h i s
i s explained i t i s a Booleanvalued i n t e r p r e t a t i o n , and by CGS21 the usual i n t e r pretation i s impossible. Now we can t r a n s l a t e i t t o ( A ) : l i s t the reasonable axioms f o r the monadic theory of the real order (considered as a twosort model).
S. SHELAH
160
REFERENCES
CEHMRl P. Erdos, A. H a j n a l , A. Mate and R. Rado, C o m b i n a t o r i a l s e t t h e o r y , N o r t h H o l l a n d P u b l . Co.
CF11
H. Friedman, On t h e necessary use o f A b s t r a c t s e t t h e o r y , Advances i n Mathem a t i c s 41 (1981), 209280.
TF21
H. Friedman, T r a n s l a t a b i l i t y and r e l a t i v e c o n s i s t e n c y .
C FMS 1 H. Friedman, K. McAloon and S.G. Simpson, A f i n i t e c o m b i n a t o r i a l p r i n c i p l e which i s e q u i v a l e n t t o t h e I  c o n s i s t e n c y o f p r e d i c a t i v e a n a l y s i s . CGSll
Y. Gurevich and S. Shelah, The monadic t h e o r y and t h e n e x t w o r l d . I s r a e l J.
Math. CGS21 Y. Gurevich and S. Shelah, A r i t h m e t i c cannot be i n t e r p r a t e d i n monadic theory o f 8. [MI
A. M a c i n t y r e , Ramsey q u a n t i f i e r s i n a r i t h m e t i c , Proc. o f a L o g i c Symp. .(Karpacz 1979) ed. L. P a c h o l s k i and A. W i l k i e , S p r i n g e r V e r l a g L e c t u r e Notes i n Mathematics.
[PHI
J . P a r i s and L. H a r r i n q t o n , A mathematical incompleteness i n Peano a r i t h m e t i c , Handbook o f Mathematical L o g i c , ed. Barwise, N o r t h  H o l l a n d Publ. Co.., 1977, 11331142.
CSh 1
S. Shelah, The monadic t h e o r y o f o r d e r , Annals of Math. 102 (1975), 379419.
CSSl
S.G. Simpson and J.Schmer1, On t h e r o l e o f Ramsey q u a n t i f i e r s i n f i r s t o r d e r a r i t h m e t i c , J. Symb. L o g i c .
LOGIC COLLOQUIUM '82 G. Lalli, G. Long0 and A . 'Marcia (editors) 0Elsevier Science Publishers B. V. (NorthHolland), 1984
161
CONTINUOUS TRUTH I Nonconstructive Objects Michael P . Fourman Department of Mathematics Department of Pure Mathematics Columbia University Uni vers i t y of Sydney New York, N . Y . 10027 N.S.W. 2006 U.S.A. Australia
W e g i v e a general theory of the l o g i c of p o t e n t i a l l y i n f i n i t e o b j e c t s , derived from a theory of meaning f o r statements concerning these o b j e c t s . The paper has two main p a r t s which may be read independently but a r e intended t o complement each o t h e r . The f i r s t p a r t i s e s s e n t i a l l y philosophical. In i t , we d i s c u s s the theory of meaning. We b e l i e v e t h a t even t h e s t a u n c h e s t r e a l i s t must view p o t e n t i a l i n f i n i t i e s o p e r a t i o n a l l y . The second p a r t i s formal. In i t , we consider t h e i n t e r p r e t a t i o n of l o g i c i n t h e gros topos of sheaves over t h e category of separable l o c a l e s equipped with t h e open cover topology. We show t h a t general p r i n c i p l e s of c o n t i n u i t y , l o c a l choice and l o c a l compactness hold f o r t h e s e models. We conclude with a b r i e f discussion of the philosophical s i g n i f i c a n c e of our formal r e s u l t s . They allow us t o reconc!le our explanation of meaning w i t h the "equivalence thesis , t h a t 'snow i s white i s t r u e ' i f f snow is white.
PROLEGOMENON Classical mathematics i s based on a p l a t o n i c view of mathematical o b j e c t s . The meanings of mathematical statements a r e determined t r u t h  f u n c t i o n a l l y . T h i s Fregean explanation of meaning j u s t i f i e s c l a s s i c a l l o g i c . The d e f i c i e n c i e s of such a view a r e amply discussed by Dummett C19781. A c o n s t r u c t i v e mathematician r e j e c t s t h e completed i n f i n i t i e s of classiGa1 mathematics. For h i m , t h e objects of mathematics a r e e s s e n t i a l l y f i n i t e . The meaning
of q u a n t i f i c a t i o n over i n f i n i t e domains is given o p e r a t i o n a l l y i n terms of a theory of c o n s t r u c t i o n s . T h e r e s u l t i n g l o g i c includes Heyting's p r e d i c a t e c a l culus and o t h e r p r i n c i p l e s ( e . g . choice p r i n c i p l e s ) .
As Dummett has s t r e s s e d , one t a s k of any philosophy of mathematics i s t o explain the a p p l i c a b i l i t y of mathematics. The p o t e n t i a l i n f i n i t i e s of experience exceed t h e f i n i t e o b j e c t s of t h e s t r i c t c o n s t r u c t i v i s t . They demanda mathematics of inf i n i t e objects. Naive a b s t r a c t i o n leads t o the i d e a l i n f i n i t e o b j e c t s of c l a s s i c a l mathematics. This i d e a l i s a t i o n has enjoyed remarkable success. However, the meaning of statements .of c l a s s i c a l mathematics remains problematic. Brouwer C19811 introduced t o mathematics p o t e n t i a l l y i n f i n i t e o b j e c t s such a s f r e e choice sequences. Consideration of t h e s e j u s t i f i e d , f o r Brouwer, i n t u i t i o n i s t i c l o g i c , including various choice and continuity princip2e.s. W e s h a l l consider a general notion of nonconstructive o b j e c t . For us, t o present such a notion i s t o give a theory of meaning f o r statements involving nonconstructive o b j e c t s .
Our nonconstructive o b j e c t s a r e not t h e p l a t o n i c ideal o b j e c t s of c l a s s i c a l mathematics nor t h e f i n i t a r y o b j e c t s of pure constructivism. They a r e p o t e n t i a l l y
M.P. FOURMAN
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i n f i n i t e o b j e c t s r e l a t e d t o t h e l a w l e s s sequences o f K r e i s e l 119681and t o Brouwer's f r e e  c h o i c e sequences ( T r o e l s t r a 119771). The meanin s o f s t a t e m e n t s about t h e s e o b j e c t s cannot be g i v e n i n terms o f t r u t h c o n d i t i o n s ?as f o r c l a s s i c a l P l a t o n i s t mathematics) o r i n terms of c o n s t r u c t i o n s ( a s f o r n a i v e c o n s t r u c t i v i s m ) . The essence o f t h e s e n o n  c o n s t r u c t i v e o b j e c t s l i e s i n t h e i r i n f i n i t e c h a r a c t e r . They a r e n o t , i n g e n e r a l , t o t a l l y grasped. They a r e g i v e n i n terms o f p a r t i a l d a t a which may l a t e r be r e f i n e d . Meaning f o r statements a b o u t n o n  c o n s t r u c t i v e o b j e c t s i s g i v e n b y s a y i n g what d a t a j u s t i f i e s a g i v e n a s s e r t i o n .
To d e s c r i b e a p a r t i c u l a r n o t i o n o f n o n  c o n s t r u c t i v e o b j e c t i s t o d e s c r i b e t h e t y p e o f d a t a on which i t i s based. We c o n s i d e r v a r i o u s such n o t i o n s . Each c o n c e p t i o n o f d a t a g i v e s an e x p l a n a t i o n o f meaning w h i c h extends t h e range o f meaningful statements and may b e viewed as i n t r o d u c i n g new o b j e c t s i n t h a t i t a s c r i b e s meani n g t o new forms o f q u a n t i f i c a t i o n . I n f a c t f o r each t y p e o f d a t a we i n t r o d u c e a c o n c r e t e r e p r e s e n t a t i o n o f t h e n o n  c o n s t r u c t i v e o b j e c t s based on i t . Such a p r o j e c t i s n o t n o v e l : B e t h 119471 i n t r o d u c e d h i s models t o p r o v i d e j u s t such an e x p l a n a t i o n o f meaning f o r c h o i c e sequences. Our models g e n e r a l i s e Beth's. Dumnett 119771 makes a l e n g t h y c r i t i q u e o f t h e view t h a t t h e i n t e n d e d meanings o f o f t h e l o g i c a l c o n s t a n t s a r e f a i t h f u l l y r e p r e s e n t e d on B e t h t r e e s . Since o u r models g e n e r a l i s e B e t h ' s t h e y appear prima f a c i e t o be s u s c e p t i b l e t o t h e same c r i t i c i s m s . However, Dummett's remarks on t h e (non)consonance o f t h e i n t e n d e d meanings o f t h e c o n n e c t i v e s w i t h t h e i r i n t e r p r e t a t i o n i n B e t h t r e e s a r e d i r e c t e d a t a d i f f e r e n t problem f r o m t h e one we address. Dummett appears t o have o i e r l o o k e d t h e p o s s i b i l i t y o f s e p a r a t i n g t h e problem o f e x p l a i n i n g t h e c o n s t r u c t i v e meaning o f statements c o n c e r n i n g l a w l i k e o b j e c t s f r o m t h a t o f e x p l a i n i n g t h e i n t u i t i o n i s t i c meaning o f statements c o n c e r n i n g c h o i c e sequences. Although we know o f no s a t i s f a c t o r y e x p l a n a t i o n o f c o n s t r u c t i v e t r u t h ( i n p a r t i c u l a r , we agree w i t h Dummett t h a t B e t h models do n o t g i v e one), such a s e p a r a t i o n appears n a t u r a l . I t i s p o s s i b l e t o c o n c e i v e o f c o n s t r u c t i v e t r u t h i n d e p e n d e n t l y o f c h o i c e sequences. Given such a c o n c e p t i o n , Beth models p r o v i d e an account o f t h e i n t r o d u c t i o n o f n o n  l a w l i k e o b j e c t s . I t i s t h i s t y p e o f account we have g e n e r a l i s e d . By way o f example we now c o n s i d e r two n o t i o n s o f d a t a c l o s e l y r e l a t e d t o Beth models. They b o t h a r i s e f r o m t h e same i n f o r m a l p i c t u r e . The Imagine r e c e i v i n g f r o m Mars an i n f i n i t e sequence a o f n a t u r a l numbers. p i c t u r e i s o f a t i c k e r  t a p e which produces an i n d e f i n i t e l y c o n t i n u e d f i n i t e i n i t i a l segment a o f t h e sequence CL. (We w r i t e CL E a t o mean t h a t a i s an i n i t i a l segment o f a . ) We want t o examine t h e consequences o f t r e a t i n g such undetermined sequences s e r i o u s l y as sequences. ( L a t e r we s h a l l i n t r o d u c e more i n t e r e s t i n g examples )
.
A n a i v e view o f t h i s example c o n s i d e r s t h e stages b y which i n f o r m a t i o n a r i s e s : a t any stage, t h e p o s s i b l e f u t u r e d a t a i s r e p r e s e n t e d b y t h e c o l l e c t i o n N(where n E N) must appear. N o t o n l y do we n o t y e t know which o f t h e s e p o s s i b i l i t i e s w i l l occur, i t i s n o t y e t determined which w i l l o c c u r . On t h e b a s i s o f t h i s d a t a we may cons t r u c t many sequences. The s i m p l e s t o f these, a i s g i v e n b y t r a n s c r i b i n g t h e d a t a as i t a r r i v e s . Thus on t h e b a s i s o f d a t a b 5 a, we a r e j u s t i f i e d i n a s s e r t i n g t h a t a i s an i n i t i a l segment o f a. We w r i t e t h i s b i t CL E a. (We o r d e r sequences by s e t t i n g b 5 a i f a i s an i n i t i a l seqment o f b s i n c e t h e n b a l l o w s fewer p o s s i b i l i t i e s f o r CL.)Another sequence B i s g i v e n by f i r s t w r i t i n g down a f i x e d f i n i t e sequence b and t h e n c o n t i n u i n g w i t h t h e incoming data. S c h e m a t i c a l l y , a i l 6 t b*a where * denotes c o n c a t e n a t i o n , and hence " o b v i o u s l y " , f o r any c 5 a, we have c i k D E b*a. We want t o make a l l such "obvious" assumptions d b o u t t h e n a t u r e of j u s t i f i c a t i o n e x p 1 i c i t ; s o we g i v e i t s two b a s i c s t r u c t u r a l p r o p e r t i e s . J u s t i f i c a t i o n should be persistent
alC $ a*blt 0
Continuous Truth I and inductive
a* < n > l t $ f o r a l l n alk @
163
E
N
P e r s i s t e n c e r e f l e c t s t h e i d e a t h a t knowledge, once j u s t i f i e d , i s secure. The i n d u c t i v e c l a u s e comes f r o m r e f l e c t i o n on t h e i n f i n i t e c h a r a c t e r o f a. Given a E a, t h e c o l l e c t i o n { a *I n c N I covers a l l p o s s i b i l i t i e s f o r f u t u r e data. I n general, i f we s t i p u l a t e b l k $ f o r b E B 5 N
ample, any monotone f u n c t i o n g: N
Then a monotone map N
+
P w i l l r e p r e s e n t a nonconstructive
We do n o t w i s h t o i n t r o d u c e t e c h n i c a l i t i e s h e r e . L a t e r we s h a l l g i v e a d e f i n i t i o n o f all 4 , f o r f i r s t  o r d e r $ , b y i n d u c t i o n on t h e s t r u c t u r e o f 9. F o r t h e moment we j u s t remark t h a t such a d e f i n i t i o n o f a l l $ can b e g i v e n and t h a t t h e i n t e r p r e t a t i o n s o f t h e c o n n e c t i v e s a r e c o m p l e t e l y determined, i n t h e c o n t e x t o f o u r requirements on j u s t i f i c a t i o n , b y r e q u i r i n g t h a t t h e r u l e s o f p o s i t i v e l o g i c be v a l i d . I n o u r p r e s e n t case t h i s would amount t o g i v i n g B e t h ' s semantics f o r i n t u i t i o n i s t i c l o g i c w i t h a s l i g h t l y m o d i f i e d n o t i o n o f " b a r " . B e t h ' s semantics a r e w e l l known t o be e q u i v a l e n t t o t h e t o p o l o g i c a l i n t e r p r e t a t i o n o v e r B a i r e space e x p l o i t e d b y S c o t t [ I 9 6 8 1 and Moschovakis 119731;our m o d i f i c a t i o n r e p l a c e s B a i r e space b y f o r m a l B a i r e space ( Fourman and Grayson C19823). We now r e t u r n t o o u r p i c t u r e o f t h e t i c k e r  t a p e . What we have done i s t o g i v e a r e p r e s e n t a t i o n o f t h e subjective e x p e r i e n c e o f r e c e i v i n g word f r o m Mars, a n ext e r n a l view o f how t h e w o r l d w i l l l o o k when d a t a a appears on t h e t i c k e r  t a p e . T h i s view i s dependent on t h e c o n t i n g e n c i e s o f what d a t a i s a v a i l a b l e . B u t mathem a t i c s s h o u l d be t i m e l e s s and a b s o l u t e . T h i s r e q u i r e m e n t appears t o exclude cons i d e r a t i o n o f p o t e n t i a l l y i n f i n i t e o b j e c t s . We now a t t e m p t t o r e s o l v e t h i s contradiction.

P i c t u r e a room w i t h a t i c k e r  t a p e , A and v a r i o u s sequences i n progress, a,p,y d e s c r i b e d above f o r example. Now suppose t h a t on t h e t a p e we have t h e ' f i n i t e sequence b. We have d u l y n o t e d t h a t a E b, 8 E b*b, y E g ( b ) . Consider now ano t h e r room A ' w i t h a t i c k e r  t a p e which, as y e t , i s b l a n k and t h r e e nonconstructive objects defined by c l t a' E b*c
CIF8 '
E
b*b*c
c ~ F Y g(b*c) ' . I n t h i s room on t h e b a s i s . o f no d a t a we can a l r e a d y n o t e t h a t a' E b, 8 ' E b*b, g ( b ) . Furthermore d a t a b*c a r r i v i n g i n room A w i l l always have t h e same conI n f a c t the sequences f o r a . 8 , ~as d a t a c a r r i v i n g i n room A ' has f o r a ' , B ' , y ' . mathematics (and l o g i c ) o f t h e two rooms, A w i t h d a t a b and A ' w i t h no data, s h o u l d b e t h e same. We want t o add t h i s t o o u r f o r m a l t r e a t m e n t . T h i s i s done by r e g a r d i n g incoming d a t a n o t as changing t h e w o r l d b u t r a t h e r as e f f e c t i n g a t r a n s f o r m a t i o n w h i c h changes o u r view o f t h e w o r l d . We c o n s i d e r n o t a p a r t i c u l a r t i c k e r  t a p e b u t r a t h e r t h e uses w h i c h c o u l d b e made o f such an i n d e t e r m i n a t e sequence t o generate n o n  c o n s t r u c t i v e o b j e c t s . Data j u s t becomes a way o f t r a n s y' 6
M.P. FOURMAN
164
forming one such process i n t o a n o t h e r , g e n e r a l l y l e s s f r e e : i t s r e s t r i c t i o n . We give a general d e f i n i t i o n of t h i s transformation a s follows: c l k $(sib) i f f b*clk $(6) (Where 6 i s a nonconstructive o b j e c t given by s t i p u l a t i n g what d a t a j u s t i f i e s $ ( 6 ) f o r various $.) For example, a l b = a ' = a; Olb = a ' ; y l b = y ' .
This change of viewpoint amount s f o r m a l l y t o a change in our r e p r e s e n t a t i o n of d a t a . Formerly we considered the p a r t i a l l y ordered set o r t r e e N < N as representing various possible s t a t e s of information. Incoming data changes t h e world i n t h a t i t places us i n a new s t a t e . Now we consider N"' a s a c o l l e c t i o n of transformations which a c t t o change our view of t h e world. Formally i t i s convenient t o r e p r e s e n t t h e data a s t h e monoid of f i n i t e sequences under concatenation; i f g: N C N P r e p r e s e n t s a nonconstructive o b j e c t y then y l b i s represented by gob where b : a b*a a c t s by l e f t concatenation. The notion of j u s t i f i c a t i o n i s t o be s t a b l e under such a change of perspective: a I b $ ' ] b i f f b*alk $ (where I b i s applied t o t h e nonconstructive parameters of $ . ) f
We now consider examples of a more general type of nonconstructive o b j e c t intend
ed t o r e p r e s e n t p o t e n t i a l i n f i n i t i e s of experience. We base our d e s c r i p t i o n , f o r t h e sake of e x p o s i t i o n , on a view o f c l a s s i c a l experimental physics which we asc r i b e t o t h e nineteenth century. B r i e f l y i t runs a s follows: Physics i s based on measurement. Experiments determine values o f parameters a t o a c e r t a i n degree of p r e c i s i o n . Generally some e r r o r i s i n e v i t a b l e b u t i t may i n p r i n c i p l e be made a r b i t r a r i l y small ( t h i s i s t h e assumption which leads us t o l a b e l t h i s a s a nineteenth century noti o n ) . Now, we refuse t o admit t h e c l a s s i c a l a s c r i p t i o n of a c t u a l values t o these parameters. A t f i r s t consideration t h i s may appear c h u r l i s h . There i s an apparent d i f f e r e n c e between a sequence determined only by t h e f r e e w i l l of a Martian and a physical value. W e leave a s i d e the question o f whether this i s an actual d i f f e r ence because t h i s question misses t h e p o i n t . The point i s t o a s k , "How can we assign meaning t o statements concerning such q u a n t i t i e s , i n p a r t i c u l a r how should we understand q u a n t i f i c a t i o n over such q u a n t i t i e s ? " Our r e f u s a l amounts t o denying the coherence of any explanation based on the assumption t h a t every sentence has a determinate truth value, e i t h e r true o r f a l s e . We r e f e r t o Dummett f o r e l a b o r a t i o n of t h i s point. The p o s s i b l e r e s u l t s of experiments a r e concrete by experiment t h a t a E U. These possible U form assumes t h a t a r b i t r a r y refinement of our methods presented by saying t h a t the V i < U representing cover U
.
however. I n general we may f i n d a poset IP. The c o n c e i t which i s i n p r i n c i p l e possible i s rea c e r t a i n degree .of refinement
For example, measurement of a q u a n t i t y c l a s s i c a l l y represented by a real parameter could be represented by taking f o r P t h e poset of r a t i o n a l open i n t e r v a l s , with t h e s t i p u l a t i o n t h a t f o r each E > 0 an open i n t e r v a l U i s covered by t h e c o l l e c t i o n of a l l s u b i n t e r v a l s of length 5 E , a l s o t h a t t h e c o l l e c t i o n of a l l proper s u b i n t e r v a l s of U covers U. In general then we consider a poset IP of p o s s i b l e outcomes f o r an experiment. We a s k , a s a technical convenience, t h a t i f p and q re resent a p r i o r i compatible r e s u l t s ( i . e . i f t h e r e i s an r with r 5 p and r 5 qp then we can consider the outcome which c o n s i s t s j u s t of g e t t i n g t h e s e two r e s u l t s ( i . e . we have an infinium p A q E P). We a l s o consider no information as a possible r e s u l t ( i . e . Ip has a
Continuous Truth I t o p element T ) . The p o s e t demand t h a t t h i s b e
IP
i s equipped w i t h a n o t i o n o f c o v e r i n g f a m i l y .
r e f Zective
stabZe
165
We
i p } covers p I f U covers p and q
monotone
5
p t h e n Iq
w Iw
A
E
U l covers q
I f V 2 U covers p t h e n V covers p.
The n o t i o n o f a c o v e r i n g f a m i l y i s c r u c i a l t o o u r e x p l a n a t i o n o f meaning f o r i n complete o b j e c t s . I t f o r m a l i s e s t h e sense i n which t h e y a r e p o t e n t i a l l y i n f i n i t e . We a v o i d t h e metaphor o f W r i g h t C19811 which r e p r e s e n t s such a c o v e r i n g f a m i l y as embodying t h e r e c o g n i t i o n t h a t t h e s t a t e o f i n f o r m a t i o n i s capable o f e f f e c t i v e enlargement t o one o f t y p e a*because i t seems t o l e a v e open t o us t h e c h o i c e o f n o t p e r f o r m i n g t h i s enlargement. The i d e a we have i s t o i n t r o d u c e c o n s i d e r a t i o n o f a p a r t i c u l a r t y p e o f i n c o m p l e t e o b j e c t b y s p e c i f y i n g t h e t y p e o f d a t a which generates i t . T h i s s p e c i f i c a t i o n i n c l u d e s a n o t i o n o f c o v e r i n g f a m i l y . D i f f e r ences o v e r w h i c h i s t h e p r o p e r c o l l e c t i o n o f c o v e r i n g f a m i l i e s do n o t a f f e c t t h e b a s i c c o n c e p t i o n b u t m e r e l y l e a d t o d i f f e r e n t types o f data. We a r e n o t as mathematicians o r l o g i c i a n s i n t e r e s t e d i n t h e r e s u l t o f a p a r t i c u l a r experiment. Rather, we a r e i n t e r e s t e d i n t h o s e p r o p e r t i e s which would remain i n v a r i a n t no m a t t e r what t h e outcome o r methodology o f a p a r t i c u l a r experiment. It i s n o t t h e r e s u l t b u t t h e uses t o which t h e r e s u l t m i g h t be p u t i n d e f i n i n g mathem a t i c a l q u a n t i t i e s which i n t e r e s t us. Were t h e temperature s c a l e n o n  l i n e a r , o r t h e t i m e s c a l e g i v e n by t h e unequal t i m e o f t h e sun, p h y s i c s would be d i f f e r e n t ( i t was). B u t mathematics and l o g i c s h o u l d be immune t o such v a g a r i e s . Our s o l u t i o n i s s i m i l a r t o t h a t we employed i n g i v i n g an o b j e c t i v e view of. open data. The p o s s i b i l i t y we envisage i s t h a t o f a change o f s c a l e which i n some sense r e f i n e s o u r p o s s i b i l i t i e s f o r measurement. The measurements o f t h e o l d cont e x t s h o u l d be meaningful i n t h e new one b u t t h e new one may a f f o r d f i n e r d i s t i n c t i o n s . To d e s c r i b e such a change o f s c a l e i s t o say which new o b s e r v a t i o n s q E Q a r e t o b e viewed as r e f i n i n g an o l d o b s e r v a t i o n p E IP. We w r i t e t h i s r e l a t i o n q 5 f * ( p ) and demand t h a t i t be
monotone
p q
muZtip Zicative
pi I i
continuous
{q I q
f*(p) q 5 f*(p') q 5 f*(p A p ' )
5
E
p'
5
I
r
covers p
f*(pi)
some i
E
5
f*(p)
11 covers r
.
The m o t i v a t i o n f o r t h e f i r s t two i s c l e a r . C o n t i n u i t y may b e viewed as t h e r e quirement t h a t a p r e v i o u s c o n v i c t i o n t h a t a c e r t a i n f a m i l y covers, cannot be overt u r n e d . The change o f v i e w p o i n t induced b y such a t r a n s f o r m a t i o n f i s g i v e n by
M a t h e m a t i c a l l y , o u r n o t i o n o f d a t a g i v e s a p r e s e n t a t i o n o f a ZocaZe. Change o f s c a l e i s r e p r e s e n t e d by a continuous f u n c t i o n between l o c a l e s . A b s t r a c t l y we w r i t e such a change f: Y X. f
We now c o n s i d e r a supplement t o o u r n o t i o n o f j u s t i f i c a t i o n . Suppose, we consider, t h a t c o n s i d e r a t i o n o f a p a r t i c u l a r t y p e o f d a t a would j u s t i f y 0, t h e n C$ i s j u s t i f i e d . T h i s i s t h e r e f l e c t i o n on w h i c h o u r whole p r o j e c t i s based: t h a t we can j u s t i f y t a l k o f i n c o m p l e t e o b j e c t s by r e f l e c t i n g on hypothetica2 i n d e f i n i t e l y c o n t i n u e d processes.
M.P. FOURMAN
166
We s h a l l f o r m u l a t e t h i s b y s a y i n g t h a t i f f : Y + X r e p r e s e n t s t h e i n t r o d u c t i o n o f new d i s t i n c t i o n s independent o f those r e p r e s e n t e d b y X t h e n
o r t h a t such an f i s a cover. Our f i n a l problem o f f o r m a l i s a t i o n i s t o c h a r a c t e r i s e t h e i n t r o d u c t i o n o f independent data. A s i m p l e example i s , g i v e n P a n d Q w i t h n o t i o n s o f c o v e r i n g , t o consider P x Q t h e product poset w i t h coverings
I I
1 1
E I } covers p i E I} covers q . The p r o j e c t i o n g i v e n by5 n * ( p ' ) i f f p s p ' r e p r e s e n t s t h e i n t r o d u c t i o n o f d a t a o f t y p e Q i n d e p e n d e n t l y o f t h e d a t a IP under c o n s i d e r a t i o n . We s h a l l r e q u i r e t h a t a l l such p r o j e c t i o n s be covers.
II
i
E
i
E
I 1 covers I } covers
p.q
when {pi
p,q
when
qi
i
I n general t h e r e a r e two c o n d i t i o n s we r e q u i r e t o view a change o f s c a l e as t h e i n t r o d u c t i o n o f independent data. The f i r s t i s obvious: no new covers s h o u l d be i n t r o d u c e d between e x i s t i n g o b s e r v a t i o n s . qi s f * ( r i )
{qi 1 i
{ri I i
E
E
I } covers each q I } covers r
5
f*(r)
The second i s s u b t l e : no new c o n d i t i o n a l r e l a t i o n s h i p s s h o u l d b e i n t r o d u c e d between e x i s t i n g o b s e r v a t i o n s . We e x p l a i n : i f w E Q i s such t h a t
r 5 w f*(p) r 5 f*(q) (we view w as e s t a b l i s h i n g a c o n d i t i o n a l r e l a t i o n s h i p between f * ( p ) and f * ( q ) ) , we demand t h a t w 5 f * ( s ) f o r some s E I P such t h a t r
5
r s p
r s s r s q
Technically, these require( t h a t t h e r e l a t i o n s h i p be a l r e a d y e s t a b l i s h e d i n IP). ments amount t o demanding t h a t t h e continuous map f : Y + X be a s u r j e c t i o n and t h a t i t be open. The s t r u c t u r e o f d a t a we have a r r i v e d a t may b e viewed as t h e c a t e g o r y o f l o c a l e s equipped w i t h t h e t o p o l o g y o f c o v e r i n g b y open maps. Before t u r n i n g t o a formal examination o f t h e i n t e r p r e t a t i o n o f l o g i c over t h i s s i t e , we sum up o u r i n t e n t i o n s . We i n t r o d u c e n o n  c o n s t r u c t i v e o b j e c t s b y e x p l a i n i n g t h e meanings o f t h e connecti v e s f o r statements c o n c e r n i n g them. T h i s i s not a m a t t e r o f c h a r a c t e r i s i n g a domain o f q u a n t i f i c a t i o n . We have t o e x p l a i n t h e c o n n e c t i v e s anew i n terms o f t h e way such an o b j e c t i s g i v e n t o us. Moreover, i t i s n o t s u f f i c i e n t t o m e r e l y paraphrase t h e new q u a n t i f i e r s Wa and 3u. Such a paraphrase e n t a i l s a r e v i s i o n o f t h e i n t e r p r e t a t i o n s o f + and v . Our aim i s t o show t h a t i t i s p o s s i b l e t o d e r i v e r i g o r o u s l y p r o p e r t i e s o f v a r i o u s domains o f i n c o m p l e t e o b j e c t s by g i v i n g a f o r m a l r e p r e s e n t a t i o n o f t h e d a t a which p r e s e n t s them as a s i t e . We c o n s i d e r t h a t t h e passage f r o m an i n f o r m a l n o t i o n o f data t o t h e c o r r e s p o n d i n g s i t e i s s i m p l e and n a t u r a l . (Indeed, f o r us, t o have a c l e a r c o n c e p t i o n o f a t y p e o f d a t a i s t o be a b l e t o d e s c r i b e t h e c o r r e s p o n d i n g s i t e . ) Once t h i s passage i s made, t h e d e r i v a t i o n o f p r o p e r t i e s ( c h o i c e and cont i n u i t y p r i n c i p l e s , f o r example) i s a mathematical m a t t e r . Our hope i n p r e s e n t i n g these modeis i s L e i b n i t z i a n : t o e l i m i n a t e f u r t h e r d i s c u s s i o n o f t h e j u s t i f i c a t i o n o f such p r i n c i p l e s b y r e d u c i n g t h e m a t t e r t o c a l c u l a t i o n . I n o u r paper " N o t i o n s o f Choice Sequence" C19821 we presented v a r i o u s n o t i o n s o f
Continuous Truth I c h o i c e sequence, i n c l u d i n g ones purpose. U n f o r t u n a t e l y , as t h e i s i n t h e eye o f t h e b e h o l d e r . a t i o n o f our informal notion o f
167
s a t i s f y i n g t h e axioms o f LS and CS, w i t h t h e same l i t e r a t u r e on c h o i c e sequences makes c l e a r , c l a r i t y Hence t h e p r e s e n t a t t e m p t a t a more c a r e f u l explann o n  c o n s t r u c t i v e o b j e c t and i t s f o r m a l i s a t i o n .
CONTINUOUS TRUTH We s t a r t w i t h a c o n c r e t e p r e s e n t a t i o n o f t h e i n t e r p r e t a t i o n o f h i g h e r  o r d e r l o g i c i n a Grothendieck topos. T h i s m a t e r i a l (5613) i s wellknown t o cognoscenti ( tautologously), b u t i s otherwise accessible only through a study o f scattered r e f e r e n c e s . We g i v e some o f these sources b u t make no s y s t e m a t i c a t t e m p t a t a complete l i s t . Many i m p o r t a n t and h i s t o r i c a l l y s i g n i f i c a n t c o n t r i b u t i o n s a r e n o t mentioned. Our account i s f u l l e r t h a n i s l o g i c a l l y necessary f o r t h e sequel i n o r d e r t o p o i n t o u t some connections between d i f f e r e n t approaches. I t i s not, however, e x h a u s t i v e . 51
Frames and Locales
A frame i s a complete l a t t i c e w i t h f i n i t e A d i s t r i b u t i v e o v e r 1.1 D e f i n i t i o n . a r b i t r a r y V. Frame morphisms, "andor maps", a r e maps p r e s e r v i n g these o p e r a t i o n s ; T,A,V. 1.2 Example. The l a t t i c e O(X) o f open subsets o f a t o p o l o g i c a l space i s a frame. I f f: Y + X i s a continuous map t h e n t h e inverse image f*: O ( X ) r O ( Y ) i s an A,Vmap. 1.3 D e f i n i t i o n . The c a t e g o r y o f Zocalos o r g e n e r a l i s e d spaces i s t h e dual o f t h e c a t e g o r y o f frames. We c a l l t h e morphisms continuous maps f: Y + X and w r i t e f*: U(X) + O ( Y ) f o r t h e c o r r e s p o n d i n g i n v e r s e image maps between t h e frames of opens o f X and Y ( a s i n t h e t o p o l o g i c a l case). Example 1.2 g i v e s a f u n c t o r 6: Top maps t o l o c a l e s .
+
LOC f r o m t o p o l o g i c a l spaces and continuous
1.4 D i s c r e t e spaces. S p a t i a l l y P(A) corresponds t o t h e d i s c r e t e t o p o l o g y on A. p o i n t space w i t h O( ll) = P( ll) . 1.5
Definition.
1.6
Lemma.
An example i s t h e one
A t o p o l o g i c a l space X i s sober i f f Top[Il,Xl
1 LocCll,pX1.
On t h e f u l l subcategory o f sober spaces 6 i s f u l l and f a i t h f u l .
We t a c i t l y r e s t r i c t o u r a t t e n t i o n t o sober spaces and h e n c e f o r t h o m i t mention o f 8. We view l o c a l e s as generalised spaces. (The r e l a t i o n s h i p between LOC and Top i s b e t t e r expressed i n terms o f t h e r i g h t a d j o i n t , p t : LOC + Top, t o 6.) Q u o t i e n t maps o f frames i n d u c e congruences: i f f*: O(X) + U(Y) i s has a c a n o n i c a l p a q i f f f * p = f * q . Each congruence c l a s s !PI r e p r e s e n t a t i v e j p = V t q I p a q } . The maps j : O(X) + O(X) a r i s i n g i n t h i s way a r e monotone P 2 jp j2 = j idempotent ,
muttip l i c a t i v e
j(p
A
9) = j p
A
jq
.
Such maps a r e c a l l e d n u c t e i . The q u o t i e n t may b e i d e n t i f i e d as t h e image ( o r f i x e d p o i n t s ) o f j . The q u o t i e n t s o f O(X) a r e i s o m o r p h i c ( a s posets) w i t h t h e n u c l e i on U(X). S p a t i a l l y we view these q u o t i e n t s as g i v i n g r i s e t o subspaces o f x.
M.P. FOURMAN
168
1.8 Surjections. Dually, we view i n j e c t i v e inverse image maps as giving r i s e to surjections o f spaces. Each frame map f* has a r i g h t adjoint f,,
1.9 Right a d j o i n t s . given by The map q
p
A
f*p = V{q q has a r i g h t a d j o i n t r p ~ q s r iff
1
f*q
+p
where p + r = v!q I p A q 2 r } . morphisms are d i f f e r e n t ) .
d i r e c t image,
.
pl r defined by q s p + r 5
f
Thus frames are complete Heyting algebras ( b u t the
Y i s o en i f the inverse image map 1.10 Definition. A map of spaces f : X f*: O(y) + O ( X ) has a l e f t a d j o i n t 3,: U ( X ) + O ( V 7 commuting with A : 3f(f*(Y) A x ) = Y A j f ( X ) or, equivalently, i f f* preserves +. f
1.11 Proposition. The category o f locales i s complete and cocomplete. surjections a r e s t a b l e (under pullback).
Open
The theory of locales i s developed extensively by Joyal and Tierney C19821. Johnstone 119821 uses locales systematically and has a comprehensive bibliography. 52
S i t e s a n d Sheaves
2.1 Definitions. Let 0 be a small categor . A cribte K of A E ! E l i s a suboil : t h a t i s , f o r each B E 181 a s e t functor of the representable functor A E S' K ( B ) 5 IB,AI, s t a b l e under composition; f o r each f E K ( B ) and g : C B in C , the composite f g E K ( C ) . f
0
2 . 2 Lemma. The c r i b l e s of A form a frame, P ( A ) . If f : B + A i n B we have an inverse image map f*: P(A) + P(B) given by f o r the correspondf*K = Ig f g E K} f o r K E P ( A ) . By abuse we write f : ing continuous map. This map i s open.
B A
0
f
2.3 Definition. (LawvereTierney) A Grothendieck topology j on is a family of nuclei j A : P(A) P(A), natural i n A: t h a t i s f * o j A = jBof*for f: B
+
f
A.
Lemma. I f j i s a Grothendieck topology on 0 , the quotient frames n(A) have induced inverse image maps f*: n(A) + n ( B ) and the corresponding map of locales, which we write f : B j +A', i s open. 2.4
2.5 Definitions. which i s
A pretopotogy J on 0 i s a family J(A)
A
reflerrive
K
multiplicative
K
stable
(For example, l e t K A crible K
E
E
E
E
J(A) i f f j K
J(A)
E
J(A) K n L J(A) f*K =
E
L E J(A) J(A) f: B + A
E
J(B)
T).
P(A) i s inductively c t o s e d f o r J i f f
c
P ( A ) f o r each A
E
B
169
Continuous Truth I f: B + A
f*K
E
J(B)
.
f c K
As A i s c l o s e d and an i n t e r s e c t i o n of c l o s e d c r i b l e s i s closed, each c r i b l e K E P(A) has a cZosure j A K . T h i s g i v e s a t o p o l o g y j on C. We say K inductiveZy covers A iffj K = A, and w r i t e t h i s K E J(A). U(X) be a frame viewed as a p o s e t viewed as ( I d e n t i f y i n g B + A w i t h i t s domain.) Then be a s m a l l c a t e g o r y o f l o c a l e s c l o s e d under f i n i t e l i m i t s (2) Let i n c l u s i o n s . L e t K E J(A) i f f K c o n t a i n s some f a m i l y Ifi: Bi + A l i E
2.6
Exam l e s . (1)
___R_ Let K J A i f f VK E
maps such t h a t
V
Let
= A.
= A.
3fi(Bi)
a category. n(X) 1 O ( X ) . and open 11 o f open
The c r i b l e generated b y each open i n c l u s i o n U T h i s assignment g i v e s an A;V map r*: U(A)
i s closed f o r t h i s topology.
+
4
A
n(A),
s p a t i a l l y a s u r j e c t i o n r : AJ + A . Each c l o s e d c r i b l e c o n t a i n s a l a r g e s t open i n c l u s i o n . T h i s assignment g i v e s an A V map i*:n(A) + U(A), s p a t i a l l y we have an i n c l u s i o n i: A c+JJ. Furthermore, adjoint retract o f
AJ
r
o
i = i d A and r * i * s id,(A)
so A i s an
(Fourman C19821).
2.7 D e f i n i t i o n . A presheaf on C i s a f u n c t o r X: Cop + Sets. I f f: B + A E C and a E X(A) we use t h e n o t a t i o n a l f , "a r e s t r i c t e d a l o n g f " , f o r X ( f ) ( a ) E X(B). Note t h a t a l f l g = a l f o g and a l i d = a. The a p p r o p r i a t e morpkisms between presheaves F: Y + X a r e n a t u r a l t r a n s f o r m a t i o n s , maps FA: Y(A) + X(A) w h i c h commute w i t h r e s t r i c t i o n s FB(a,lf) = ( F A a ) l f . 2.8
Exam l e s .
( 1 ) The r e p r e s e n t a b l e f u n c t o r [,A], ( o r by abuse, j u s t i f i e d by i s a presheaf. R e s t r i c t i o n s are by composition g l f = gof. Yoneda's lemma t e l l s us t h a t f o r any o t h e r presheaf, X, we have X(A) [A,X]. In
Yone 7 a+ s emma, +A)
p a r t i c u l a r t h e embedding C + Stop i s f u l l and f a i t h f u l . Each c r i b l e K E P(A) i s a subpresheaf K H A . ( 2 ) P and n a r e presheaves w i t h r e s t r i c t i o n s g i v e n b y i n v e r s e images K l f = f * ( K ) . 2.9 D e f i n i t i o n . A p r e s h e a f X on C i s a sheaf f o r t h e ( p r e ) t o p o l o g y J i f whene v e r K E J ( A ) , each n a t u r a l t r a n s f o r m a t i o n x: K + X has a unique e x t e n s i o n a l o n g K >+ A . E q u i v a l e n t 1y, i f K E J(A) and we have a f a m i l y x f E X(B) f o r f: B f A E K such t h a t xgf = x f l g f o r each g: C
xf = x l f f o r each f Grothendieck topos.
E
K.
+
8, t h e r e i s a unique x
E
X(A) such t h a t
The c a t e a o r v of sheaves and n a t u r a l t r a n s f o r m a t i o n s i s a
There i s , as y e t , no s t a i s f a c t o r y i n t r o d u c t o r y t e x t on topos t h e o r y . r e f e r e n c e s a r e SGA4, W r a i t h C19751, Johnstone C19771, F r e y d '19721.
53
The b a s i c
Forcing over a s i t e
Here we d e s c r i b e J o y a l ' s p r e s e n t a t i o n o f i n t e r p r e t a t i o n s i n t o p o i i n terms o f a n o t i o n o f f o r c i n g . L e t C and a ( p r e ) t o p o l o g y J b e f i x e d . The b a s i c s t r u c t u r e s we c o n s i d e r a r e diagrams o f preskeaves on C. Each p r e s h e a f A i n t e r p r e t s a type o r s o r t o f v a r i a b l e . A morphism f : A1 x ... x An + B i n t e r p r e t s an nary operation.
A subobject R

A1
x
...
x
An i n t e r p r e t s an n  a r y r e l a t i o n .
3.1 D e f i n i t i o n s . L e t L be a f i r s t  o r d e r language ( p o s s i b l y manysorted) w i t h e q u a l i t y . An i n t e r p r e t a t i o n o f L i s g i v e n b y a s s i g n i n g t o each s o r t A o f L a p r e s h e a f A, t o each o p e r a t i o n F f r o m A1,. . .,An t o B a n a t u r a l t r a n s f o r m a t i o n F: A1 R
*
x
A1
... x An B, and t o each r e l a t i o n R on A1 x ... x An .. . x An. Given such an i n t e r p r e t a t i o n , f o r U E J C C J +
x
a subfunctor we l e t
LU b e t h e
M.P. FOURMAN
170
expansion o f L o b t a i n e d by a d d i n g c o n s t a n t s o f t h e a p p r o p r i a t e s o r t s f o r t h e e l e ments o f A(U). I f f : V + U t h e n f o r any term T o r f o r m u l a Q o f LU we o b t a i n a term
[lTnu
o r f o r m u l a @ I of f LV by r e s t r i c t i n g any new c o n s t a n t s which occur.
T l f
for U
E E
IC1 we d e f i n e f o r each c l o s e d term A(U) b y i n d u c t i o n : ucnu
Note t h a t
U T l f l "
=
c
=
. . ,Tn) 1,
[IF( TI,.
for =
c
E
A(U)
..,iTnj) .
F(lT,n,.
Now we d e f i n e i n d u c t i v e l y t h e r e l a t i o n , U f o r c e s Q,
[TDulf.
UIk Q f o r 4 a sentence o f Lu. INDUCTIVE DEFINITION OF FORCING
Vflt
a l l f: V f
@If
+
K K
E
J(U)
UIt @
f o r a l l f: V
+
U, i f VIE $ I f t h e n V l t $If V l t @ $ +
f o r a l l f: V
+
U, f o r a l l c E A(U), VIE ~ l f C c / x l U l t wx.+
We now g i v e some " d e r i v e d r u l e s " f o r f o r c i n g : 3.2
(PI
Now
o f s o r t A o f LU an interpretation
T
Lemma.
Basic properties o f f o r c i n g UIt 0
f : V + U
V l t +If
171
Continuous Truth I
K
f o r each f i n some
f o r each f i n some
K
E
E
U I t $ y Ji J(U) e i t h e r V f l t $ 4 f o r V f l t $ l f
Ult 3x.g j ( U ) we have V f l t $ C c / x l f o r some c
+
*
V l t 8 U i f V l t g l f t h e n VIE +If +
f o r a l l f: V
A(Vf)
E
VIE W X . $ f o r a l l f: V
(Atomic)
U and a l l c
f
f o r each f i n some K
E
E
j(U)'we
A(U), we have Vlk $ l f [ c / x l
hav;
...,
[Tnllf>
E
R(Vf)
Our p r e s e n t a t i o n h e r e i s nonstandard i n t h a t t h e d e f i n i t i o n o f f o r c i n g i s u s u a l l y g i v e n b y s t i p u l a t i n g b o t h p o s i t i v e and n e g a t i v e r u l e s f o r each c o n n e c t i v e , ( I ) and (P) a r e t h e n d e r i v e d . The r e s u l t i n g r e l a t i o n i s t h e same. 3.3 D e f i n i t i o n . pretation iff
r
A sequent

r 1 $)
i s uaZid ( w r i t t e n
UI~$CS(X)/XI a l l q u I t $CS(X)/Xl
E
i n the given i n t e r 
r
where E i s an i n t e r p r e a t i o n o f t h e v a r i a b l e s o f L by elements o f t h e a p p r o p r i a t e A(U). I f each s o r t i s i n t e r p r e t e d by an inhabited p r e s h e a f (each 3.4 P r o p o s i t i o n . A(U) i s i n h a b i t e d ) t h e n t h e axioms and r u l e s o f H e y t i n g ' s p r o p o s i t i o n a l c a l c u l u s a r e v a l i d f o r k.
( A d a p t a t i o n s f o r domains w h i c h a r e n o t i n h a b i t e d a r e discussed i n Fourman r19771, S c o t t 119781, J o y a l & B o i l e a u C19811, Makkai & Reyes 119771.) 3.5
Definitions.
A p r e s h e a f A i s separated i f f
'Ita
=
a = b
A subobject R
t+
f o r a,b
E
A(U)
for a
A(U).
A i s cZosed i f f U I t R(a)
FaquJ
E
A h i g h e r  o r d e r t y p e  t h e o r y i s m e r e l y a manysorted f i r s t  o r d e r t h e o r y w i t h some s t r u c t u r e on t h e c o l l e c t i o n o f s o r t s and c e r t a i n d i s t i n g u i s h e d o p e r a t i o n s and r e l a t i o n s . One o f t h e i n s i g h t s due t o Lawvere and T i e r n e y i s t h a t t o p o i have such h i g h e r  o r d e r s t r u c t u r e . We c o n s i d e r languages where f o r any two s o r t s A and B we can f o r m t h e product A x B w i t h a p p r o p r i a t e pairing and p ro j e c t i o n operations, t h e f unction space BA w i t h an evaluation operation ( ), and a l s o t h e power t y p e
M.P. FOURMAN
172
P(A) w i t h a membership reZation E . An i n t e r p r e t a t i o n i s standard i f a l l t h i s s t r u c t u r e i s i n t e r p r e t e d by t h e c o r r e s p o n d i n g s t r u c t u r e on Sh((C). 3.6 P r o p o s i t i o n . I n any s t a n d a r d i n t e r p r e t a t i o n t h e f o l l o w i n g schemata, which combine comprehension and extensionality, a r e v a l i d . x

A 3!y E B.@(x,y) 3 ! z E P(A) W X
E
E
3!f
A
B A WX
E
(X
z
E
++
E
A.$(x,f(x))
@(x)).
0
Thus powertypes and f u n c t i o n spaces behave as t h e y should. The c a t e g o r i c a l c h a r a c t e r i s a t i o n of t h i s h i g h e r  o r d e r s t r u c t u r e i n terms o f a d j o i n t s i s v e r y simple, p r o d u c t s a r e c a t e g o r i c a l p r o d u c t s ,
We s h a l l n o t d e s c r i b e t h i s s t r u c t u r e i n general h e r e . We s h a l l b e d e a l i n g p r i m a r i l y w i t h s o r t s i n t e r p r e t e d by r e p r e s e n t a b l e s . These a r e p a r t i c u l a r l y s i m p l e t o deal w i t h because t h e y have generic elements. A wellknown consequence o f t h i s i s t h e Yoneda Lemma: OP F(U) [U,FI f o r F E I S c I and U E / c C 1 . We use t h i s t o c a l c u l a t e some examples o f t h e h i g h e r  o r d e r s t r u c t u r e . F o r t h i s e x e r c i s e , we suppose t h a t ci: has f i n i t e p r o d u c t s and t h a t each r e p r e s e n t a b l e f u n c t o r i s a sheaf. 3.7
Lemma.
(1) F'(U)
I f F i s a s h e a f and U,V a r e r e p r e s e n t a b l e F(U
C)
x
U and E
E
F(U
V) w i t h U l k a
E
R i f f Rl = V .
w i t h e v a l u a t i o n f o r u: V (2)
(PU)(V) = n(U
Proof.
U
F (V)
x
[V,F
(PU)(V)
U
1
+
V) g i v e n by ~ ( u )= sI.
x
[UxV,FI 2 F(UxV)
CV,PUI
0
S u b ( U x V ) = 6?(UxV).
A l o g i c a l c o u n t e r p a r t t o Yoneda's lemma i s t h e f o l l o w i n g . 3.8
Lemma.
Generic elements for representables.
I f U i s representable then
VIkWx
E
U.$
iff
V
Ulk $IT~[T~/XI.
x
Proof. + 
V
x
I n one d i r e c t i o n t h i s i s immediate f r o m I n t h e o t h e r , suppose V w i t h a: W + U E U(W) t h e n cf,a>: W + V x U and, b y p e r s i s t e n c e , i f U(k$fn1C.rr2/xl t h e n Wk I $ l f [ a / x l . So b y ( W ) ' we have V[kWx.$ . 0
We g i v e an example of t h e use o f g e n e r i c elements i n t h e s i m p l e case of a c a t e g o r y o f presheaves. 3.9 Proof.
Proposition.
Choice h o l d s f o r r e p r e s e n t a b l e s i n c a t e g o r i e s o f presheaves.
L e t U be a r e p r e s e n t a b l e and suppose
Vlk
WX
E
u . 3 ~E F.@(x,y)
then
U X VIk 3~
F.$IT~(T~,Y)
U x V l k $ 1 ~ ~ ( . r r ~ , Cf o) r some 6
E
F(UxV)
173
Continuous Truth I U r e g a r d i n g 5 as an element o f F (V) t h i s g i v e s u x
Since
51= 5 .
Vlt
~l+1,(51n2)(~l))
vit
wx
.
Thus U.4(X,E(X)),
t
and s o
vlt
0
3f.Vx.+(x,f(x)).
From a c a t e g o r y  t h e o r e t i c v i e w p o i n t t h i s r e s u l t i s wellknown i n t h e form, "Representables a r e i n t e r n a l l y p r o j e c t i v e " .
54
POINTS, LOCAL CHOICE, CONTINUITY
Now we l e t CC be a c a t e g o r y o f l o c a l e s c l o s e d under f i n i t e l i m i t s and open i n c l u s i o n s , equipped w i t h t h e open c o v e r t o p o l o g y , J. We w r i t e E f o r t h e topos Sh(C,J). F o r each l o c a l e X we d e f i n e an i n t e r n a l l o c a l e X b y
O(X)(u);
O(XX U J ) .
T h i s i s generated i n t e r n a l l y b y t h e b a s i s g i v e n b y B()#)(U)E
U)
O(Xx
or even b y t h e c o n s t a n t b a s i s Bo(X)(U) = O(X),
with t h e i n c l u s i o n s go(#) 4 X
x
UJ
f
X
x
U
f
X.
B(X)
4 I)()#)
induced b y t h e p r o j e c t i o n s
( I n t h e t e r m i n o l o g y o f J o y a l & T i e r n e y X = P*(X).)
The i n t e r n a l space o f p o i n t s o f X i s g i v e n b y ( p t X ) ( U ) ,z cruj,x1. T h i s i s t h e space o f Evalued models o f X. iff
F o r a: UJ
+
X and W
E
O(XxUJ)
= T.
'(w)
s i n c e i*:o ( u ~ )+ U ( U ) r e f l e c t s 4.1
Proposition.
Proof.
F o r any X
We must show f o r W
6
B I t h e i n t e r n a l l o c a l e X has enough p o i n t s
E
O(X) t h a t
U l t K covers p t W
U l t K i s an i n d u c t i v e c r i b l e E K
UIFW
.
We assume t h e hypotheses, and l e t
M
=
{Wi
x
Ui
I
UiItWi
C l e a r l y IK i s a d o u b l y i n d u c t i v e c r i b l e o f O(W) We show t h a t I K covers W x U t h a t i s t h a t W x U s i n c e t h e n U l t W E K.
By p e r s i s t e n c e W
x
UlF KAn2
covers n l
E
x E
KIUil U(U), t h a t i s an open o f W x U. K, which i s e v i d e n t l y s u f f i c i e n t
M.P. FOURMAN
174
that i s W
U(k3V
x
Kln2.rl
E
V.
U(W) we have Wi
x
Ui
E
so IK"
= {Wi
x
U i l f o r some Vi
c
It Vi
E
K1r2
A
rl c Vi}
covers W x U. [Because, i f p: X + Y i s an open s u r j e c t i o n and Xlk V E K J P A e l p E V then YlkV E K A e E V: t h e b a s i c opens o f )# a r e constant and thus descend open s u r j e c t i o n s . 1 B u t now we c l a i m IK* IK because, by d e f i n i t i o n wi x ui I t n1 E vi i f f Mi 5 Vi and, as p r o j e c t i o n s a r e covers, Wi x Uilb Vi
E
KIn2 i f f
UiIkVi
E
KIUi.
0
Special cases o f t h i s a r e worthy o f mention. When X i s B a i r e space NN, Cantor N space 2 , Dedekind r e a l s R , t o say t h a t X has enough p o i n t s i s t h e i n t e r n a l statement of Bar i n d u c t i o n , Fan theorem, Heine Bore1 theorem ( r e s p e c t i v e l y ) . For these cases i t i s s u f f i c e i n t t o t a k e t h e topology on Q generated by covering f a m i l i e s of open inclusions: s i n c e each o f these spaces X has a p o i n t t h e proe c t i o n s X x U + X are covers f o r t h i s topology. We c a l l t h i s topology the open inclusion topology. We introduce some more general spaces. L e t f: X + U i n LOC. We consider t h e i n t e r n a l l o c a l e X / f defined a t U by t h e b a s i s U(V) w i t h a l l i t s standard covers. More p r o p e r l y f o r 9: W + U we d e f i n e
=
W/f)lg 0(9*X) given by p u l l i n g back f along g. Any commuting t r i a n g l e
x
F Y
induces an i n t e r n a l map o f l o c a l e s 5 : X / f + Y/h defined a t U. Given by 5l on b a s i s elements, t h i s c l e a r l y takes b a s i c covers t o covers. Furthermore, i f 5: X + Y i s open (and s u r j e c t i v e ) then 5: X / f + Y/h i s open, s i n c e i t s u f f i c e s t o d e f i n e comnuting w i t h A on b a s i s elements, (and
4
s u r j e c t i v e since i f 5: X + Y i s an open s u r j e c t i o n then so a r e a l l i t s pullbacks, so i n t e r n a l l y 5l r e f l e c t s b a s i c open covers). These spaces i n c l u d e t h e spaces )K we introduced e a r l i e r as U1lX
(X
x U)/v
.
We now s p e c i a l i s e t o t h e case where t h e o b j e c t s o f b a r e T.I an isomorphism (ptX)(U) CCUj,Xl CCU,Xl
Then U
4
U j induces
so t h a t X represents t h e f u n c t o r p t X . This happens i n p a r t i c u l a r f o r t h e spaces N, NN, pN, R and t h e i r b a s i c opens (see Fourman 119831.) Furthermore, any element o f p t ( X / f ) d e f i n e d a t U induces a comnuting t r i a n g l e
\/
UJ
175
Continuous Truth I
X
which f o r TI
spaces
X corresponds t o a section o f g
U
U
So we have a p r e s e n t a t i o n correspond t o commuting t r i a n g l e s
pt(X/f)'g
/ / d l /
W
.U
9
w i t h r e s t r i c t i o n g i v e n b y composition. We e x t e n d o u r e a r l i e r lemma on g e n e r i c elements: 4.2
Lemma.
I f objects o f
(I:
a r e TI
then
ulk Vx
E
Pt(X/f).$
iff
Xlk
0
($lf)(id).
O f course these g e n e r a l i s e d r e p r e s e n t a b l e s can b e d e f i n e d i n t e r n a l l y i n any Grothendieck topos and t h i s r e s u l t h o l d s .
e4.3
I f t h e o b j e c t s o f C a r e T, t h e n f o r any X
P r o o s i t i o n.
I C I and any
+
A
Wx
+
Proof.
E p t ( X ) . 3 a c A.$(x,a) 3 open c o v e r p: Z >> X and a f u n c t i o n f: p t Z
E
WX E A.WZ E p t ZCpz = x As p t X i s r e p r e s e n t a b l e ,
+
such t h a t
$(x,f(z))l.
U I k V x 3 a o(x,a) iff
X i f f f o r some open c o v e r p: Z
zlk
ulk 3a
x
X
>
x
$ln2(nl,a)
U
$ln20~(nlo~,S)
f o r some 5 c A ( Z )
iff
Ult'Jz
E
z
$(P(Z),dZ)).
0
We do n o t know under what c o n d i t i o n s 6 descends t o g i v e a f u n c t i o n d e f i n e d on a c o v e r b y open s e t s . We can ensure t h i s b y c o n s i d e r i n g t h e open i n c l u s i o n t o p o l o g y on C i n w h i c h case we o b t a i n
1 Wx +
Wx
E
p t X.3a
E
3 open c o v e r Ui E
A.$(x,a) E
Ui.$(X'fi(X))
U(X) and f u n c t i o n s fi:
Ui
+
A such t h a t
.
We now c o n s i d e r c o n t i n u i t y . 4.4.
Proposition.
I f X,Y
a r e TI
then
1Vf:
pt
)#
+
p t W,
f is continuous.
M.P. FOURMAN
176
Proof. I f Uik f : p t )# + p t W t h e n f i s r e p r e s e n t e d b y 5 : X V E ( ) ( Y ) a b a s i c open o f W , w: W + U and x: W + X we have
Wkl iff
[S
0
(t;lw)(x)
E
<x,W>ll(v)
w It x
iff
1 regarding 5 V
O(Xx U) as an open o f
E
5 1(V) i s open.
Thus
Ulk
55
Iteration
E
U
+
Y in
(c.
For
v w
=
1 1 <x,w> 5 (V) =
iff
x
w
c51(v)lwl d e f i n e d a t U.
)#
0
We r e t u r n f o r a w h i l e t o c o n s i d e r a t i o n o f a general Grothendieck topos B = Sh(O,J). We c o n s i d e r t h e i n t e r n a l c a t e g o r y (I i n E g i v e n b y (E(U)
(c/u
w i t h r e s t r i c t i o n s g i v e n b y p u l l i n g back. [For those who w o r r y about coherence (one s h o u l d w o r r y ) , we remark t h a t a conc r e t e c a t e g o r y i n E w i t h an e q u i v a l e n t c a t e g o r y o f s e c t i o n s o v e r U i s g i v e n by c o n s i d e r i n g V / f t o be r e p r e s e n t e d as t h e element S o f (PV)(U) determined b y W / ~ V E S i f~f ~ f o v = g . So & i s an i n t e r n a l s m a l l f u l l subcategory o f E whose o b j e c t s a r e s u b f u n c t o r s o f representables.] We g i v e C_ a t o p o l o g y b y l e t t i n g
xi x
\/
Now f o r A
E
I E I we d e f i n e
w i t h r e s t r i c t i o n s f o r g: V
and f o r 5: Y/h
+
cover X / f i n
A, E +
X/g i n a/U,
+
i f Xi
+
X cover X i n
ShE(C,J) b y UkA_(X/f) A(X) U given b y r e s t r i c t i o n along f*g
by r e s t r i c t i o n along 5 Y
Any morphism A
&
B i n E induces
A
X
U +,B i n ShE($,J).
c.
Continuous Truth I
177
For those who p r e f e r g l o b a l d e s c r i p t i o n s , we associate t o A functors 6/U + E/U n a t u r a l i n U (i.e.
comnuting w i t h g* f o r g: V
+
E
If[ (pseudo)
U) as f o l l o w s :
where
For Y
'
,E/U
nf
U X
a, B
+
X
31
we have nh
P
.
npE whence
nhS* * nf
(as E.*
4ng)
U
and nhAy
* nPx (as
Ay)
E*Ax
.
This gives t h e r e q u i r e d arrow nhayA functor
+
What we o b t a i n i s an ( i n t e r n a l )
nfAXA.
C+EC
OP *
We s h a l l show t h a t t h i s preserves f i r s t order l o g i c . liere we work c o n c r e t e l y f o r t h e sake o f computations. A simple b u t more a b s t r a c t treatment w i l l appear i n Fourman and K e l l y C19831. We now consider a f i r s t  o r d e r language L w i t h s o r t s f o r t h e o b j e c t s o f E and operations symbols f o r i t s morphisms. I n f a c t t o avoid s i z e problems, we consider an a r b i t r a r y small f r a g ment o f such a language. We may consider L a l s o as a language i n K as a constant object ( v i a A ) . Working i n E we consider t h e i n t e r p r e t a t i o n o f L given by i n t e r p r e t i n g t h e s o r t A by A and each o p e r a t i o n f: A + B by t h e corresponding morphism 4 + &. 5.1
Lemma.
For f: X
+
U and g: X
~ l xk/ f k
9
+
V
iff
vlt
X/gl!
+
Ulk X/flk *g i s defined t o mean 0 f o r a l l g: X v As no r u l e decreases t h e complexity o f then IF i s closed under t h e r u l e s o f 9 we say assume t h a t t h e r e s u l t holds f o r subformulae o f 9.
Proof.
By i n d u c t i o n , i t s u f f i c e s t o show t h a t i f
v ~ XF/ g l k
+
.
Only (+)+ and ( W ) ' present any d i f f i c u l t i e s . r e s u l t f o r @ and $.
Me consider (+)+, and suppose t h e
Suppose t h a t f o r a l l E: W U and a l l h: Z + g*X, i f W Z/(E*f h) Ip*01(f*E 0 h) Then i f n: W ' + V and h ' : Z + rr*X a r e such then W l k Z / ( E * f h ) I p d ( f * E  h ) . that W ' Z ' / ( n * g h ' ) IF @ l ( g * n h ' ) then by i n d u c t i o n hypothesis +
0
0
M.P. FOURMAN
178
It
U k  Z ' ( f 0 g*no h ' ) * $1(g*no h ' ) whence ( l e t t i n g 5 = i d and h = g*n h ' ) we have 0
U
Z'/(
f
0
g*n
0
It * $1 (g*n
h'
h') So V l k X / g / k ~  t $ . The p r o o f
i n particularW'IkZ'(n*goh')lkJil(g*qoh').
0
I
for V 5.2
i s similar.
Theorem.
Proof.
0
F o r Q a f o r m u l a o f L w i t h a p p r o p r i a t e parameters
U IF'' X/flk Q" i f f xlk Q . F i r s t l y , t h i s i s w e l l formed: Parameters f o r Q a t X / f a r e elements o f which a r e g i v e n as elements o f A(X) and a r e t h u s parameters f o r $ a t X.
m)
We proceed by i n d u c t i o n .
T h a t i s , we show t h a t i f we d e f i n e
It* i n t e r n a l l y
It
Ulk X / f * Q iff X l t Q c l o s e d under t h e d e f i n i n g c l a u s e s o f l k i n t e r n a l l y , (whence UIk X / f 1 1@ X\k $) and i f we d e f i n e \I by + X $ i f f Ulc X / f Ik @ t h e n i s c l o s e d under t h e d e f i n i n g c l a u s e s o f (whence Xlk Q *VIE X / f l t  Q). then
by

it* i s
It+
\kt
As t h e o p e r a t i o n s A + B a r e j u s t t h o s e i n h e r i t e d f r o m E, terms a r e i n t e r p r e t e d a l i k e i n b o t h contgxts: Thus i f [ T I = Uo] t h e n UlkUrl = Dull, so i s closed under ( = ) + and i f Ulk U ~ l l= Uol t h e n UIk T = a,
11'
so
IF*
i s c l o s e d under ( = ) +
It and \I* a r e c l o s e d under ( A ) ' , (v)', (3.)' i s t r i v i a l . F o r I, suppose 1 1 ' $Ifi f o r fi: Xi X i n some cover o f X t h e n X I 1 Xi/fi $Ifi and by I i n t e r n a l l y Xik X / i d l k $. I n t h e c o n t r a r y d i r e c t i o n , suppose Ulk Xi/g fi IF* $Ifi f o r some c o v e r of X as above. Then Xi $Ifi so Xlk Q t h a t i s Ulk X/g Q. F o r (+)+, f i r s t suppose t h a t f o r a l l f: V U i f V I   + ~ lt hfe n V I k + ~ l f Then . we c l a i m U I U / i d l k @ + I$,because f o r a l l g: W + U and a l l h : V + W , i f W @1g h, t h e n V It+$ l g h so V IF $19 h, t h a t i s W V/h v/h Jilg h. Conversely, iff o r a l l g: W + U and a l l h: Z + g*X, where f: X + U, i f WIE Z / g * f h \I* $ l f * g h , t h e n X I k @ + $, because f o r h: Z X i f Z \ k $ l h then U l t Z / f h It* $ l h so Ulk Z / f h It*Jl?h which g i v e s Zlk $Ih, so Ulk X / f I/* Q Ji. That
+
Xi
+
0
+
IF
0
o
0
0
+
t
0
0
0
+
0
The p r o o f f o r W+ i s s i m i l a r .
0
We view t h i s thorem as a s s e r t i n g t h a t i n t h e topos E t h e n a i v e n o t i o n o f t r u t h g i v e n by t h e e q u i v a l e n c e t h e s i s i s consonant w i t h t h e t h e o r y o f meaning g i v e n b y t h e n o t i o n o f f o r c i n g o v e r t h e s i t e &. O f course t h i s may seem vacuous as i t appears t h a t B i s manufactured w i t h t h i s r e s u l t i n mind. However, i n t h e case o f p r i m a r y i n t e r k t f o r t h i s paper, t h e r e s u l t s o f 84 a l l o w us t o r e g a r d (I i n t e r n a l l y as a f u l l subcategory o f Loc(E) equipped w i t h t h e open cover t o p o l o g y . I n f a c t , i f Q i s t h e c a t e g o r y o f s e p a r a b l e l o c a l e s , we may i d e n t i f y (I as a c a t e g o r y o f s e p g r a b l e l o c a l e s i n E. We s h a l l deal w i t h t h i s , among o t h e r t h i n g s , i n a sequel t o t h i s paper. Given f: X
f
U we may view an element a o f A(X) as a f u n c t i o n : U
It a:
X/f
+
A,.
T h i s a l l o w s us t o r e p h r a s e o u r theorem. 5.3
Corollary.
ulkX/flk $(a)
iff
Ulk~tE
X/f@(a[t)).
0
We view t h i s as a g e n e r a l f o r m o f t h e e l i m i n a t i o n theorem ( c f . T r o e l s t r a C19771
Continuous Truth I
179
The appropriate theory o f continuous t r u t h CT has an axiom f o r each pp.33,79). clause i n t h e d e f i n i t i o n o f X/f/k$(a). For example, t h e clause f o r 3 gives the axiom o f l o c a l choice Y t E 3 y $ ( a ( t ) , y ) i f f 3 open cover p: Z >> X and continuThe t r a n s l a t i o n T $ o f a formula $ w i t h ous f: Z + Y such t h a t W z $ ( a ( p ( z ) ) , f ( z ) ) . o u t f r e e lawless v a r i a b l e s i s given by T$ :def/k $.
X
CODA A general n o t i o n o f nonconstructive o b j e c t i s given by i n t e r p r e t a t i o n s i n Grothendieck t o p o i . The process o f i t e r a t i o n described i n 55 shows how we may view ( i n t e r n a l ) t r u t h i n t h i s i n t e r p r e t a t i o n as given by a nonstandard theory o f meaning. The clauses d e f i n i n g t h i s g i v e axioms f o r the corresponding theory o f continuous t r u t h CT and an " e l i m i n a t i o n " t r a n s l a t i o n . By construction, CT tf T$ and f o r formulae i n t h e l a w l i k e p a r t o f t h e language T $ 5 $. The p r o o f t h e o r e t i c content o f t h e e l i m i n a t i o n ;
I$
CT
$
iff
ID
T$,
requires f o r m a l i s a t i o n o f our treatment i n an appropriate theory I D o f i n d u c t i v e d e f i n i t i o n s . We do n o t undertake t h i s here. A f i n a l example o f an u n f i n i s h e d o b j e c t i s t h i s paper. Some o f t h e r e s u l t s , i n p a r t i c u l a r c o n t i n u i t y p r i n c i p l e s i n sheaves over s i t e s , go back t o 1978 and were much i n f l u e n c e d by discussions w i t h S c o t t and Hyland. Some r e s u l t s are s t i l l being r e f i n e d . Other p e r s i s t e n t i n f l u e n c e s have been those o f Joyal and Lawvere on t h e one hand and o f K r e i s e l , T r o e l s t r a and Dummett on the other. This research has been supported a t various times by the N.S.F. (U.S.A.), the S.R.C. (Netherlands), and t h e A.R.G.S. ( A u s t r a l i a ) , and made e a s i e r (U.K.), t h e Z.W.O. by t h e h o s p i t a l i t y o f many people n o t a b l y C h r i s t i n e Fox, I r e n e Scott, Karen Green, and Imogen K e l l y . I am g r a t e f u l .
REFERENCES A r t i n , M., Grothendieck, A., Verdier, J.L., ThGorie des Topos e t Cohomologie, E t a l e des Sch6mas (SGA4), (Lecture Notes i n Math. 269, 270, SpringerVerlag, B e r l i n , 1972). Beth, E.W., Semantical Considerations on I n t u i t i o n i s t i c Logic, Indag. Math., 9(1947), p.5727. Boileau, Andr6 & Joyal, Andr6, La logique des topos, J.S.L.
46(1981), p.616.
Brouwer, L.E.J., Cambridge Lectures on I n t u i t i o n i s m , D. van Dalen, ed. (Cambridge U n i v e r s i t y Press, 1981). Dummett, Michael, Elements o f I n t u i t i o n i s m , (Oxford U n i v e r s i t y Press, 1977). Dummett, Michael, T r u t h and
o t h e r enigmas, (Duckworth, London, 1978).
Fourman, Michael P., The l o g i c o f Topoi, i n Handbook o f Math. Logic (ed. Barwise, J.), (NorthHolland, 1977), p.105390.Fourman, Michael P., Notions o f Choice Sequence, Proc. Brouwer Symposium, (ed. T r o e l s t r a , A. and van Dalen, D.), (NorthHolland, 1982). Fourman, Michael P. & Grayson, Robin J., Formal Spaces, Proc. Brouwer Symposium, (ed. T r o e l s t r a , A. and van Dalen, D.), (NorthHolland, 1982). Fourman, Michael P.,
T1 spaces over t o p o l o g i c a l s i t e s , JPAA,
( t o appear), 1983.
180
M.P. FOURMAN
Freyd, P e t e r , Aspects of Topoi, Bull. A u s t r a l . Math. SOC., 7(1972), p.176. I s b e l l , John, Atomless p a r t s of spaces, Math. Scand., 31(1972), p.532. Johnstone, P e t e r T . , Topos Theory, (Acad. Press, London, 1977). Johnstone, Peter T . , Stone spaces, (Acad. Press, London, 1982). J o y a l , Andre, & Tierney, Myles, An extension of the Galois theory of Grothendieck, p r e p r i n t , 1982. Kreise!, Georg, Lawless sequences o f natural numbers. p .22248.
Comp. Math. 20(1968),
Makkai , Michael & Reyes, Gonzalo, FirstOrder Categorical Logic, (Lecture Notes in Math. 611, SpringerVerlag, 1977). Moschovakis, Joan R., A topological i n t e r p r e t a t i o n o f secondorder i n t u i t i o n i s t i c a r i t h m e t i c , Comp. Math., ( 3 ) , 26( 1973), p.26175. S c o t t , Dana S., Extending t h e topological i n t e r p r e t a t i o n t o i n t u i t i o n i s t i c a n a l y s i s , Comp. Math. 20(1968), 22248. S c o t t , Dana S . , I d e n t i t y and Existence i n I n t u i t i o n i s t i c Logic, Proc. Durham Symposium, (ed. Fourman e t a l . ) (Lecture Notes i n Math. 753, SpringerVerlag, 1978) , p. 66096. T r o e l s t r a , Anne S . , Choice Sequences, (Oxford University P r e s s , 1977). Wraith, Gavin C . , Lectures on elementary t o p o i , Model theory and t o p o i , (ed. Lawvere F.W. e t a l . ) , (Lecture Notes i n Math. 445, SpringerVerlag. B e r l i n , 1975), p. 114206. Wright, Crispin, W i t t g e n s t e i n ' s Philosophy of Mathematics, (Duckworth, 1981).
LOGIC COLLOQUIUM '82 G. Lolli, G. Longo and A . Marcia (editors) 0 Elsevier Science Publishers B. V. (NorthHolland), 1984
181
HEYTINGVALUED SEMANTICS R.J. Grayson
*
Institut fur mathematische Logik und Grundlagenforschung Einsteinstrafle 6 4 ,
4400 Munster, West Germany
Introduction. Chapter I.
The Logic o f HSets.
5 1 . Complete Heyting algebras. § 2. Interpretations of propositional logic
5 5 9
3. Hsets. 4.
Interpretations of predicate logic.
5 . Number systems.
§ 6. Complete Hsets.
5
7. Interpretations o f higherorder logic.
Chapter 11. Mathematics in HSets. § 8.
5
Some internal constructions.
9. Internal topologies.
§ l0.Choice principles.
9
11.Continuity principles.
References
Introduction. In this paper we develop a semantics for intuitionistic systems in which sentences are given "truthvalues'' in complete Heyting algebras (cHa), just as sentences of classical set theory are given values in complete Boolean algebras ([MD], for example). T h e use o f the lattice of open subsets of a topological space t o interpret intuitionistic propositional logic goes back t o Tarski ([Ta,RS]). Extensions t o predicate logic were made b y Beth and Kripke (ID]) and applied t o metamathematical results for arithmetic by Smorynski ([Tr]). Further interest was drawn t o the area by the topological interpretations of analysis in [Sl,Mo,VD], where it was shown that "Brouwer's Theorem", on the continuity o f all functions between reals o r the Baire space, could be modelled in this way. In addition, Bishop's book ([Bi]) showed the feasibility of constructivism and gave new impetus t o the investigation of constructive and intuitionistic systems. A t the same time, interest has arisen from the theory of topoi,
*
Research Fellow of the AlexandervonHumboldtFoundation
R.J. GRAYSON
182
which can be seen as a categorytheoretical formulation of intuitionistic higherorder logic ( [ F l l , for example). Other kinds of semantics are also suggested by this approach, for example, sheaves over sites ([MR]). However, the level of generality of Heytingvalued semantics seems to provide a natural stoppingpoint: the notion of cHa is simply an algebraicisation o f the notion of "truthvalue" for intuitionistic predicate logic, staying within the conceptual framework of topological, Beth and Kripke models. The general theory of sheaves over a cHa (here called Hsets) is worked out in great detail in [ F S ] , where it is shown how they model intuitionistic higherorder logic (the extension to set theory is made in [Gl]). This paper is designed as a selfcontained introductory exposition of the basic definitions and results, which it is hoped will enable the interested reader then to come to grips with more detailed treatments as well as with more specialised papers in this area. The paper falls into two chapters. In Chapter I we describe successively the interpretations of propositional, predicate and higherorder logic over a cHa. In Chapter I1 we develop some analysis and topology in these models, with particular emphasis on topological models and on the interpretation of various principles of choice and continuity. We close with Joyal's very elegant proof, using topological models, of a derived rule of local continuous choice for intuitionistic higherorder logic. I have not attempted on the whole to assign credit too exactly, beyond references to the literature, but I should like to acknowledge here the contributions of Dana Scott, whose influence on the whole treatment should be clear, and of Mike Fourman and Martin Hyland, who have stimulated my interest in the subject over the years. I thank the AlexandervonHumboldtFoundation, Bonn, for financial support, and the Institut fur mathematische Logik und Grundlagenforschung, Miinster, for their hospitality.
CHAPTER I.
5
THE LOGIC OF HSETS
1 . COMPLETE HEYTING ALGEBRAS
We begin by defining the structures which are to act as our domains of "truthvalues". Although we will be mostly concerned with topological examples, this more general, algebraic setting seems to make the essential features clearer, besides providing further examples (see 9 . 7 for example). Much information on the classical theory of complete Heyting algebras (cHa) may be found in [ R S ] and on the constructive theory in [ F S , Chapter I]; for we want to be handle our models "constructively" too (see 7.8 for further discussion of this point). 1 . 1 Definition. A complete Heyting algebra
lattice (H,Z), with finitary and infinitary by h , l \ , v , V ,
is a complete
meet
and join denoted
satisfying the distributive law, for pEH and ASH, phVA
E
V(phq1qEA).
Hereafter H will always denote a cHa, with elements p,q,... also the notation T for VH. the "top" element, and 1 for "bottom" one.
AH,
.
We use
the
HeytingValued Semantics
Logically, the order relation tion. 
5
183
is read as the relation of implica
In addition one may define in any complete lattice an
implication operation by
5
(p9) = V I r t p A r
q}.
AS a special case we have negation ~p defined as (p+I), which equals V{r I pAr=I) 1.2 Lemma. In any cHa H the implication operator is characterised by the adjunction rZ(p+q)
iff
(phr)lq.
Proof. If ( p n r ) ~ q ,then rL(p+q) always holds, by definition of implication. If H is a cHa and rl(p+q), then the distributive law gives pAr
5 =
PAVCSIPAS5 V{pAS [ P A S
q}
5 qj
5 4.
Proof. (i)

(iii) follow at once from Lemma 1 . 2 .
Since ~p(~p,
(iii) gives ph7p=l and then p5,.p. From (i) and qATq=l we obtain ph(p+q)h~q=I, hence applying 1 . 2
(p+q)
5
~ ( p h ~ qand ) (ptq1Aq 5 TP, by
again gives (p+q)
5
(iii)i
(q+p). The remainder is left as
an exercise. 1.4
Examples. a) The open subsets O(T) of any topological space T
form a cHa under inclusion, 5 . A , V , V are the settheoretic fl,U,u while hA=Int ( O A ) and T
(U+V) = IntftI tEU + tEV).
is T , I is the empty set
In this context we use u , V ,
d , and
...
rU is Int(T'U1.
for elements of O ( T ) , and s , t , . . .
for elements of T. We call such cHa topological; ways of obtaining
R.J. GRAYSON
184
nontopological examples may be found in [FS,S2]. b) As special cases of topological cHa we have those arising from partial orders ( K , c ) , where K is given the topology of upwards closed subsets (that i s , O ( K ) consists of those P such that Vi,jEK.j)iEP + j E P ) . This provides the connection between Kripkemodels based on partial orders and semantics with "truthvalues" in topological cHa (see 3 . 3 (c))
.
c) For a similar connection with Bethmodels based on a partial order ( K , Z ) , one takes T to consist of all maximal chains a in K , with O(T) having as subbasis the sets {UlfEa) for iEK. 1.5 Heyting Algebras. A lattice equipped with an implication having the property of Lemma 1.2 we may call simply a Heyting algebra. These are treated in [RSI under the name of "relatively pseudocomplemented" lattices; it is shown there that all such lattices satisfy & t Jfinitary distributive laws, as well as the infinitary one for such joins as exist.
For the purposes of 2.5 it is useful to note the following simple completion process for any Heyting algebra H: Let O(H) be the topology of downwards closed subsets of H (compare 1 . 4 (b)), and let J be the Joperator ( I F S , 2 . 1 1 ] ) defined by J ( U ) = the set of all joins of subsets of U which exist in H. Then p W [PI = {qlq
(i) (ii) (iii) (iv) (v)
.
T=pVrp T=,r+r (Tnlr) 5 (T+r) t(pAq) 5 lpvq r+(pvq) 5 (rrp)v(rrq)
These follow straightforwardly from the q=lp and ~ r = @ .
observation
that p=lq,
2. INTERPRETATIONS OF PROPOSITIONAL L O G I C
The interpretation of intuitionistic propositional logic in a general cHa (or even, Heyting algebra1.5) can perhaps hardly be counted a s "interpreting" at all; it i s more a matter of algebraicising logic, as i s made clear in [RS]. 2.1 Definition. An interpretation of a propositional language in a cHa H assigns an element "PI1 of H , the "truthvalue'' of P , to each propositional letter P. Symbols t and f , for "true" and "false", are included in the language, and we require "t11 =T and [[f]l=I.
Given an interpretation we extend the evaluation to give a value [[A]] in H to each formula A of the language as follows: AAB 11 = [[ A 11 A [[ B and similarly for v,+ and
7 .
11
HeytingValued Semantics
185
2.2 Definition. A propositional formula A is valid in an interpretation iff [[A T. Further, A is universally valid iff it is valid in all interpretations.
I=
2.3 Definition. The system IPL of intuitionistic ro ositional logic is given by the following axioms and rules (takenPfrEm [Tr, 1.1.311, rules being indicated by the double arrow
*.
PL PL PL PL PL PL PL PL PL
1) 2) 3) 4) 5) 6)
7)
8) 9)
A+A A,A+B * B A+B,B+C i, A+C AhB+A, AhB+B, A+AVB, B+AVB A+C,B+C AVB+C A+B,A+C * A+BhC AhB+C 9 A+(B+C) A+(B+C) * AhB+C f+A,A+t
2.4 Soundness Theorem. Every propositional formula provable in IPL is universally valid. Proof. Straightforward using Lemma 1.2 and its Corollary. Firstly, validity of A+A means that “A 11 + [[ A 11 = T, that is, that [[A]] 5 [[A]]. Closure under PL 2 means that, if [[A]] = T and “A11 + “ B l I = T , then “ B l I = T ; but “A11 + “ B l I = T iff “A 11 5 [[B]]. P L 3 is just the transitivity of 5 in H , while PL 46 express that A and V are respectively meet and join in H. PL 78 correspond exactly to Lemma 1.2, and PL 9 results from the requirements [[ t I] =T , [[ f 11 = 1. Note: By the soundness theorem we can give counterexamples to the provability of various assertions, by the method of 1.6. 2.5 Completeness Theorem. Every universally valid propositional formula is provable in IPL. Proof. We construct in fact a “universal” interpretation for which validity is exactly provability. Denote provability in IPL by Iand consider the “LindenbaumHeyting“ algebra of equivalence classes of formulae under the equivalence relation AB
iff
I A
++
B,
with the order relation given by [A]
5 [B] iff k A + B .
NOW complete this Heyting algebra as in 1.5 and interpret “PI] as the (image of the) equivalence class [PI. Then [[A]] = [A] for every formula, whence A is valid iff [A]=T=[tl iff IA.
s
3. HSETS
Preparatory to interpreting predicate logic over a cHa H, we describe the objects which are to provide the domains of interpretation of variables. They are sets with an Hvalued equality relation, [I.= #I], which is not required to be reflexive, the value [[a=a]] rather giving a measure of the “existence” of an element a. Further discussion of these “partial” objects and their logic will be found in [FS,§41 and [S2].
R.J. GRAYSON
186
3.1 Definition. An Hset (given a cHa H) is a s e t A with a function [[. =*I] : AxAG satisfying, for all a,b,cEA, (i) (ii)
[[ a=b]l [[ a=bll
= A
[[ b=a]l [I b=cl1
5
[[ a=c]l
We do not require “a=a]l= T, by [[Ea]] = [[a=a]]. If [[Eall
=
.
but define the existence predicate E T , a is called global.
A s motivation for this definition we have the following basic example. 3 . 2 Definition. Given topological spaces X and T , the O(T)set X consists of all continuous functions a:U+X for UEO(T), with equality defined by
“a=b]]
= Int{tla(t)=b(t)}.
(Here external equality is also taken t o be “strict”, so that a(t)=b(t) implies tEdom(a) I7 dom(b) . ) Then [[ Eall = [[ a=a]] = dom(a), as this is open by definition. T h u s the existence predicate has a very natural interpretation as the “domain of definition” o f an object. Furthermore, equality on X is “local”, since tE[[ a=b]] iff a and b agree on some neighbourhoox o f t.
3.3 Further examples. a) F o r any set A the trivial or constant Hset f h a s A as its underlying set with “a=b
11
i
=
T
i f a=b
I
if a+b.
(For a constructive version, when equality o n A i s not decidable, one puts “a=bIl =V{Tla=b}.) If X,T are topological spaces, the constant O(T)set ? can be identified with the subset o f XT consisting of all constant functions G=At.x, for xEx, since in x T ’ [[ $=$]]= {t I X=y} = V { T l x=y}. b) The product (A x...xA
)
o f Hsets A
* *
*,An is defined t o be
their settheoretic product with equality
a=gll
=
& [ [ 
ai=bill
(Note that this differs slightly from the definition in [FS,4.8]: we are not concerned here t o k e e p the product “separated“ in the sense o f [FS,4.6].)
HeytingValued Semantics
187
c) The domain of a Kripkemodel based on a partial order ( K , O ([D,Tr]) is given by assigning a set A(i) to each iEK, in such a way that j ’ i implies A(i) 5 A(j). Then with A = ,u A(i! we have iEK a natural existence predicate [[ E()]] : AbO(K), K having the topology of 1.4(b), given by [[Ea]]
= {ilaEA(i)}.
Furthermore, if the model is equipped with equivalence relations we have an evaluation i o n each A(i), with i 5 j for
icj,
[[ a=bl]
=
(ilai
b}
making A an O(K)set. The point we want to make here i s that “partial” objects are already latent in the growing domains of Kripkemodels. 3.4 Definition. A predicate on an Hset A is a function [[ P()]] : AbH, which is strict and extensional, that is, for a,bEA, (i) (ii)
tt
~ ( a ) l l 5 [kall P(a) 11
A
“a=bl]
5 “P(b)
11
A relation on an Hset (or on several Hsets) is taken to be a predicate on the appropriate product. 3.5 Remarks. a) The requirement of strictness for predicates, which says that a predicate can hold only of existing objects, is found to be technically convenient in handling the models. It should be noted though that (strict) predicates are not (in general) closed under logical operations, for example, negation: Xa,b. [[a=b]] may very well not be strict. This point is discussed further in [52,3.11. b) In [FS] and [52], use is made of a (nonstrict) relation of equivalence f , which may be defined in any Hset by “a~b]]
=
([hall
V
[[Ebll
b
[[ a=bll )
and expresses (the value of) “a and b are equal insofar as either of them exists”. This relation is useful for talking about partial functions, and the logic of partial elements given in S 4 may also be neatly axiomatised using E and Einstead of =: in particular, in any Hset, equality can be reeovered by “a=bll
= [hall
A
“a=bl]
However we will make no further mention of this relation.
4.
INTERPRETATIONS OF PREDICATE LOGIC
We are now ready to formulate the notion of an interpretation in Hsets and to prove soundness for a formal system of predicate logic with “partial existence”. For a completeness theorem for this logic we fall back on the wellknown theorem for Kripkemodels. 4.1 Definition. An interpretation of a firstorder (relational) predicate language, with equality and existence predicate, over a cHa H consists of an Hset A together with a relation on A (in the sense of 3.4) for each nonlogical relation symbol of the language. (In general, constant and function symbols have to be interpreted as special kinds of relations; see 4.78.)
R.J. GRAYSON
188
Given an interpretation in an Hset A, we define the value “C]] in H for each sentence C o f the firstorder language extended by adding constants for the elements of A , as follows: (i)
The atomic cases are given by the interpretation, equality and existence being interpreted by the basic structure of A as an Hset (3.1).
(ii)
The propositional connectives are dealt with as in 2.1.
(iii)
The evaluation of quantifiers is defined by Ujx. cII = V { “Ea A c [a/xlIl laeA} “Vx. C11
=A{
“Ea+
C[a/x]]l
laen}.
(Note: this i s the first point where w e have needed completeness o f H.) The idea in evaluating quantified formulae is that we only quanitify over existing objects; in other words, the quantifiers are relativised to the predicate E. On the other hand, free variables need not refer t o existing objects, and we make the following definition. 4.2 Definition. A formula C with free variables x in an interpretation in an Hset A iff
.
V = E A ~ .[[ c [ a l / x l ,. . ,an/xnI 11
=
T
I
.
.
. .xn
i s valid
.
C i s universally valid iff it i s so in all interpretations. We return to give examples of interpretations and o f the evaluation of formulae in § 5. Before that w e want to deal briefly with a formal system (essentially that o f [S2]) which i s natural for the above notion of interpretation. 4.3 Definition. T h e system IQL o f predicate logic with equality and existence consis t s o f the propositional system I P L (2.3) together with (i)
 
For equality and existence the axioms El) Ex ++ x=x E 2 ) x=y y=x E3) x=y A y = z x=z E4) R(5) + EX h...hEX 1 E5) R(x)  A x l = y l A Axn=yn + R(x)
...
(ii)
For Ql) Q2) Q3) Q4)
(for each symbol R)
the quantifiers the axioms and rules E X A C + 3x.C EX AVX.C + C ( E X A C  D ) * (3x.CD) ( D A E x + C ) 9 (D+VX.C)
(In Q3 and Q 4 , x should not be free in D.) 4.4 Soundness Theorem. Every predicate formula provable in IQL i s universally valid. Proof. The validity o f axioms El5 is simply built into the definition o f interpretation, since (by 1.3) an implication CD is valid 5 “011 T o check Q 1 , for example, in an interpretation iff [[C]] in an Hset A , we need to show that
.
VaEA.
“Ed]
A
“C[a/xll]
<_ “3X.Cll
(assuming for convenience that x i s the only variable free in C).
HeytingValued Semantics
189
But this is immediate from the definition of all the terms on the left.
I]
“3x.C
as the join of
To show closure under Q4 note first that, for any p in H , p<[[’dx.cn
VaEA. I[DlI A whence
“Dl1
5
iff VaEA.pA [[Ed]
<_ “Vx.C]]
[[
c [a/xlIl
again. Now the validity of the hypothesis
making use of Lemma 1 . 2 of Q4 reads
;
5
“Ed
[[ C [a/Xll]
that is, (D+Vx.C) is also valid.
4.5 Completeness Theorem. Every universally valid predicate formula is provable in IQL. We do not give a proof of this result, since it follows straightforwardly from the completeness theorem for Kripkemodels, which can be seen as special cases of interpretations in our sense (according to 1.4(b) and 3.3(c)). 4.6 These interpretations extend immediately to manysorted languages, with sorts a , B , each interpreted by an Hset Aa,A B , . . .
...
acting as the range f v riables xa,xB,. fiers of the form Vx8,3x
’,...
..
of each sort, with quanti
Actually, as may already have been noticed, we are tailoring our logic to our interpretations, rather than viceversa. So we will tend to start with Hsets A,B, with various relations (and functions  see 4 . 7 ) on them, and interpret the language of these structures, with a sort for each Hset, writing, for example, the quantifiers now as VxEA,3yEB,
...
...
4.7 Function Relations. In order t o interpret predicate languages with constant and function symbols we need in general to treat them as special kinds of relation, namely singletons and functional relations respectively. A singleton on an Hset A is a predicate P such that
Va,bEA. [ [ P(a) 11
A
“P(b)
11
<_ “a=bll
.
A functional relation between Hsets A and B is a relation R on AxB such that VaEA Vb,b’EB. “R(a,b)
11
A
“R(a,b’)
11 <_
“b=b’l]
These conditions clearly correspond respectively to the validity of Vx,yEA (P(x) AP(y) + x=y) and VxEA Vy,zEB (R(x,y) A R ( x , z ) + y=z). In general, functional relations will serve to interpret partial functions (and singletons partially existing constants). To interpret total ones the relation R should also be total, that is, VaEA.
[I
Eall
<_v{ “R(a,b) 11
I bEB);
in other words, VxEA i’yEB.R(x,y) i s valid. 4.8 Functions. In particular cases, however, function symbols may be interpreted by functions F:A+B which are strict and extensional, that is, for a,a‘EA, (i) (ii)
“E(Fa)ll [[ a=a’I1
5 A
“Eall [[ E (Fa)I1
.(
[[ Fa=Fa’I1
F then gives rise to a functional relation R by “R(a,b)]] = “Fa=b]l, which is total just in case for all a
=
R.J. GRAYSON
190 (iii) [[ Eall
5
E(Fa)
11 .
The property of completeness for B , which ensures that every functional relation arises from a function in this way, is discussed in
5
6.
4.9 If function symbols occur in a manysorted firstorder language and are interpreted as in 4.8, we now have a class of terms u , ~ , . . . each of a particular sort; these are evaluated as elements R u 11, [[TI] of the Hsets interpreting those sorts, by the clause
,...
“F(U l,...,Un)ll
bn]]
= F ( ~ ~ U l ~ ~ , . . ) ,~ ,
and atomic sentences containing terms are evaluated by
,...
~ ~ R ~ U l , ~ .= . “R(“UIII ~ U n ~ ~ ~
“On]]
)I1
The logic IQL should then be extended by the addition of a rule of substitution. Full discussion of this logic may be found in r.521.
5
5. NUMBER SYSTEMS
In this section we consider structures which interpret the firstorder theories of the natural numbers, integers, rationals and real numbers in Hsets. The higherorder theory (in particular, the induction axiom for the natural numbers) will be dealt with in 8, where we also see that these structures are the (standard) interpretations of the number systems.
s
G,?,$
5 . 1 We start by giving the constant Hsets (3.3(a)) the trivial or constant firstorder structure of the natural numbers, integers and rationals. That is, on each we use the standard arithmetic functions to interpret Successor, sum, product (as in 4.81, while the order relation (on Q, say) is given by
Then it is an easy induceion to show that in each case the terms are interpreted by themselves, ~[u]]=u, and the sentences by
For a constructive treatment we show “CIl= VITlC), and for this H should be an open cHa in the sense of [G4]. In any case this indicates that these interpretations have in themselves no interest for intuitionistic mathematics; they are needed however to provide a basis for the more interesting interpretations of analysis. 5.2 T o interpret the firstorder theory of real numbers we specialise (as defined in 3.21, where to the case of O(T)sets and consider R R denotes the (external) real numbers. ‘?his O(T)set has a very rich structure, as first exploited (essentially) by [Sl]; see 5.7 for a discussion of his model. We consider first relations of order and apartness on R [[a
T:
= {tla(t)< b(t)l =
{tEdom(a)
n dom(b) 1
a(t)
b(t))
HeytingValued Semantics
191
the righthand sets being always open as a,b are continuous. As in 3.2 the occurrence of < on the right is taken to be “strict“; for a constructive treatment we put on the right too in the definition of apartness. Here is a picture in the case T=R:
[[ a
T
5.3 Proposition. The intuitionistic theory of order and apartness on the reals (eg [He,Sl]) is valid in every RT. That is, the following are valid: (i) vx,y. x=y ++ 7x*y (ii) Vx,y. xty P yylx (iii) Vx,y,z. xPy + (x#z v zky) (iv) Vx,y. xyy ++ ( x < y v y<x) (V) vx,y,z. X < y A y < Z + X
7
“arb]]
“Ed]
n
[[ Ebll
Since [[a=b]] n [[a*]] is clearly empty, we obtain the inclusion from left to right (Cor. 1.3). On the other hand, if U I7 aabl] = @ and tEU n dom a Il dom b, we have r(a(t) # b(t)), so a(t)=b(t), and the other inclusion follows. For (iv) we need [[ affb]] E [[ aCb]] u “Ma]] b(t) iff a(t) < b(t) v b(t) < a(t). a(t)
+
, which follows from
For (vi), let tEdom(a) n dom(b) f l domtc) and a(t) < b(t); a(t)< c(t) or c(t)< b(t), so tE[[a
then
For (viil , if act\ < b(t\ , L e t x be chosen so t h a t a \ t > < x( b \ t ) . Then for the constant function x s . x we have t € “ a < k b I l . Hence I cERT} = [[3z.a
=
(tl,to)
= (to,t2)
“atbll
= (tl,to) U (to,t2)
“a=bll
=
@.
192
R.J. GRAYSON
Hence the following sentences C are not valid in both cases): (i) (ii)
(in fact to @ "Cll
a
Furthermore, by "shifting" a and b about we can arrange to omit any point of T , so that the universally quantified forms of (i) and (ii) get value @, and we obtain the stronger result that (i)' rVx,y. x
V
yCx V X=y x#y
+
are both valid in RR. 5.5 Arithmetic structure on RT is also given "pointwise", by functions into RT, namely a+b = At.(a(t)+b(t)) a.b = At. (a(t).b(t)) a
1
= ~t.a(t)'
In each case the arithmetic functions on the right are taken to be "strict" also, so that dom(a+b) = dom(a) fl dom(b) = dom(a.b) and dom(a') = ftEdom(a)la(t)+O), taking the external inverse function to be defined just at nonzero reals. The inverse function i s thus a typical example of a partial function, and satisfies in R V X . E ( X  ~ ) ~ + +x
8.
Together with the other usual identities this shows that R models the theory of an apartness field [He], with unit elements The constant functions 8 and I. AS
in 5.4 we can give counterexamples in R
Let a be At.max{O,t) and b be At.max{O,t). a(t).b(t) = 0, so that [[a.b=OI]
R: Then, for all t ,
= R
while [ [ a s v b a l l = (,O) U ( 0 , ~ )so that (a.b=O is not valid.
+
a=O
V
b=O)
A
5.6 A s in 3.3(a) we can treat the O(T)set Q as the subset of R T consisting of all constant functions $ for ~ E Q .We can now prove one half of a cheorem characterising RT as the Dedekind cuts in the (The other half is proved in 8.5.) rationals, interpreted as
3.
Theorem. The subset 6 i s dense in RT and each element of RT acts as a Dedekind cut in 6 . T2at is, the following are valid (using p,q,... to range over Q , and x,y,... over R ) :
T
(i) (ii) (iii) (iv) (V)
(vi)
vx,y. xX 3q. X
Proof. The proof of ti) proceeds exactly as that of 5.3(vii), except that we now choose a rational p between a(t) and b(t). The remaining assertions may be straightforwardly deduced from the properties in 5.3 and the denseness of Q , arguing within the system IQL.
HeytingValued Semantics
193
To explain our sense of "cut" here briefly: thinking of the rationals below (resp. above) an element x as the lefthand (resp. righthand) elements of a cut in Q , these conditions say successively that the two halves of the cut are inhabited, disjoint, closed downwards (resp. upwards), open (in Q) , and close together. This notion of cut is due originally to Tierney. 5 . 1 It may be asked why we consider artial functions at all in our models RT, and not only total ones, t s was done in [Sll (and, for Baire space, in h o ] ) . The point is that in those papers only special cases of the space T are considered, for which every partial function is locall extendable to a total one; that is, for each a and tEdom(a), theri is a total b w i t h a = b ] ] .
In general there may be very few total continuous functions but many partial ones, so that the total ones do not at all provide a representative picture. For example, take T = R U { * I , where R has its usual topology and has asneighbourhoods just the complements of finite sets. Then all total elements of RT are constant, while there are many nonconstant elements defined just on R itself.
*
s
6
COMPLETE HSETS
It is immediate that the O(T)sets XT are always complete in the following sense, which we formulate for an arbitrary Hset A.
6.1
(i) For each aEA and pEH there is a (unique) restriction, alp, of a to p, with the property VbEA. [[ alp=b]]
=
[[ a=b]I
A
p.
From this follows [[ E(a1p) 11 = [[ Ea]] A p = [[ a=alp]I , that is, the "existence" of a is restricted to p , but where alp exists it equals a. In X the restriction alU is simply the settheoretic restriction of a toTa smaller domain. (ii) For each compatible subset B of A , i.e. such that Vb,b'EB. [[ Ebll A [[ Eb'N 5 [[ b=b']I , there is a (unique) join, V B , in A, with the property VaEA. [[ a=VB]I
=
v f "a=b
v{[[
11 I
bEB).
.
EbII 1 bEB), and for bEB, From this follows [[ E(VB) 11 = [[ Eb]] 5 [[ b=VB]], that is, V B is a "glueing together" of the elements of B. A subset B of X is compatible iff all its elements agree pairwise on the intersecTions of their domains. Then the join of B i s the settheoretic union of B, which is again an element of X T' 6 . 2 Remarks. a) Alternative definitions of "completeness" and proofs that the notions all coincide may be found in [FS,§4]; in particular, complete H'sets are equivalent to sheaves over H. In [FS] also the completion of an Hset is constructed. Our position here is that completeness is useful (as in 6 . 3 ) when it arises naturally, but not worthwhile introducing specially.
b) Constant Hsets are never complete (unless H is trivial), since all elements are global, so that the "nonexistent" element (a7 1) is missing, at least. However, when X is treated as a subset of X T as in 3.3(a), we can identify its completion as the locally constant elements of X In particular, this is useful for understanding the completions 8 and Q^ as subsets of RT.
OF
.
R.J. GRAYSON
194
6.3 The consequence (indeed, equivalent) of completeness that interestsus most here is that, when B is complete, every (total) functional relation R on AxB, for any A , arises from a (total) function F:A+B (as in 4.8), related to it by VaEA,bEB. [[ R(a,b) 11
=
[[ F(a)=bll.
Namely, given R , define F(a) = v { b l "
R(a,b)ll I b€BI.
The compatibility of the set on the righthand side, for each a , is precisely the functional character of R. This representation of arbitrary functional relations will be especially useful in 9.8. 6.4 Another interesting equivalent of completeness is that, if A is a complete Hset, we can evaluate description terms (1x.C) ("the x such that C " ) in them, according to [[ Ix.CI1
since "Vx.x=a such that C".
++
=
V{al [[ Vx.x=a
++
Cll 1 a€A}
C " expresses exactly that "a i s the unique element
The corresponding logic of descriptions [52,§61 has the one extra axiom I)
Vy[y=Ix.C
f*
Vx(x=y
ft
Cll.
As an example, if R is a functional relation on AXB, one obtains the the corresponding funytion (6.3) as F(a) = [[I~€B.R(a,x)ll. In particular,'the inverse a in RT (5.5) is obtained as (the interpretation of) IX. (a.x='i). These are examples where existence of a solution may be partial, but uniqueness is guaranteed. T o give an example of "partial uniqueness" we refer back to the picture in 3.2 and consider the term Ix.(x=avx=b): here a solution exists on the interval (to,t,) but is unique only on (t1, t2)
.
5
7
INTERPRETATION OF HIGHERORDER LOGIC
The final step, before doing "mathematics in Hsets" properly, is to interpret intuitionistic higherorder logic. We will here be exclusively concerned with "standard" interpretations, in that powersets will contain "all possible" subsets. Thus the interpretations will no longer be complete for the logic, and certain problems will necessitate recourse to other kinds of interpretation (realisability, On the sheaves over sites etc), to which we refer briefly in 1 1 . 1  2 . other hand, the mere fact of soundness of the interpretations is quite powerful, as will come out most clearly in 11. This is also the point at which one begins to reap the benefits of the generalisation from Kripkemodels; as observed in the introduction to [VD], standard interpretations of, say, Baire space in Kripkemodels over a partial order only yield constant structures, and nothing i s gained.
s
7.1 Definition. The powerset P(A) on an Hset A consists of all predicates (3.4) P,Q,... on A , with equality defined by P=Qll
= I \ { " Pa]]
+* [[ Qall
I aEA)
In this context we write [[ a€P]] for [[Pa 11 , and interpret bounded quantifiers 3x€P, etc., in the obvious way. Note that every element of P(A) i s global ( [ [ EP]] = T ) , so that P(A) is (almost) never complete (compare 6.2(b)).
195
Heyt ingValuedSemantics
7.2 Proposition. P(A) acts as an extensional powerset of A, satisfying full comprehension and with E strict (as a relation on AxP(A)). That is, the following are valid: (i) (ii) (iii)
VX,YEP(A). X=Y .++ VxEA(xEX ++ xEY) 3XEP(A) VxEA. xEX ++ C , for each formula C. xEX b E x h E X .
Proof. (i) and (iii) are simply the definition of P(A). For (ii), given a formula C (in which we have assigned constants from some Hsets to all the free variables except x), we set
E[
a E ~ l 1= [[ Ea
A
~ [ a / x l l l,
so that P is clearly a predicate satisfying
(ii).
7.3 Definition. A manysorted higherorder language is one in which, for each sort a , we have a power sort P(a), and, for any sorts a l , ...,a n , we have a product sort ( a x...xan). 1
In addition there should be function symbols for tupling and projection for the product sorts, a relatioE symbol for membership, E , on each a x P ( a ) , and abstraction terms {x IC} of sort P(a)for each formula C. The system IHL of intuitionistic higherorder logic is a manysorted (4.6) version of the system IQL (4.3) with the addition of standard equations for tupling and projection, and axioms of extensionality, comprehension and strictness of E for the power sorts, as formulated in Proposition 7.2. The system IHLN is obtained by adding a sort N for natural numbers with a symbol for the successor function, and satisfying Peano's axioms including full induction (which we formulate below in 8.2). 7.4 Standard Interpretations. A standard interpretation of a manysorted higherorder language is one in the sense of § 4 in which the Hset A assigned to a power sort is always the powerset P(A ) P (a)
of that assigned to a, and in which the membership relation is interprehed as in 7.1. In particular then the interpretation [[{x lC}]]of an abstraction term is given as in the proof of 7.2(ii) by the predicate la.[[ EahC[a/x]]1 on Aa. Soundness of standard interpretations for the system IHL is immediate from 7.2; the extension to IHLN is dealt with in 9 8 . A more detailed description of a system similar to IHL may be found in [S2,§7] and of standard interpretations in Hsets in [FS,§5 and s71. 7.5 Types. A type in a manysorted higherorder language is a term whose sort is a power sort. In particular, each abstraction term i s a type. We may think naively of types as " s ts" % =xa '} and use the notation Each sort a we identify with the type {x'lx VxEa etc. for the quantifiers. Conversely, each type can be treated in as a new sort, using the restricted quantifiers VxEu, 3xEr, the obvious way, and.relativising power sorts, product sorts and abstraction according to P(u) = {XEP(a) IXc_u} for u a term of sort P(a), and so on (see 152,571).
...
7.6 Interpreting Types. In parallel to the treatment of types as new sorts, we want to interpret each type (in a given standard interpretation in nsets) as an Hset: A term u of sort P(a) is already interpreted as a predicate [[o]] on the Hset Aa. The Hset A is then defined as having the same
R.J. GRAYSON
196
but with equality relativised to [[u]]
underlying set as A [[ a=b]I
[[ a=bll a
=
aEol1
A
:
,
denoting by the subscripts u and a evaluation in A
and A
a'
Now an induction over the terms and formulae of the language shows that interpreting the relativised quantifiers etc. as speaking about the Hsets A amounts to the same thing as interpreting them as defined in tge original language, within the Hsets A a' 7 . 7 Exponents. A basic example of a type is the exponent U,B,
B a of sorts
given by the term {xEp(axB) lVxEa 3!yED.<x,y>Ex).
The interpretation of this term as an Hset, according to 7.6, is then the exponent of A and A B , which consists of all relations R,S on AaXAB with [[ E(R) 11
=
[[ R a total functional relation]]
Ba
and [[ R=Sl]
=
[[ E(R)II
Ba
aa
A
{ [ [ R(a,b)ll
aEA,,
*+
[[ S(a,btll 1
b€AB)
7.8 Now we ask the reader to look back over this first chapter and see that the definitions of Hsets and validity in them, and the proof o'f soundness of standard interpretations, can all be carried out within the system IHLN of 7 . 3 . This means, for example, that we can iterate the construction of the models inside any universe of Hsets, just as forcing is iterated in classical set theory (see [FS,§9] for example). More interestingly, perhaps, we can use the provable soundness of the interpretations to obtain derived rules for the system. We give an example of this, due to Joyal, in 1 1 . 5 ; other examples may be found in [Be,H2,FJ]. So in Chapter I1 we will be concerned to note, as we did in 5, what principles are needed to prove the validity of various assertions in various models, arguing so far as possible "constructively", i.e. within the system we are interpreting. In order to distinguish what is assumed to hold "on the outside" (or "in the ground model") from that which is valid in the interpretations, we use the terms external and internal.
The above cor.siderations all extend mutatis mutandis to systems of with the powerset axiom and full comprehension, as formulated in [Gl] and exploited in [HZ]. The general problem of interpreting a set theory, with only the axiom of exponents, within such a theory is dealt with in [G3]; applications of this are made in [Be].
set theory ~
CHAPTER 11.
5 8.
MATHEMATICS IN HSETS
SOME INTERNAL CONSTRUCTIONS
We are now ready to interpret constructions within the system IHLN of higherorder logic with a sort for natural members (7.3) in Hsets. The integers, rationals, real numbers, functions etc. appear as types (7.5) in this language, which we want to interpret as Hsets according to 7 . 6 . Such characterisations will generally be
HeytingValued Semantics
197
only "up to isomorphism", in the following sense. 8.1 Definition. An isomorphism between Hsets A and B is a total functional relation ( 4 . 7 ) on AxB which is internally oneone and onto. As in 4.8 and 6 . 3 , in particular cases an isomorphism may be given by a function from A to B which is internally oneone and onto, or even (for example, when both A and B are complete) by a pair of functions F:A+B and G:B+A which are inverse to one another:
VaEA.
[[ Ea]]
VbEB. [[ Eb]]
and
5 5
[[ a=G(Fa) 11 [[ b=F(Gb) 11
.
The extension to isomorphisms of structures is made in the obvious way. As in classical mathematics there is only one structure (up to isomorphism) satisfying Peano's axioms for arithmetic in any standard interpretation fn Hsets. We do not prove this fact but only show that the Hset N with constant structure ( 5 . 1 ) does satisfy the axioms, and hence can serve as the interpretation of the sort N. 8 . 2 Proposition. f j with the standard successor function S satisfies Peano's axioms for arithmetic, including induction in the form
VXEP(2). O E X A v x E X . SXEX
+
VXEN. XEX
Proof. Since the interpretation of firstorder sentences is always absolute (5.11, the firstorder axioms are trivial. To prove inguction one shows, by an external induction, for any predicate P on N. that OEP
if
q =
then
VnEN. q
By definition q ( " of 4.4,
5
5
5
:
95" VxEP.SxEP11,
[[ nEP11
A
[[ nEP11, q
VXEP. SXEP]], nEP11
OEPII and VnEN. q
hence, if q
A
5
that is, as in the proof
[[ SnEPl];
[[ SnEPl].
8.3 We leave the reader to check that, for some standard definitions of the integers and rationals as types obtained from products of N, the corresponding Hsets, according to 7.6, are isomorphic to the constant Hsets 2 and Alternatively 5 and with constant structure (5.1), can be shown to be the unique Hsets (up to isomorphism) with certain properties (e.9. 6 is a countable dense linear order without endpoints).
a.
0,
We are now ready to formulate and prove the "converse" of Theorem 5.6, giving a characterisation of real numbers in topological models. 8.4 Definition. A Dedekind cut (in the rationals) is a pair (L,U) of subsets of Q which are inhabited, disjoint, closed downwards (resp. upwards), open (in Q), and close together: that is, (i) (ii) (iii) (iv) (V)
3pEL A 3pEU L n u = @ (pqEU + pEU) (PEL + 3qEL. q>p) A (PEU + 3qEU. q
The conjunction of (i)(v) we abbreviate as Cut (L,U). The type R of Dedekind reals (defined in IHLN) is the set of such cuts, with order, for example, defined by
R.J. GRAYSON
198
(L,u) < (L',u') iff 3pEu
n
L'.
R is understood to carry its order topology, with basis the rational
open intervals, the rationals being embedded in R by p +b (Iqlqpl). The main point about this notion of "real number" is that its interpretation in O(T)sets is RT, as we now show; we consider reals given by sequences of rationals in 10.3. In the nontopological case the representation of the reals is not so concrete [FH,521. 8.5 Theorem. The standard interpretation of the type of Dedekind reals in O(T)sets is isomorphic to the O(T)set RT, with structure as in 5.2 and 5.5. (For a constructive treatment we regard the external reals as defined in the same way, as Dedekind cuts.) 2 Proof. A s a term, R is interpreted as the predicate C u t on P(Q) ; so, as a type, it is interpreted as the O(T)set of pairs (L,U) of predicates on with [[ E(L,U) 11 = Cut ( L , U )11 and "extensional" equality (relativised to "Cut"):
6,
(L,u)
=
(L',u')II
=
[[ cutc~,u)ll
A
[[ L=L'IIA
uU=u'i,
the latter being evaluated in P(Q). Now, by Theorem 5.6, every element a of RT determines predicates L a' Ua on Q , for which [[ Cut(La,U )
and
11
Eall, according to
=
[t PEL,]]
=
[[ p
= {tlp
PEU,II
=
[[ a
= {tJa(t)
[[
Conversely, for any predicates L . U .
and
Lt = Ut =
if we set
IpltE"pELl11 IPltE" PEUIII,
we find that, for t€[[ Cut(L,U)l], Cut (Lt,Ut) holds; for example, if t€[[ 3pEL]], then, for some p,tE[[pEL]I, so 3pELt, and so on. Then, if we set a(t) = (Lt,Ut) for t€[[ Cut(L,U)]], we obtain an element a of RT with [[ Ea]] = [[ Cut (L,U)11. that, for all p,q,
To check continuity of a, observe
{tlp
n [tpE~lI n
[[ qEul1
which is open as L,U are predicates. We leave it to the reader to check that the two functions between O(T)sets defined above constitute an isomorphism in the sense (at the end) of 8.1, which preserves all firstorder structure (in particular, order). We now give a similar representation for Baire space in O(T)sets. 8.6
Theorem. Baire space NN is interpreted in O(T)sets by
N (N ) T'
Proof. NN is defined as the exponent type ( 7 . 7 ) , which is interpreted by the O(T)set of all predicates R on ,'?i with [[ E(R) 11 = = [[ R a total functional relation]] and "extensional" equality. Now, as for the reals in 8.5, since NN is topologised by the subbasic opens V = {xlx(m)=n? each element a of (NN)T'determines a n,m
199
HeytingValuedSemantics
predicate R a , for which [[ R to [[ Ra(m,n)II
total functional]] =
= [[ Ea]]
,
according
{t(a(t)(m)=nI.
Conversely each predicate R determines a in (NN)T with domain [[ R total functional]] according to a(t) (m)=n iff tE" R(m,n) 11
.
8 . 7 Remarks. a) Theorems 8.5 and 8 . 6 are special cases of a general result for the spaces of models of arithmetically defined infinitary geometric propositional theories; this general theory is described in [FGI. b) In proving 8 . 6 we had to dealwith functional relations rather is far from complete than actual functions from 6 to N. since (6.23). Indeed functions from fi to 6 are simply standard, external functions from N to N , which ive rise just to the constant elements of (NN),. But, in case T is Nw itself, there are many nonconstant elements, for example, the identity function; it is this richness which is exploited in [Mo,vD] (who however consider only the total elements  see 5.7).
a
5
9.
INTERNAL TOPOLOGIES
In this section we give each O(T)set X a natural internal topology, as an example of how higherorder srructures are interpreted in our semantics. In the cases of the reals and Baire space we obtain the usual internally defined topologies. This approach is . generalised in [FS,§81 and exploited in [G2] to give results in general topology. 9.1 Notation. Let X and T be topological spaces. We use the variables s,t,... to range over T ; x , ~ , . . . over X ; a r b , over X (3.2); over O(T) ; V , V ' , over O ( X ) : W,W',.. over O ( T x X ) T the U,U',... product topology on TxX.
...
.
...
9.2 Definition. For each open set W in the product topology on TxX we define a predicate W on X T by " ~ E ~ I =I {tj (t,a(t))EwI, this set being open as a is continuous. The O(T)set O(X ) conT sists of all such predicates, with "extensional" equality defined as for P(X1 (7.1). In particular, each element is global.
9.3 Proposition. O(X
T
)
is internally a topology on X
Proof. For any W,W' and a in X [[
defining
zflc
I1
=
[[a Ex]]
T'
T
n
[[ a EL'
I1
=
aEznw' 11,
as an abstraction term in the standard way. Hence [[Wnw' = = T , while W' is again an element of O(X T 1 ; thus O(X ) is internally closed under intersections. Also the whole spzce XT is clearly represented by the predicate (E).
a']]
200
R.J. GRAYSON
To verify closure under unions we consider an arbitrary predicate P on O(XT), and define the open set
Wo = U{WplWEO(TxX)
wP
where
=
wn
( r r WEPII XX) .
Then iff iff iff iff Thus
"W0=
UP]] = T , as required. N
special cases we obtain i n this way topologies on R T and ( N )T' which turn out to coincide with the usual topologies defined internally, when we construe them as the internal reals and Baire space as in 8 . 5  6 . 9.4
As
This follows fairly immediately from the observation that, if B is a basis for O(X), the sets {(UxV) IVEB, U E O ( T ) 1 form a basis for O(TxX). Then, writing VT for the predicate (E). the elements V , for V in B, form an internal basis for O(XT), since, if (UxV)sWT u g VTGw]3. Now for the reals the basic opens are the rational intervals (p,q), , as basic opens of O ( R T ) ; which get interpreted as ( ~ , q ) ~hence similarly for Baire space. 9.5 Metrics. To show how topological structure on X carries over to X let X be a metric space with metric d, and define
d'
:T'2 XT + R T
by d'(a,bl
= At. d(a(t1 ,b(t)).
Thus d' reads off the distance between a and b pointwise, producing a real number in O(T)sets, by 8.5. It is an easy exercise to check that d' is internally a metric function on X , and that the corresponding. metric topology coincides with that xefined in 9.2. 9.6 Compactness. A topological space is compact iff every open cover has a finite subcover, the constructive sense of "finite" used here being that of "enumerable by the natural numbers less than some natural number".
Proposition. I f Proof
x
is compact, so is XT'
Let P be any predicate on O(X ) and suppose that UP]]; then, by the proof 0 f ~ 9 . 3 , {t}xX 5 U{W,lWEO(TXX)}.
5
Thus for xEX, by the definition of the product topology, we can find U,V,W with tEU, xEV and (UxV)5Wp, that is, UxVcW and Uc_[[WEP]]
.
Now, if X is compact, we can find finitely many such opens U.,V 1 iewi for i=l,. n , with X E iynVi. Then with U = iTnUi we find
..,
that (UxX) tEU
5
at t.
[[ XT
5
iynWi and U
5
n [[ WiEP]l .
i bn iynWi icnwiEPn, thus A
Hence
giving a finite subcover of P
9.7 Remarks. The above proof is constructive. Furthermore, since, by an extension of Theorem 8 . 5 , the closed unit interval [0,11 is ' interpreted in O(T)sets as [0,1lT, compactness of the unit interval
201
HeytingValuedSemantics
is valid in all topological models (when assumed externally); similarly for Cantor space 2N, which is interpreted as (2N)T. These principles are acceptable to intuitionists of the Brouwer school, but regarded as uncertain by the Bishop school of constructivism. Our models show at least that they are constructively "consistent", for example, with the completeness of intuitionistic predicate logic. That they are also inde endent of IHLN is shown by the nontopological countermodels
9.8 Continuous Functions. We take as the constructive definition of a continuous function F between spaces X and Y that, for VEO(Y), F~(v)Eo(x). We now want to give an external representation of internal continuous functions between spaces X and Y T'
Since YT fs complete (6.1) we can treat arbitrary internal functional relations to YT as external functions ( 6 . 3 ) . So let F : X + YT be a function of O(T)sets which is total and continuous T
"over U " , that is, U = F total continuous]]. We represent F by a continuous function f : UxXtY given by the equation
(*I
f(t,a(t))
=
~ ( a(t), ) for t E U n d o m a.
First we must show that f is welldefined by ( * ) , that is, if a(t)=b(t), then F(a) (t)=F(b)(t); for this we need to assume that Y space. is a T Let VEg(Y); then V (as in 9.4) belongs to O(Y ) , so U C Fl(VT) open]], hence we cxn find WEO(TxX) with U 5 [[$=Fl(V )IT. Then we have the following chain of equivalences; for t E U I7 do; a , iff
(t,a(t))EW tE [[ a Ewll
iff
tE[[F(a)EvTl1
iff
F(a) (t)EV.
Thus, if a(t)=b(t), F(a) (t)EV iff F(b) (t)EV, for any VEO(Y); so the T property ensures ~ ( a(t)=F(b) ) (t). Furthermore, for the special case of a=R, the above equivalences give (t,x)EW iff f(t,x)EV, whence fl(V) = WEO(TxX); so f is continuous. Conversely, given a continuous f : UxXtY, the equation ( * ) clearly defines a function F : X + Y T with U = [[ F total]]. Furthermore, for 1
VEO(Y), since W = f (V] is open, we obtain equivalences as above showing that U C_ [[ W=F (V )]I. But by 9 . 4 the elements VT form a basis for O(YT), so F is ayso continuous over U. This representation allow us to draw "pictures" of arbitrary internal continuous fbnctions as continuous Yvalued surfaces over the TxXplane. In this way one can draw simple counterexamples to classical theorems such as the Intermediate Value Thebrem, or the attainment of bounds on a closed interval. 9.9 Brouwer's Theorem. The principal result of [Sl] is that, for T the Baire space, the O(T)set R T satisfies the socalled "Brouwer's Theorem":
All functions from reals to reals are continuous. A
similar result for (NN), is in [Mo,vDI and a result for more
R.J. GRAYSON
202
general spaces T is proved in [G2, 5 his theorem [Br] from stronger forms we discuss in S 11; he obtained also closed intervals, which holds too in such intervals are compact.
8.21. Brouwer himself deduced of continuity r.L;nciple, which uniform continuity on all our models since by 9.6 all
The scheme of Scott's proof is as follows: Given F : R rR with U = [[ F total]] , define f : U x R + R by the equation ( * ) as in 9.8, now using the special properties of T to show that f is welldefined and continuous. This then implies continuity of F over U , as in 9.8.
5
10
CHOICE PRINCIPLES
A good deal of the "mathematics of Hsets" is now available in the literature [BM, vD, F H , F S , G12, MO, Mu, R , Sl], so having outlined the framework of the theory we concentrate on two types of principle of importance in intuitionism, principles of choice and continuity. We will find that validity in all topological models (in particular, in the models over Euclidean spaces) corresponds to a certain kind of continuity property, which conflicts with even the weakest of countable choice principles, denoted by ACNN: ACNN: VmEN 3nEN. R(m,n)
+
3f
:
N  + N VmEN. R(m,f(m)).
Thus the theory generally interpreted in these models turns out to be rather different from the traditional intuitionistic one, or that presented in Bishop's book [Bi]. We start however with a " po si tive " re s u 1t
.
10.1 Proposition. In O(NN)sets the principle of (relativised) Dependent Choices holds: that is, for any O(NN)set A, DC(A):
VxEA 3yEA. R(x,y) + VxEA 3f:N+A[f(O)=x
A
VmEN. R(f(m),f(m+l))].
Proof. See tMo,S3] for example. The proof uses the property of Baire space that every open cover has a disjoint refinement. It is of interest that this proof can be made constructive assuming Dependent Choices and Bar Induction externally. By contrast with the preceding result for O(NN)sets, where internal Baire space 1"( is large and rich, we find that the local connecNN N tedness of the reals make (N trivial. R N 10.2 Proposition. Every element of (N )R i s locally constant, so ("1,
is just the completion of the constant set ("1
(see 6.2 (b))
.
N
Proof. If a is a continuous function from R to N , the image under a of any rational interval (p,q) contained in dom(a) must be connected in N1, hence a singleton; that is, a must be constant on (Pt9). 10.3 Cauchy Reals. Now the reason for real numbers as cuts in the rationals lence classes of) Cauchy sequences of intuitionism and constructivism (e.g.
our choice of definition of (8.4) rather than as (equivarationals, as is more usual in [Bi]), becomes apparent:
Let us define a (Dedekind) real x to be Cauchy iff it can be approximated by a sequence of rationals, that is
203
HeytingValued Semantics 3f
:
N + Q VnEN.lxf(n) I < l/n.
Then in O(R)sets, since all sequences of natural numbers, hence also of rationals, are (locally) constant, every Cauchy real has to be (locally) constant (as an element of R ) , whereas R has a multitude of nonconstant elements, for examplg, the identiay function Xt.t. In particular, since every Dedekind real x clearly does satisfy VnEN 3qEQ.Ixql
u c_
R f converges to all,
and fE(QN),
satisfies
then, for each tEU, f(t) converges to a(t), which equals t. Thus we must have found an approximating sequence f(t) continuously in the parameter t , and this is what 10.2 shows to be impossible. Thus failure of ACNN over R is seen to fact that (even classically) one cannot quence to each real number continuously The relevance of local continuity comes example.
correspond to the simple choose an approximating sein the real (even locally). out more clearly in the next
10.5 Roots of Cubics. A wellknown fact from elementary analysis is that one cannot find a total continuous function of t giving a root of the cubic x3x+t, because one has to make a "jump" somewhere. On the other hand, one can easily choose a root locally continuously, in the sense that, for each t , there is a neighbourhood of t on which one can choose a root continuously.
What is perhaps less wellknown is that one cannot choose a root of the cubic x3+sx+t even locally continuously in both s and t. A s the picture below of the surface x3+sx+t=0 over the (s,t)plane indicates, there is no continuous choice of root on any neighbourhood of the origin.
27
To interpret this in our models: the parameter space is now R 2 and we have two "generic" elements of R 2 , a=X(s,t) . s and b=X(s,t) .t. The failure of continuity in parameaers then shows that 3 ( 0 , O ) B [[ 3x. x +ax+b=Oll,
R.J. GRAYSON
204
2 so that, in O ( R )sets, the reals are not realclosed (in the simplest sense); on the other hand, the principle ACNN suffices to prove realclosure. 10.6 A Derived Rule. We can also apply the above considerations in a more positive direction to any assertion of the form 3y. R(x,y), where R is polynomial equation. If this is valid in all topological models, it must be valid in the model over the parameter space of the parameters s , and hence a solution y must exist locally continuously in the parameters. Furthermore, all this is provable in the system IHLN, which leads to a derived rule of local continuous choice for this system ; we prove a general result of this form for arbitrary R in 11.5, but it seems helpful to have these simple examples in the background for motivation. On the other hand our cubic example shows that ACNN prevents such a derived rule even for the special case of polynomial equations.
Vz
9
11
CONTINUITY PRINCIPLES
Among the most positive and "anticlassical" tenets of intuitionism are continuity principles of various degrees of strength. In this final section we indicate to what extent these can be interpreted in our models, and prove a derived rule of local continuous choice for our basic system IHLN using only the soundness theorem for topological models ( a proof due to Joyal). 11.1 Weak Continuity. We have already seen that the continuity of all functions from reals to reals, or from Baire space to Baire space, holds in certain topological models ( 9 . 9 ) . In intuitionistic treatments these are sometimes derived from the principle of weak continuity for N ~ WC, , namely wc: 
VUENN 3nEN. A(a,n) t VUENN 3 m , n E N vgENN[6(m)=p(m) tA(a,n)1
In topological terms this says that every countable cover of Baire space has an open refinement. In case the formula A is a formula of analysis without parameters other than a , WC can be shown (classically) to be valid in O(NN)sets [vD]. In its full generality however WC can never hold in topological models [ G Z , s 8.11. It is an open question whether WC might hold over some nontopological cHa, but Krol' [K] has given a permutation submodel of the "full" model over O(NN) in which WC is valid (see also [ G Z , Appendix]). 11.2 Continuous Choice. A much stronger group of principles are those of continuous choice, which we may formulate generally as CC(X,Y) for (definable) spaces x and Y. VxEX 3yEY. A(x,y) CC(X,Y) : 3f
:
X
*
N
+
Y VxEX. A(x,f ( X I .
The special case CC(N ,NN) is also known as Va 3Bcontinuity. This is inconsistent with Kripke's Schema and hence fails in the model . of Krol'. Fourman [F2] has shown how to model this principle in sheaves over a site; as worked out in [HM] one obtains in this way in fact a model for the f u l l theory CS of choice sequences. The
205
HeytingValued Semantics
consistency of such strong principles had of course already been shown by other means, for example, realisability [Tr]. On the other hand, realisability does not appear so useful in dealing with continuity principles for the real numbers, which we now consider. 11.3 Local Continuous Choice. The principle CC(R,R) is simply inconsistent as is shown by the first example f 10.5, since one may easily prove (without ACNN) that Vt 3x.xqx+t=0. The more traditional counterexample is given by the provability of VxER jnEN.x
+
%Y
Vx'EU.A(x',f(x')).
Fourman's models in sheaves over sites IF21 show the consistency (relative to IHLN) of LCC(X,Y) for any complete separable metric spaces X,Y (definable in IHLN). On the other hand, the cubic example of 10.5 shows LCC(Rp) to be inconsistent with ACNN. Discussion of the relations between various continuity principles (in the presence of countable choice) may be found in [Be]. 11.4 Derived Rules. T o each continuous choice principle CC(X,Y) or LCC(x,Y) there corresponds a continuous choice rule, that provability of the hypothesis implies provability of the conclusion. We denote these rules by CCR(X,Y) and LCCR(X,Y). Derived rules of this kind are proved in [Be], using realisability, and [HI] by prooftheoretic means, for various systems. We give here an exceedingly elegant proof of LCCR(X,Y) for IHLN, for any definable complete separable metric spaces X,Y, which is due essentially to Joyal and uses just the provable soundness (7.8) of topological models for IHLN; other applications of this technique are t o appear in [FJ]. The addition of ACNN prevents LCCR(R,R), of course, as noted in 10.6. 11.5 Theorem. (Joyal, Hayashi) The system IHLN is closed under the rule LCCR(X,Y) for any definable (provably) complete separable metric spaces X,Y. Proof. This will be an informal proof within IHLN starting from the ( c 1 0 s ed) as sumption
1 VxEX 3yEY.A(x,y). By the soundness theorem, provable in IHLN ( 7 . 8 ) , this is provably valid in all topological models. We now define the particular space T , over which we want to use this validity: T = Xx{O,l]
and
wo w1
with open sets those WST for which
= {xExl<x,o> =
E wl
{XEXl<X,l> E
E O(X)
w} 2 wo
This has the effect that X is homeomorphic to the closed subspace T =Xx{O} of T , while X disc (the set X with the discrete topology) is homeomorphic to the open subspace T =Xx{l). as a "glueing" of x disc to X along the'identiy
T can be regarded map.
The assumptions on X and Y ensure that, just as for R and NN in 8.5 and 8.6, when we interpret the definitions of X and Y in O(T)sets,
R.J. GRAYSON
206
we obtain the O(T)sets XT and Y with topologies as in 9.2. (The general theory appears in [FG, 3T81.1 Now we apply the internal validity of VxEX 3yEY.A(x,y), in O(T)sets, to the projection a=(X<x,i>.x), which belongs to XT and so is treated internally as an "element of X". Then we get [[ 3bEYT.A(a,b)]1 = T, hence, for any point x of X, we may find b in YT with <x,O> E [[ A(a,b)]]
fI dom(b) = W ,
where we may suppose without I D S S that W =W,=U€O(X). W e next want to transfer into the universe over X disc, i.e. P(X)sets,
a) Since
into
for which we need three observations. Xdisc is (homeomorphic to) the open subspace T 1 of T,
evaluations [[
lldisc
in P(X)sets
are obtained simply by
"restriction" to T 1 of evaluations over O(T)
.
b) Since Y is a T1space, for any x'EU, we find that b(<x',O>) = b(<x',l>), so that the restrictizn of b to T gives rise to a continuous function 2 from U to Y with b(x') = b(<xl , l > ) . Similarly the restriction of a gives rise simply to the identity function id on X. Hence (a) yields 'disc
C 
A
[[ A(id,b)lldisc

c) Finally, interpretations of formulae over discrete spaces are always obtained "pointwise" in terms of "external truth", in particular [[ A(id,b)]Idisc
= {x'EXIA(id(x') ,%(xu))).
Together with (b) this shows thzt we have found a neighbourhood U of x and a continuous function b:U+Y such that Vx'EU. A(x',%(x')), giving the conclusion of LCCR(X,Y).
REFERENCES. [Be] M. Beeson: Principles of continuous choice, Annals of Math. Logic 1 2 (19771, 249322 [Bi] E. Bishop: Foundations of constructive analysis, McGrawHill, 1967. [ B r ] Brouwer's Cambridge Lectures on intuitionism, ed. D. van Dalen, Cambridge University Press, 1981.
[BM]
Burden and C . Mulvey: Banach spaces in categories of sheaves, in Applications of Sheaves, Springer Lecture Notes 753 (1979), 169 196.
C.
[vD] D. van Dalen: An interpretation of intuitionistic analysis, Annals of Math. Logic 1 3 (19781, 143.
207
HeytingValued Semantics
[D]
M. Dummett: Elements of intuitionism, Oxford University Press, 1977.
[Fi] M.P. Fourman: The logic of topoi, & Handbook of Mathematical Logic (ed. J. Barwise), NorthHolland, 1977, 10531090 [F2]
:
Continuous truth, to appear (1982).
e Proceedings
[FG] M.P. Fourman and R.J. Grayson: Formal spaces, of the Brouwer Symposium, NorthHolland, 1982.
[FH] M.P. Fourman and J.M.E. Hyland: Sheaf models for analysis, & Applications of Sheaves, Springer Lecture Notes 753 (1979), 280301. [FJ] M.P. Fourman and A. Joyal: Metamathematical applications of sheaf theory, to appear. [FS] M.P. Fourman and D.S. Scott: The logic of sheaves, Applications of Sheaves, Springer Lecture Notes 753 (1979). 302401. [Gl] R.J. Grayson: Heytingvalued models for intuitionistic settheory, 9Applications of Sheaves, Springer Lecture Notes 753 (1979), 402414. [G2]
            :C oncepts of general topology in constructive mathematics and in sheaves, Annals of Math. Logic 20 (19811, 141. Ditto, 1 1 , to appear in the Annals of Math. Logic.
[G3]
:
[G4]
. Constructive properties of complete Heyting algebras and related structures, preprint (1982).
[Hl]
S . Hayashi: Derived rules related to a constructive theory of metric spaces, Annals of Math. Logic 19 (19801, 3365.
[H2]
          :A
F orcing in intuitionistic systems without powerset, to appear in Journal of Symbolic Logic (1981).
note on the bar induction rule, & Proceedings of the Brouwer Symposium, NorthHolland, 1982.
[He] A. Heyting: Intuitionism, An Introduction, NorthHolland, 1956. rHm3 G. van der Hoeven and I. Moerdijk: Sheaf models for choice sequences, preprint (1982). [Hy] J.M.E. Hyland: Aspects of constructivity in mathematics, Oxford Logic Colloquium '76 (eds. Gandy and Hyland), NorthHolland, 1977. [K]
M.D. Krol': A topological model for intuitionistic analysis with Kripke's Schema, ZMLG 24 (19781, 427436.
[MR] M. Makkai and G. Reyes: Firstorder categorical logic, Springer Lecture Notes 611, 1977. [MDI R. Mansfield and J. Dawson: Booleanvalued set theory and forcing, Synthese 3 3 (1976), 223252.
208
R.J. GRAYSON J.R. Moschovakis: A topological interpretation of secondorder intuitionistic arithmetic, Comp. Math. 26 ( 1 9 7 3 ) , 2 6 1  2 7 5 . C.J. Mulvey: Intuitionistic algebra and representations of rings, in Mem. Amer. Math. S O C . 1 4 8 ( 1 9 7 4 1 , 3  5 7 . H. Rasiowa and R. Sikorski: The mathematics of metamathematics, Warsaw, 1 9 6 3 . Rousseau: Topos theory and complex analysis, in Applications of Sheaves, Springer Lecture Notes 7 5 3 ( 1 9 7 9 ) , 6 2 3  6 5 9 .
C.
D.S. Scott: Extending the topological interpretation to intuitionistic analysis  I , Comp. Math. 2 0 ( 1 9 6 8 1 , 1 9 4  2 1 0 .  11, & Intuitionism and Proof Theory (eds. Kino, Myhill, Vesley), NorthHolland ( 1 9 7 0 ) .
_______.
. Identity and existence in intuitionistic logic, Applications of Sheaves, Springer Lecture Notes 7 5 3 ( 1 9 7 9 ) , 660696.
A. Tarski: Der Aussayenkalkul und die Topologie, Fund. Math. 3 1 ( 1 9 3 8 ) , 1 0 3  1 3 4 . A.S. Troelstra: Metamathematical investigation of intuitionistic arithmetic and analysis, Springer Lecture Notes 3 4 4 ( 1 9 7 3 ) .
Address for correspondence: Church Cottage, Benenden, Cranbrook, Kent, England.
LOGIC COLLOQUIUM '82 G . Lolli, C.Long0 and A. Marcia [editors) 0 Elsevier Science Publishers 8.V. (NorthHolland), 1984
209
LAMBDA C A L C U L U S ANU I T S MODELS
Henk Eatendhegt M a t h e m a t i c a l I n s t i t u t e , Budapestlaan 6 3508 TA U t r e c h t , The N e t h e r l a n d s .
INTRODUCTION The Lambda c a l c u l u s was i n t r o d u c e d by Church around 1930 as a f o r m a l t h e o r y about r u l e s ( i . e .
f u n c t i o n s as g i v e n by a l g o r i t h m s ) .
The r e l a t e d t h e o r y o f c m b i 
n a t o r s was i n i t i a t e d by S c h S n f i n k e l and C u r r y s m e y e a r s e a r l i e r . The t h e o r y was c o n c e i v e d as t y p e f r e e : a l l o b j e c t s can be used b o t h as argument and as r u l e t o be a p p l i e d t o o t h e r o b j e c t s . Perhaps t h e subconscious w i s h was t o have a u n i v e r s e
U such t h a t a l l ( o r a t l e a s t many) f u n c t i o n s f r o m U t o U b e l o n g t o U. S i n c e by Cantors theorem t h e c a r d i n a l i t y o f Uu i s l a r g e r than t h a t o f U,
i t was n o t c l e a r
how t o c o n s t r u c t such a U. I n s p i t e o f t h i s , t h e r e were i n t e r e s t i n g r e s u l t s i n t h e s u b j e c t . Kleene showed t h a t t h e r e c u r s i v e f u n c t i o n s can be r e p r e s e n t e d i n t h e A  c a l c u l u s .
Rosser c l a r i 
f i e d t h e r e l a t i o n between t h e A  c a l c u l u s and t h e t h e o r y o f c o m b i n a t o r s . The cons i s t e n c y o f t h e A  c a l c u l u s was p r o v e d v i a t h e ChurchRosser theorem. As a consequence o f t h i s c o n s i s t e n c y t h e r e a r e t h e open o r c l o s e d t e r m models c o n s i s t i n g o f t h e open or c l o s e d terms modulo p r o v a b l e e q u a l i t y . n a t o r s , e.g.
I n t e r e s t i n g work on t h e c m b i 
by BZjhm and h i s s c h o o l , can be viewed as r e s u l t s on t h e t e r m models.
I n 1969 S c o t t c o n s t r u c t e d n o n  s y n t a c t i c a l models o f t h e A  c a l c u l u s . A l t h o u g h t h e f u l l f u n c t i o n space Uu c a n n o t be i s o m o r p h i c t o U, some s u b s e t can be, e.g. t h e s e t o f c o n t i n u o u s f u n c t i o n s w i t h r e s p e c t t o sane c o n v e n i e n t t o p o l o g y . Because o f S c h l i n f i n k e l s i d e n t i f i c a t i o n o f Uuxu w i t h (UU)'
i t i s n a t u r a l t o use a class o f
t o p o l o g i c a l spaces t h a t f o r m a C a r t e s i a n c l o s e d c a t e g o r y ( c c c ) . F o r t h i s reason S c o t t worked w i t h i n t h e c a t e g o r y o f with c o n s t r u c t e d an o b j e c t ,D
i s o m o r p h i c t o , ,D:
c o n t i n u o u s maps and thus y i e l d i n g an e x t e n s i o n a l model o f
the Acalculus. Some r e l a t e d C a r t e s i a n c l o s e d c a t e g o r i e s a r e a l s o o f importance. F i r s t t h e c o n t i n u o u s l a t t i c e s have a more n a t u r a l r e l a t i o n between t h e i r l a t t i c e s t r u c t u r e and t o p o l o g y ; e.g.
the topology o f a product i s the product o f the respective
t o p o l o g i e s , something t h a t i s f a l s e f o r c o m p l e t e l a t t i c e s . Then t h e r e a r e t h e ( c p o ' s ) o f w h i c h t h e r e a r e many more than t h e complete l a t t i c e s . P l o t k i n s model T W i s a cpo and n o t a c o m p l e t e l a t t i c e . Another u s e f u l c a t e g o r y i s t h a t o f fospaces as d e f i n e d by Ershov. These o b j e c t s have t h e advanP
H. BARENDREGT
210
t a g e o f n o t h a v i n g t o be complete,
e.g.
the set o f r.e.
s e t s p a r t i a l l y o r d e r e d by
i n c l u s i o n i s an f space.
0
I t t o o k some t i m e a f t e r S c o t t gave h i s model c o n s t r u c t i o n u n t i l t h e r e was an agreement what i s t h e g e n e r a l n o t i o n o f a model o f t h e A  c a l c u l u s .
See Koymans
[1983] f o r t h e h i s t o r y . P r e s e n t l y one c o n s i d e r s two k i n d s o f models, v i z . t h e A a l g e b r a s and t h e Amodels.
The A  a l g e b r a s s a t i s f y a l l p r o v a b l e e q u a t i o n s o f t h e
A  c a l c u l u s and form an e q u a t i o n a l c l a s s ( a x i o m a t i z e d by k x y = x , s x y z = x z ( y z ) and t h e f i v e c o m b i n a t o r y axioms o f C u r r y ) . T h e r e f o r e t h e A  a l g e b r a s a r e c l o s e d under s u b s t r u c t u r e s and homomorphic images. The Amodels on t h e o t h e r hand s a t i s f y a l l p r o v a b l e e q u a t i o n s and moreover t h e a x i o m o f weak e x t e n s i o n a l i t y Vx(M=N)
*
Ax.M=Ax.N.
I t t u r n s o u t t h a t Amodels can be d e s c r i b e d by some f i r s t o r d e r axioms, b u t n o t
by e q u a t i o n s .
Indeed Amodels a r e n o t c l o s e d u n d e r s u b s t r u c t u r e s n o r u n d e r homo
morph i c images. Next t o t h e f i r s t o r d e r d e f i n i t i o n o f A  a l g e b r a s and Amodels,
there i s a
s y n t a c t i c a l and a l s o a c a t a g o r i c a l d e s c r i p t i o n o f t h e s e c l a s s e s . The s y n t a c t i c a l d e s c r i p t i o n i s c o n v e n i e n t when c a l c u l a t i n g t h e i n t e r p r e t a t i o n o f terms i n a model The c a t e g o r i c a l d e s c r i p t i o n o f A  a l g e b r a s
i s r a t h e r n a t u r a l and u n i f i e s t h e two
I t c o n s i s t s o f a C a r t e s i a n c l o s e d c a t e g o r y t t o g e t h e r w i t h a so c a l l e d U r e f l e x i v e o b j e c t U E Q , i . e . U i s a r e t r a c t o f U: t h e r e a r e maps F:U+Uu and
concepts.
G:UU+U
such t h a t FOG = idUU. A s shown i n Koymans [19831,
i n t h i s context a
A
model i s a A  a l g e b r a t h a t a r i s e s f r o m a c a t e g o r y t w i t h an o b j e c t U t h a t has "enough po i n t s"
.
Because of t h e p r e s e n t d e s c r i p t i o n o f lambda c a l c u l u s models, c h a p t e r
5 of
B a r e n d r e g t 119811 becomes somewhat o u t o f d a t e . T h i s p a p e r may be c o n s i d e r e d as a replacement o f t h a t c h a p t e r . U s i n g t h e c a t e g o r i c a l d e s c r i p t i o n o f t h e A  c a l c u l u s models, S c o t t [ 1 9 8 0 ] makes t h e f o l l o w i n g p h i l o s o p h i c a l remarks. 1 . The models f o r t h e t y p e f r e e A  c a l c u l u s come f r o m c c c ' s w i t h a r e f l e x i v e o b j e c t . The
CCC'S

themselves c o r r e s p o n d t o t h e t y p e d A  c a l c u l u s .
There
f o r e t h e t y p e d A  c a l c u l u s has p r i o r i t y o v e r t h e t y p e f r e e t h e o r y .
2. Let
E
be a ccc w i t h r e f l e x i v e o b j e c t U. By t h e Yoneda lemna d: can be em
bedded i n t o a t o p o s D = Setcop, U s i n g t h e K r i p k e  J o y a l semantics, i n s i d e PJ i t i s s a t i s f i e d t h a t Uu i s t h e f u l l f u n c t i o n space o f U and t h e r e f o r e
t h e axiom o f weak e x t e n s i o n a l i t y i s s a t i s f i e d by U i n
ID. The p r i c e one
has t o pay i s t o use i n t u i t i o n i s t i c l o g i c , s i n c e c l a s s i c a l l o g i c i s n o t sound f o r t h e K r i p k e  J o y a l
interpretation.
Some comnents. A s t o 1 , t h e r e a r e c e r t a i n l y n i c e r e s u l t s i n t h e t y p e d A  c a l c u l u s , f o r i n s t a n c e Statman [19801,
[19821. However we d i s a g r e e w i t h S c o t t s grounds f o r
21 1
Lambda Calculus and its Models
c o n c l u d i n g t h a t t h e typed t h e o r y has p r i o r i t y o v e r t h e t y p e f r e e one. Even i f t h e r e a r e f o r example more semigroups t h a n groups,
i t does n o t f o l l o w t h a t t h e
t h e o r y o f semigroups i s more fundamental t h a n t h e t h e o r y o f groups. As t o 2, S c o t t s s u g g e s t i o n t o make t r u e t h e o l d dream o f Church and C u r r y , namely U u E U , i n s i d e a topos,
i s indeed v e r y i n t e r e s t i n g . One has t o w a i t and see what a p p l i 
c a t i o n s t h i s can g i v e . T h a t A  a l g e b r a s a r e i n t e r n a l l y a l r e a d y Amodels does n o t mean t h a t t h e c l a s s o f A  a l g e b r a s
i s d e v o i d o f i n t e r e s t . Compare t h i s w i t h t h e
n o t i o n o f a r e g u l a r r i n g . Viewed i n s i d e a t o p o s , r e g u l a r r i n g s a r e ( i n t u i t i o n i s t i c ) f i e l d s . B u t r e g u l a r r i n g s m e r i t a t t e n t i o n b y themselves and n o t j u s t as g l o b a l s e c t i o n s o f a f i e l d i n a topos. The same a p p l i e s t o A  a l g e b r a s . models a r e A  a l g e b r a s ,
b u t n o t Amodels
Closed term
i n g e n e r a l , due t o wincompleteness.
N e v e r t h e l e s s t h e s e s t r u c t u r e s have i n t e r e s t i n g p r o p e r t i e s , e.g.
t h e y a r e precom
p l e t e numbered s e t s i n t h e sense o f Ershov, see V i s s e r [19801. N o t a t i o n s and r e f e r e n c e s t h a t a r e n o t g i v e n i n t h i s paper may be found i n B a r e n d r e g t [19811. I n p a r t i c u l a r we u s e t h e v a r i a b l e c o n v e n t i o n t h a t i d e n t i f i e s terms d i f f e r i n g o n l y i n t h e names f o r t h e i r bound v a r i a b l e s (e.g.
Xx.x=hy.y)
and r e q u i r e s t h a t a bound v a r i a b l e i n some m a t h e m a t i c a l c o n t e x t i s d i f f e r e n t f r o m the free variables i n t h a t context.
Sopfie
a b s t r a c t i o n i n t h e rneta language.
I thank K a t L b t KoljmuMn f o r many u s e f u l d i s c u s s i o n s on t h e sub
Acknowledgements. j e c t and
1 denotes
WUM
Stakenbu%
51.
COMBINATORY ALGEBRAS
1.1
DEFINITION.
(i)
I=(X,.)
f o r her carefu1,nice typing o f the manuscript.
i s an a p p l i c a t i v e s t r u c t u r e i f
.
i s a b i n a r y oper
a t i o n on X. ( i i ) Such a s t r u c t u r e i s e x t e n s i o n a l i f f o r a , b E X one has (VxEX
Notation.
+
( i ) As i n a l g e b r a , a.b
( i i ) If
a=b.
i s u s u a l l y w r i t t e n as ab.
+
If b=bl,
..., bn.
then
.. ( a b l ) b 2 . . . b n ) .
a b = a bl...bn=(
1.2
*
a.x=b.x)
I=
(X,.)
DEFINITION. L e t
t h e n we w r i t e a E W i n s t e a d o f a E X .
I be
an a p p l i c a t i v e s t r u c t u r e .
( i ) The s e t o f terms o v e r m , n o t a t i o n S b ) , i s i n d u c t i v e l y d e f i n e d as f o l l o w s . vo’vl’v2’‘’’ aE l A,BES@) N o t a t i o n . A,B,
. . . denote
E
SC)
*
ca
+
(AB)ESC)
€%(I)
(va r i ab 1 es ) (cons t a n t s )
a r b i t r a r y terms and x,y,
...
arbitrary variables i n
Sh).
H. BARENDREGT
212
ID?
( i i ) A valuation i n
i s a map p : v a r i a b l e s + 1.F o r a v a l u a t i o n p i n m t h e
i n t e r p r e t a t i o n o f A € % @ ) i n m under p ( n o t a t i o n (Ap'or
(A)
P
P
o r (A)m i f m o r p
i s c l e a r from t h e c o n t e x t ) i s i n d u c t i v e l y d e f i n e d as u s u a l :
( i i i ) A=B i s true i n U ! t under t h e v a l u a t i o n p ( n o t a t i o n m , p C A = B ) i f
m
n
( A l p = (B)p. i s t r u e i n J!R ( n o t a t i o n 9l
(iv) A = B

( v ) The r e l a t i o n
b A = B ) i f 8 ? , p I= A = B f o r a l l v a l u a t i o n s p.
i s a l s o used f o r f i r s t o r d e r f o r m u l a s over!U?. The d e f i 
n i t i o n i s as u s u a l . FV(A) i s t h e s e t o f ( f r e e ) v a r i a b l e s i n A. v a l u e s of p on FV(A). a t i o n (A)
1.3
P
C l e a r l y (A)
I n p a r t i c u l a r f o r c l o s e d A ( i . e . FV(A)
P
depends o n l y on t h e
=0)
the interpret
i s independent o f p and may be denoted by ( A ) .
DEFINITION ( C u r r y ) . An a p p l i c a t i v e s t r u c t u r e 1 i s a c o m b i n a t o r y complete i f
f o r e v e r y A € % @ ) and xl...xn 3 f Vx,
...xn
w i t h FV(A)5{xl
f x l...x
,...,x
one has i n 1
= A
Note t h a t an e x t e n s i o n a l a p p l i c a t i v e s t r u c t u r e i s c o m b i n a t o r y complete i f f f o r a l l A € % @ ) one has
fz = A(;).
3 1 f Vz
1.4
NOTATION.
( i ) L e t p be a v a l u a t i o n i n m and l e t a E 1 . Then p ( x : = a )
i s the
valuation p' with
P'(x) = a
,
~ ' ( y )= P ( Y ) (ii)
if ygx.
+
f
If x = x
a r e d i s t i n c t and a = a , , lI.'Xn p ( x : = a ) = p(xl :=al) (X : = a 1. n n
( i i i ) A[x:=B]
1.5
...,an,
then
...
f
i s t h e r e s u l t o f s u b s t i t u t i n g t h e t e r m B f o r x i n A.
LEMMA. L e t m b e an a p p l i c a t i v e s t r u c t u r e and A , A ' , B , B ' € S @ ? ) . ( i ) (A[x : = B I ) ,
=
:= (B)p)
(ii)wCA=A'hB=B' Proof.
(i)
Then
* 1 k
A[x:=BI
=A'[x:=B'].
I n d u c t i o n on t h e s t r u c t u r e o f A.
( i i ) By a s s u m p t i o n (A) (A[x : = B l )
P
=
P
=(A')
P
and (8)
,
:= (B)p)
= (A')p(x:=
(Bl)
P
= (8')
P
for a 1 p.
by ( i ) , = (A'[x:=B']) P
P
I t follows t h a t
213
Lambda Calculus and its Models
0
and we a r e done.
1.6
DEFINITION. L e t m = (X,.)
be an a p p l i c a t i v e s t r u c t u r e and l e t cp: X " + X
be
a map. ( i ) cp i s r e p r e s e n t a b l e o v e r !IN i f 3 f E X
V z € X"
+
+
f a = cp(a).
( i i ) cp i s a l g e b r a i c o v e r m i f t h e r e i s a t e r m A€X(!IR) w i t h
. . .x,l such 6 cp(2) = (A)p(;:=;).
FV(A) z { x l , . (1)
that
( C l e a r l y ( 1 ) does n o t depend on p ) .
Combinatory completeness says that all algebraic functions are representable. The converse is trivial. Schonfinkel showed that combinatory completeness follows from two of its instances. 1.7
DEFINITION. A cornbinatory a l g e b r a i s an a p p l i c a t i v e s t r u c t u r e 9 J = (X,.,k,s)
w i t h d i s t i n g u i s h e d elements s a t i s f y i n g k y = x sxyz = x z ( y 2 ) .
1.8
DEFINITION. L e t m be a c o m b i n a t o r y a l g e b r a . ( i ) Define the f o l l o w i n g constants: K = c k , S = c s , I ( i i ) For AE%(M) and a v a r i a b l e x , d e f i n e h * x . A E S m )
= c i with i
=
skk.
i n d u c t i v e l y as f o l l o w s :
,
X*X.X
= I
X*x.P
= KP,
i f P i s a v a r i a b l e g x o r a constant,
h*x.PQ = S(X*x.P) (h*x,Q).
+
( i i i ) L e t x = x,
1.9
, . . . ,x .
PROPOSITION. ( i ) FV(X*x.A) ( i ) (A*x.A)x=A, ( i i ) (X*;.A)z=A,
Proof. 
(i), (ii).
(ii )
ay
+
Then A*x.A
= ( h*xl
= FV(A)
. . . (X*xn.A).
. .) .
 {XI.
i n e v e r y cornbinatory a l g e b r a . i n e v e r y cornbinatory a l g e b r a .
I n d u c t i o n on t h e s t r u c t u r e o f A. Note t h a t l x = S K K x = K x ( K x ) = x .
(ii).
1.10 THEOREM. An applicative structure m i s combinatory completE iff it can be expanded to a combinatory algebra [by choosing k,s). Hence every combinatory algebra is combinatory complete.
Proof. By proposition 1.9(iii).
0
214
H. BARENDREGT REMARKS.
1.11
(i.e.
( i ) Note t h a t a c o m b i n a t o r y a l g e b r a % = (X,.,k,s)
iff k f s .
Card(W)>l)
i s non t r i v i a l
Indeed, k = s i m p l i e s a = s ( k i ) ( k a ) z = k ( k i ) ( k a ) z = i
f o r a l l a, S O W i s t r i v i a l ( i i ) When c o n s i d e r i n g c o m b i n a t o r y a l g e b r a s , we u s u a l l y t a c i t l y assume t h a t t h e y a r e non t r i v i a l .
1.12 D E F I N I T I O N .
Then c p : X
+X2
Let2Qi = ( X i , . i , k i , s i ) ,
i = 1 , 2 , be two combinatory a l g e b r a s .
i s a homomorphism ( n o t a t i o n cp:Wl+211)
2
1 and k and s , i .e. cp(x.,y)
i f cp p r e s e r v e s a p p l i c a t i o n
cp(y), cp(kl) = k 2 and cp(sl) = s 2 .
= cp(x).,
( i i) I +I2 l i f cp : w1 +I2 f o r some cp. i s ernbeddable i n m 2 (9111GW2) i f cp:YJ?,+?D2 f o r some i n j e c t i v e cp.
( i i i ) Wl (Wl (iv)
i s a s u b s t r u c t u r e o f 2112 (Ilc2R2)
W,
i f cp:Wl+W2
w i t h cp t h e i d e n t i t y . )
i s i s o m o r p h i c t o m 2 @J?l~9112)i f c p : 1 1 + W 2 f o r some b i j e c t i v e cp.
1.13 DEFINITION. ( i ) Q i s t h e s e t o f terms o f c o m b i n a t o r y l o g i c ,
i.e. applicative
terms b u i l t u p from v a r i a b l e s and K , S o n l y . Go = { P € Q l F V ( P ) = 0 } . ( i i ) L e t I be a c o m b i n a t o r y a l g e b r a . Then T h m ) = { P = Q l % C P = Q ,
1.14 PROPOSITION. L e t UJ:W,+SJI,. '
I
P,Q€&
0
1.
Then f o r P , Q € % @ ? * )
L
I 1 ( i ) cp(l[PJJ ) = [cp(P)? , w h e r e cp(P) r e s u l t s i r o m P by r e p l a c i n g t h e conP WP stants c a by 'cp(a). 0 ( i i )W, P = Q * W + s ( P ) = q ( Q ) , provided P , Q E & o r cp i s s u r j e c t i v e .
+
( i i i ) Th(9111)
2
Thb2).
( i v ) Th(9111) = Th@J$), Proof. 
p r o v i d e d t h a t cp i s i n j e c t i v e .
( i ) I n d u c t i o n on t h e s t r u c t u r e o f P€X(9ll).
k P=Q
( i i ) Wl
* *

[ P I p = UQn,
UPII,,
=
* W2 C I f P,Q€Q
0
f o r a l l p,
UPDcpop= ~ Q ~ c p of po r a l l P by ( i ) ,
[Qn,,
f o r a l l p ' i f cp i s s u r j e c t i v e ,
P=Q.
t h e n t h e i r v a l u e s do n o t depend on a p.
( i i i ) By ( i i ) .
0
( i v ) As f o r ( i i ) .
The f o l l o w i n g r e s u l t i s due t o Grzegorczyk.
1.15 THEOREM. Consider t h e f o l l o w i n g f i r s t o r d e r t h e o r y (CL) i n t h e language o f combinatory algebras.
1
Vxy
(CL)
Kxy=x
,
vxyz sxyz = xz(yz) K+S.
,
215
Lambda Calculus and its Models
( i ) (CL) i s e s s e n t i a l l y u n d e c i d a b l e ,
i .e. has n o
consistent decidable
extension. ( i i ) (CL) has no r e c u r s i v e models. Proof. 
(i)
I f T i s a consistent extension o f
A = IP
I
( C L ) , then
P=SET}
i s a non t r i v i a l s e t ( S E A , K ~ A o) f terms c l o s e d under p r o v a b l e e q u a l i t y . But then as i n B a r e n d r e g t [19811, theorem 6 . 6 . 2 ( i i )
i t f o l l o w s t h a t A and t h e r e f o r e
T i s not recursive. then T h b ) = { P = Q l W t = P = Q I i s a
( i i ) I f W i s a r e c u r s i v e model o f (CL), r e c u r s i v e c o n s i s t e n t e x t e n s i o n o f (CL),
contradicting ( i ) .
The axioms f o r c o m b i n a t o r y a l g e b r a s a r e i n s p i r e d by t h e a n a l y s i s o f r e c u r s i v e p r o c e s s e s , n o t by a l g e b r a . The f o l l o w i n g shows t h a t t h e s e s t r u c t u r e s a r e i n fact algebraically pathological.
1.16 PROPOSITION. Combinatory a l g e b r a s ( e x c e p t t h e t r i v i a l one) a r e ( i ) n e v e r commutative, ( i i ) never a s s o c i a t i v e , ( i i i ) never f i n i t e , ( i v ) never recursive. Proof. 
( i ) Suppose i k = k i . Then k = i k = k i , hence a = k a b = k i a b = i b = b f o r a l l a,b
and t h e a l g e b r a i s t r i v i a l . ( i i ) S i m i l a r l y t r i v i a l i t y follows from ( k i ) i = k ( i i ) . = kk,.
( i i i ) D e f i n e kl = k , kn+l ( i v ) By l . l s ( i i ) .
Then t h e kl,k2
,...
are a l l distinct.
0
The f o l l o w i n g r e s u l t , due t o B a r e n d r e g t , Dezani and K l o p , shows t h a t combinatory algebras a r e universal f o r recursive a p p l i c a t i v e structures.
1.17 THEOREM. Given a c o m b i n a t o r y a l g e b r a 8. Then e v e r y r e c u r s i v e a p p l i c a t i v e s t r u c t u r e !XI can be embedded i n t o 8. Proof. Let 
?I be g i v e n and !XI = ( I N , . )
a
= [A,'n']
with
.
recursive.
D e f i n e i n 91
= A*z.zArn'
w i t h A E B t o be d e t e r m i n e d and ' n ' E 8
t h e nth numeral. Then
a a = [A,'n'l[A,'m'] n m = [ A , rml]Arn' = AArml'n'
= [A,F'n"m''], = [A,'n.m'],
provided A=A*pqr.[p,Frq], provided F represents
. ,
216
H. BARENDREGT
.
= a n .m The e x i s t e n c e o f F f o l l o w s f r o m t h e A  d e f i n a b i l i t y : %+‘I1
Moreover hn.a a
= a
n
m
o f the recursive functions.
i s injective:
+ ‘n’
= ‘m‘
n=m.
0
The n e x t c o r o l l a r y i s a s t r e n g t h e n i n g o f a r e s u l t o f E n g e l e r due t o F. H o n s e l l .
1.18 COROLLARY. F o r e v e r y a p p l i c a t i v e s t r u c t u r e B t h e r e i s an e x t e n s i o n a l combinatory algebra
B
such t h a t
P r o o f . Given 8 , c o n s i d e r 
91CB.
the theory
T = (CL) + Diag(21) + E x t where Diag(B) = { P = Q I P,Q€Y€(a),F V ( P Q ) = 0 , 8 b P = Q } , E x t = Vab((Vx a x = b x ) Every f i n i t e p a r t T
D
+
a=b).
o f T i s c o n s i s t e n t : by 1.17 e v e r y non t r i v i a l e x t e n s i o n a l
c o m b i n a t o r y a l g e b r a , e.g.
D,
can be made i n t o a model f o r T o . B u t t h e n by com
pactness T i s c o n s i s t e n t and has a model a l g e b r a and
BCB.
B. T h i s i s an e x t e n s i o n a l c o m b i n a t o r y
0
The c o n s t r u c t i o n i n E n g e l e r [ 1 9 8 1 ] i s m o r e i n f o r m a t i v e i n a n o t h e r sense: f o r each s e t A t h e r e i s a c o m b i n a t o r y a l g e b r a DA such t h a t e v e r y a p p l i c a t i v e s t r u c t u r e w i t h u n i v e r s e A can be embedded i n t o DA.
92.
LAMBDA ALGEBRAS AND LAMBDA MODELS. S i n c e i n a c o m b i n a t o r y a l g e b r a B a b s t r a c t i o n can be s i m u l a t e d by k and s ,
i t i s p o s s i b l e t o i n t e r p r e t e Aterms
2.1
in 8.
NOTATION. L e t C be a s e t o f c o n s t a n t s . A(C)
i s t h e s e t o f Aterms u s i n g
p o s s i b l y c o n s t a n t s f r o m C . The A  c a l c u l u s axioms and r u l e s e x t e n d i n t h e o b v i o u s way t o e q u a t i o n s M = N w i t h M , N € h ( C ) .
For these M,N we s t i l l w r i t e X I  M = N .
!lX i s an a p p l i c a t i v e s t r u c t u r e , then A(YR)
2.2
If
i s A({cala€YR}).
DEFINITION. L e t m be a c o m b i n a t o r y a l g e b r a .
(i) Let
*
: A @ R ) + S @ l ? ) be t h e map t h a t r e p l a c e s e v e r y
M* f o r *(MI x* = x , c* = c , (MN)* =
M*N*
A
by
A*,
i.e.
writing
Lambda Calculus and its Models
(Ax.M)*
217
= A*x.M*.
( i i ) For M,N€A@J?)one d e f i n e s
I k
M=N

I , p b M = N f o r a l l p.
I f I i s a c o m b i n a t o r y a l g e b r a and UAx.xcaI B
.
aEI,
then we w r i t e e.g.
Ax.xa
for
Not a l l e q u a t i o n s p r o v a b l e i n A  c a l c u l u s a r e t r u e i n a c o m b i n a t o r y a l g e b r a . E.g.
i f II)) i s t h e t e r m model o f CL, then
I#Xz.(Ax.x)z since
2.3
S(K1)I and (Xz.z)*
(Xz.(Ax.x)z)*E
DEFINITION.
= 12.7. 5
I; b u t X!Az.(Ax.x)z
=
Az.z.
( i ) A c o m b i n a t o r y a l g e b r a Yl? i s c a l l e d a A  a l g e b r a i f f o r a l l
M,N€A(!JJ~)
A
CM=N
=D
D?b M = N .
A  a l g e b r a homomorphism i s j u s t a c o m b i n a t o r y a l g e b r a homomorphism.
(ii) A
The n o t i o n o f A  a l g e b r a seems t o depend on t h e d e f i n i t i o n o f A*. of
2.4
B u t because
o t h e r d e s c r i p t i o n s , see 2 . 5 and § § 3 , 4 , t h i s i s n o t t h e case.
PROPOSITION. ( i ) I f q:iV?l+9J$,
p a r t i c u l a r q[MT
nn
= [M]m2
( i i ) Let!U?l+12. ( i i i ) 9R,C.m2 Proof. 
By 1.14.
9
thendM1:
=[q(M)IZfp
f o r M€A@).
then a l s o I 2.
Then Th@l)c_Th@J?2). So ifYl?, i s a A  a l g e b r a , Th@,)
In
f o r MEAO.
= Thm2).
0
By u s i n g C u r r y ' s c o m b i n a t o r y axioms AB one can a x i o m a t i z e t h e c l a s s o f
A
algebras.
2.5
THEOREM. L e t I be a c o m b i n a t o r y a l g e b r a . Then
f i e s the f o l l o w i n g set of equations A
(Ag)
P r o o f . By 
I
(A.I)
Ii s
a Aalgebra i f f % s a t i s 
R:
K = S ( S ( K S ) (s(KK)K)) ( K ( s K K ) ) ,
(A.2)
S = S(S(KS) (S(K(S(KS)))
(S(K(S(KK)))S)))
(A.3)
S ( S ( K S ) (S(KK) ( S ( K S ) K ) ) )
(KK) = S ( K K ) ,
(A.4)
S ( K S ) (S(KK))
(A.5)
S(K(S(KS)))
(K(K(SKK)))
= S ( K K ) (S(S(KS) (S(KK) ( S K K ) ) ) ( K ( S K K ) ) ) , (S(KS) ( S ( K S ) ) )
the fact t h a t the theories
= S ( S ( K S ) (S(KK) (S(KS) (S (K ( S (KS))
A
and CL+A
B
S) )
are equivalent,
(KS
.
i n t h e sense
H. BARENDREGT
218 that
A
I
M=N
CL+A
see B a r e n d r e g t [19811 7.3.10
I M* = N*,
B and 7.3.15.
0
The lambda a l g e b r a s u s u a l l y a r i s e as s u b s t r u c t u r e s o f a more n a t u r a l c l a s s o f A  c a l c u l u s models, t h e so c a l l e d lambda models. For t h e s e s t r u c t u r e s t h e r e i s a u n i f o r m method t o f i n d t h e elements r e p r e s e n t i n g a l g e b r a i c f u n c t i o n s , i n d e pendent o f t h e way t h e s e f u n c t i o n s a r e g i v e n (by t e r m s ) ; c f . theorem 5.8.
2.6
DEFINITION. L e t W be a c o m b i n a t o r y a l g e b r a . Ell i s c a l l e d weakly e x t e n s i o n a l
i f f o r A,B<X@i)
W C
Vx(A=B)
Ax.A = Xx.B
+
.
The c o n d i t i o n o f weak e x t e n s i o n a l i t y i s r a t h e r s y n t a c t i c a l . Meyer [19801 and S c o t t [19801 r e p l a c e i t as f o l l o w s .
2.7
DEFINITION.
( i ) I n a combinatory algebra d e f i n e 1 = S ( K I ) .
( i i ) A Amodel
W such t h a t t h e f o l l o w i n g M e y e r  S c o t t axiom
i s a Aalgebra
h o l d s i.n !JX Vx(ax=bx)
2.8
+
la = lb.
LEMMA. L e t Ell be a c o m b i n a t o r y a l g e b r a . Then i n Ell ( i ) lab = ab;
I f moreover!JX i s a A  a l g e b r a , (ii
1 = Axy.xy,
(iii
l(1x.A)
(iv
11 = 1 .
Proof. 
then
hence l a = X y . a y ;
= Xx.A,
f o r a l l A€%@)
i ) l a b = S ( K l ) a b = K l b ( a b ) = ab.
(ii
1 = S ( K I ) = (Xxyz.xz(yz)) (KI) = hyz.Klz(yz)
(iii
l ( A x . A ) = Ax.(Xx.A)x
= Ax.A,
PROPOSITION.
Proof. 
()
= 1yz.y~
by ( i i ) .
0
( i v ) By ( i i i ) and ( i i ) .
2.9
;
W i s a Amodel

i s a weakly extensional Aalgebra.
L e t m be w e a k l y e x t e n s i o n a l . Then
Vx ax = bx
* =b
Ax.ax = Ax.bx l a = l b , by 2 . 8 ( i i ) .
(*)
L e t W be a Amodel.
Vx
A = B
=b
* =b
Then
Vx(Ax.A)x = ( h x . 6 ) ~ 1 (1x.A)
= t(Xx.B)
Ax.A = hx.8,
by 2 . 8 ( i i i ) .
0
Lambda Calculus and its Models 2.10 PROPOSITION. L e t % ?be a A  a l g e b r a .
ID) i s e x t e n s i o n a l Proof. 
(*)
* *
A=B
Then
ID) i s w e a k l y e x t e n s i o n a l and s a t i s f i e s I = 1
Q
(Ax.A)x = (Ax.B)x Ax.A = Ax.B,
by e x t e n s i o n a l i t y .
Moreover I x y = x y = l x y , so by e x t e n s i o n a l i t y ( t w i c e )
(*)
By 2.9
.
W i s a Amodel
Vx a x = bx
219
*
I=1.
Hence
la = lb
0
a = b since 1 = I .
An e x t e n s i o n a l c o m b i n a t o r y a l g e b r a i s a u t o m a t i c a l l y a A  a l g e b r a .
A I
because
*
M=N
CL+ext
t
M*=N*,
This i s
see B a r e n d r e g t [19811 7 . 3 . 1 4 .
TERM MODELS, INTERIORS. 2.11 DEFINITION. L e t T be an e x t e n s i o n o f t h e t h e o r y
A,
i.e.
o f the Acalculus.
( i ) Define M = N Q T [MIT = “€A
T
h / T = {[MI, [MIT.[NlT
k M=N
; t h i s i s a congruence r e l a t i o n on
A.
I M =T N}. 1 MEA}.
= [MNIT
; t h i s i s welldefined.
The open term model o f T i s ID)m(T) =
(
A/T,.,[KIT,[SIT
).
( i i ) By r e s t r i c t i n g e v e r y t h i n g t o c l o s e d terms one d e f i n e s t h e c l o s e d t e r m

model o f
T 0
ID) (T)
=(A
0
0 0 /T,.,[KIT,[SIT
Clearly i f T i s consistent, T
# K=S,
).
i.e.
does n o t p r o v e e v e r y e q u a t i o n , then 0 I n p a r t i c u l a r ID)(A) and ID) (A) a r e
so I ( T ) and d ( T ) a r e non t r i v i a l .
non t r i v i a l s i n c e i t f o l l o w s f r o m t h e ChurchRosser theorem t h a t t h e t h e o r y
A
consistent.
2.12 PROPOSITION. L e t T be an e x t e n s i o n o f t h e A  c a l c u l u s and l e t ( i ) F o r M w i t h FV(M) = { x ,
=
[MI;
(ii) T CM=N (iii) T Proof.
t
M=N
[M [ x := +
* es
PI1
+
, . . . ,xn} (0)
* *
= ID)(O)(T).
has
.
ID)+ M = N
Wb
M = N , p r o v i d e d t h a t W = W ( T ) o r t h a t M,N a r e c l o s e d .
( i ) I n d u c t i o n on t h e s t r u c t u r e o f M * ,
(ii) TCM=N
W
and p w i t h p ( x i ) = [ P ]“’one i T
VF
TCM[Z:=bI
VP
[M[x : = P I I T = “ [ x
+
+
+
+
u s i n g TIM
+
=N[x:=P]
+
+
:=PIIT
=
M*.
is
H. BARENDREGT
220
* Vp U M ~ , = [ N J P * I + M=N . = 9i?(T). L e t Po ( x ) = [XI,. Then
( i i i ) For W
!lJ?+M=N
* U MI * [MIT *
=
1N I
= [NIT,
by ( i ) ,
T+M=N.
F o r M,N c l o s e d .
!lJ?b M = N * UMJ,
=IN]
P [MIT = [NIT,
* *
by ( i ) ,
0
TkM=N.
2.13 COROLLARY. ( i ) W(')(T)
i s a Aalgebra.
( i i ) I ( T ) i s a Amodel. P r o o f . W r i t e I= I ( T ) . ( i ) By 2 . 1 2 ( i i ) . (ii )
B? b
* IC * WC
Vx a x = bx
Vx[M]x = [ N I X where a = [ M I and b = [M][z]
*
TCMz=Nz
* *
T
*
rlx+ l a = l b .
I Az.Mz = k 1M = 1 N
T
"11~1,
=
"1,
f o r some f r e s h v a r i a b l e z ,
1z.N~
I0 (T)
Remarks. ( i ) ( J a c o p i n i [ 1 9 7 5 ] ) .
i s i n g e n e r a l n o t a Amodel. Consider 0 0 T z A a x i o m a t i z e d b y {RKZ = RSZ Z E A 1 where R ~ ( A x . x x ) ( A x . x x ) . Then V Z E A 0 T RKZ = RSZ, hence I (T) l= Vx RKx = RSx. B u t I o ( T ) # 1 (RK) = 1 ( a s ) , s i n c e
1
+
otherwise T
t RKx
= RSx, w h i c h i s f a l s e .
0
( i i ) P l o t k i n [1974] shows t h a t e v e n I
(A)
and
I0 (An)
a r e n o t Amodels.
( i i i ) By ( i ) i t follows t h a t p r o p o s i t i o n 2 . 1 2 ( i i i ) does n o t h o l d i n g e n e r a l 0 f o r I ( T ) : t a k e M=RKx, N=RSx.
conibinatory a l g e b r a . B ( n o t a t i o n B0 ) i s t h e s u b s t r u c t u r e
2.14 DEFINITION. L e t 91 be a ( i ) The i n t e r i o r o f
of
k,s. (ii)
II
is
hard i f
21'
= 21.
0 Note t h a t u p t o i s o m o r p h i s m W (T)
i s the i n t e r i o r o f %?(T).
2.15 PROPOSITION. L e t 8 be a A  a l g e b r a . (i) d(Th(8))
zWo
( i i ) L e t Th(%) = { M = N Then !$(Th(Yl))
zB.
M,N€Z(II),
c l o s e d and
B I= M = N l .
B
generated by
22 1
Lambda Calculus and its Models
Proof. 
i s a w e l l d e f i n e d isomorphism o n t o B
( i ) ( P ( [ M ] ~ ~ ( ~=~[ M ) j)'
( i ) Similarly.
0
.
0
f o l l o w s t h a t a l l A  a l g e b r a s a r i s e as a s u b s t r u c t u r e o f a Amodel.
I
2.16 PROPOSITION. ( i ) ( B a r e n d r e g t , Koymans [ 1 9 8 0 ] ) .
Every
A  a l g e b r a can be em
bedded i n t o a Amodel.
( i i ) (Meyer [ 1 9 8 1 ] ) . P r o o f . ( i ) VI%J?
0
(Th
Every X  a l g e b r a
i s t h e homomorphic image o f a Xmodel.
(E)) c m(Th (X)). 0
by t h e s u r j e c t i v e map t h a t r e p l a c e s
( ii ) Moreover %l?(Th(Z)) + %l? (Th(2)) e v e r y f r e e v a r i a b l e by say K.
0
The f o l l o w i n g i s proved i n B a r e n d r e g t and Koymans [19801. Here we s t a t e t h e result without a proof.
2.17 THEOREM. ( i ) T h e r e i s a Amodel
t h a t c a n n o t be embedded i n t o an e x t e n s i o n a l
Amodel. ( i i ) There i s a c o m b i n a t o r y c o m p l e t e a p p l i c a t i v e s t r u c t u r e t h a t cannot be made i n t o a A  a l g e b r a
(by c h o o s i n g k , s ) .
i i i ) There i s a A  a l g e b r a t h a t cannot be made i n t o a Amodel
(by changing
k,s ( i v ) There i s a Amodel CO 1
t h a t c a n n o t be made i n t o an e x t e n s i o n a l one (by
apsing i t ) . The t e r m models make i t p o s s i b l e t o g i v e t h e f o l l o w i n g p r o o f s o f some ccm
pleteness r e s u l t s .
2.18 THEOREM. ( i )
A C
M=N

M = N i s t r u e i n a l l Amodels
(or Aalgebras).
( i i ) L e t T be an e x t e n s i o n o f t h e A  c a l c u l u s . Then T+
M=N
(iii) Let (A)c
M = N i s t r u e i n a l l Amodels s a t i s f y i n g T .
94
be t h e c l a s s i c a l f i r s t o r d e r t h e o r y a x i o m a t i z e d by t h e u n i v e r 
sal closure of Kxy = x syxz = xz(yz),
KZS V x ( a x = bx)
+
Then M=N
la = lb

xt
M=N.
H. BARENDREGT
222 Proof. 
(i)
(+) By d e f i n i t i o n .
t r u e i n W ( A ) ; hence
A t
(e)I f M = N i s t r u e i n a l l Amodels,
then i t i s
M = N by 2 . 1 2 ( i i i ) .
( i i ) Similarly. ( i i i ) (a)Note t h a t W(X) C ( A ) c . T h e r e f o r e
(XIc ()
53.
t
*
M=N
W(A) C M = N
A CM=N.
+
0
Trivial.
SYNTACTICAL MODELS I n t h i s s e c t i o n a s y n t a c t i c a l d e s c r i p t i o n o f t h e A  a l g e b r a s and Amodels
w i l l be g i v e n , w h i c h i s e q u i v a l e n t t o t h e f i r s t o r d e r d e s c r i p t i o n i n 52. For some models,
i n p a r t i c u l a r t h e f i l t e r model o f B a r e n d r e g t e t a l .
[19831,
t h i s syntac
t i c a l d e s c r i p t i o n i s more c o n v e n i e n t t h a n t h e f i r s t o r d e r . The method i s due t o H i n d l e y and Longo [19801.
3.1
DEFINITION. L e t W = ( X , . )
be an a p p l i c a t i v e s t r u c t u r e .
( i ) Val @I i) s the s e t o f valuations i n
1.
l ( i i ) A s y n t a c t i c a l i n t e r p r e t a t i o n i n 1 i s a map I : h @ ? ) x V a (sn) the f o l l o w i n g conditions; 1.
nxn P
I(M,p)
i s w r i t t e n as [MD
f
X satisfying
P'
= p(x)
2. U c a l p = a
3. UPQD, = UPIIp.UQn P 4. UAx.Pl .a = UP] p ( x : = a ) P
5. pFFV(M)
EM1
= p'FFV(M)
P
= [MI
Note t h a t by t h e v a r i a b l e c o n v e n t i o n , 4
P"
i m p l i e s t h a t f o r y @ FV(M(x))
one has
41. u M ( x ) n p ( x : = a ) = [Ax.M(x)],a = UAy.M(y)l,a
:=a).
= uM(Y)np(y
( i i i ) A syntactical applicativestructure
[l 3.2
is a syntactical interpretation i n
i s of the formW=(X,.,[D)
where
W.
DEFINITION. L e t 1 be a s y n t a c t i c a l a p p l i c a t i v e s t r u c t u r e .

( i ) The n o t i o n o f s a t i s f a c t i o n i n 1131 i s d e f i n e d as u s u a l :
W,p C M=N WCM=N
[MI, Vp
= [NIP
W,pCM=N

and t h i s i s extended t o a r b i t r a r y f i r s t o r d e r f o r m u l a s o v e r t h e A  c a l c u l u s . ( i i ) IJ31 i s a s y n t a c t i c a l A  a l g e b r a (iii) i .e.
W i s a s y n t a c t i c a l Amodel i f Va 'M'p(x
: = a ) = "'p(x
:=a)
if A k M = N
(5) W C +
[Ax.MI
WC
Vx(M=N) P
= UAx.ND
M=N. f
P'
Ax.M = Ax.N,
223
Lambda Calculus and its Models
3.3
LEMMA. LetIIR be a s y n t a c t i c a l Amodel.
= Vp
(p(M,N)
Consider t h e s t a t e m e n t
6M[x : = N l I p = uMIIp(x : = [ N I P ) .
Then f o r M,NEA6J?)
*
(i) z@FV(M) (ii) ~ M , N )
cp(M,z)
W(AY.M,N);
9
( ii i ) Q(M,N).
Proof.
( i ) W r i t e M=M(x).
uwnP
then
= uM(z)np(z : = p ( z ) )
= nM(X)ip(x :=p(z))
by 4 ’ . ( i i ) F i r s t assume x B F V ( N ) . By t h e v a r i a b l e c o n v e n t i o n y s x , y f F V ( N ) . f o r p* = p ( x : = [ N ]
P
) and a r b i t r a r y aEIIR : = a ) = UM[x : = N I I p ( Y : = a )
= :N(,I
(note that I N ]
P
:=a)).
=
UAy.M[x : = N I I p *
=
:=a)(x
=
:=a)
T h e r e f o r e by
= UAy.MB
:=[NIP)’
s i n c e (D(M,N),
;
(5)
P*
and hence [Ay.M[x := N l n
P
= [Ay.M[x
=‘Ay‘M’p(x I f xEFV(N),
P
= =
( i i i ) Now cp(M,N)
:=[NIP)’
[R[x
IMLx
:= Z ] [ Z := N]]
IIR
I
P
= ”p(z
:=EN] P ) ( x
= u‘np(x
:=[NIP ) ’
by ( i ) , :=I”],)’
f o l l Q w s b y a s i m p l e i n d u c t i o n on t h e s t r u c t u r e o f M.
M=N
=P
IIRC M = N ,
i s a s y n t a c t i c a l Aalgebra,
P r o o f . By

: = z ] I p ( z :=KN],)
THEOREM. L e t W be a s y n t a c t i c a l Amodel.
A i.e.
P*
t h e n l e t z be a f r e s h v a r i a b l e . We have f o r MAy.M
[%[x := N l J
3.4
:= N l I
i n d u c t i o n on t h e l e n g t h o f p r o o f .
Then
Then
H. BARENDREGT
224 The axiom (Ax.M)N = M [ x : = N ]
f
i s sound:
= [M[x : = N ] ] Soundness o f t h e r u l e M = N
*
P'
by 3 . 3 ( i
i).
Ax.M = 1x.N f o lows from
(5). The
DEFINITION. A homomorphism between s y n t a c t i c a l A  a l g e b r a s
cp:XWl
i s a map
+XV2 such t h a t f o r a l l M € A @ ) one has
dM1,1
= I[cp(M)ncpop 2
where i n cp(M) t h e c
3.6
other rules are
0
trivial.
3.5
by 3,
(Ax.M)Nlp = !IAx.Ml,!IND,,
a r e r e p l a c e d by c
cp(a)
'
THEOREM. The c a t e g o r i e s o f s y n t a c t i c a l A  a l g e b r a s and homomorphisms and
t h a t o f Aalgebras and homomorphisms a r e i s o m o r p h i c . Moreover s y n t a c t i c a l A models c o r r e s p o n d e x a c t l y t o Amodels under t h i s isomorphism. Proof.'Easy. For a s y n t a c t i c a l X  a l g e b r a f o r c p : YX
1
+YX2
l e t Fcp = c p :
Pm,
+
XW=(X,.,II)
d e f i n e FBI = (X,.,UKl,USl);
Pmz. Then one has [MlPm
Conversely f o r a Aalgebra B = (X,.,k,s)
P d e f i n e GB = (X,.,[]')
= [ M c f o r MEAm).
above. Then F, w i t h i n v e r s e G , i s t h e r e q u i r e d isomorphism.
54.
and Gcp = cp as
0
CATEGORICAL DESCRIPTION OF THE MODELS. I n t h i s s e c t i o n t h e c l a s s o f Aalgebras w i l l be d e s c r i b e d i n a n a t u r a l c a t e 
g o r i c a l way. The Amodels a r e t h e n t h o s e A  a l g e b r a s t h a t c m e f r o m c a t e g o r i e s " w i t h enough p o i n t s " .
The method i s due t o Koymans [1983] and i s based on work
o f Scott.
4.1
DEFINITION. L e t
0
be a c a t e g o r y . The i d e n t i t y map on an o b j e c t A E U i s de
n o t e d by i d A . ( i ) 0 i s a Cartesian closed category (ccc) i f f
1.
iI
has a t e r m i n a l o b j e c t T such t h a t f o r e v e r y o b j e c t A E B t h e r e e x i s t s a
u n i q u e map ! A : A + T .
2. For A ,A
E C t h e r e i s an o b j e c t AIXAz
1 2 p i : A1xA2+Ai
u n i q u e map Notation.
(
( C a r t e s i a n p r o d u c t ) w i t h maps
( p r o j e c t i o n s ) such t h a t f o r a l l f i : C + A i
fl,fz)
: C + A 1 x A 2 w i t h p i n ( fl,f2)
If gi:Ai+B.
B1xBz, see f i g u r e .
( i = 1,2)
there i s a
= f i , see f i g u r e .
( i = l , 2 ) , then g l x g z = ( g l o p l ,
g20pz):A,XA2+
225
Lambda Calculus and its Models
I
exponcnt
pmduct
3 . For A,BE C t h e r e i s an o b j e c t
A
B E b (exponent) w i t h map e v = e v A , B :
A
3 x A + B such t h a t f o r a l l f : C x A + B t h e r e i s a u n i q u e A f : C + B A s a t i s f y i n g f = e v o ( A f X i d A ) , see f i g u r e . ( i i ) L e t C have a t e r m i n a l o b j e c t T. A
point
o f A€
i s a map x : T + A .
The
s e t o f p o i n t s o f A i s denoted by I A l . An o b j e c t A has enough p o i n t s i f f o r a l l f,g : A + B
one has f#g
*
3xE IAl
fox#gox.
Note t h a t i n a c c c one has A(hogxidB) = A(h) o g f , g ) o h = ( f o h , goh)
(
= ( f o h , gok)
fxgo (h,k)
DEFINITION. L e t C be a
4.2 o f U,
i.e.
CCC.
An o b j e c t U E B i s r e f l e x i v e i f Uu i s a r e t r a c t U and G : U + U such t h a t
t h e r e a r e maps F : U + U u
FOG = i d UU’
4.3
DEFINITION. L e t 0 be a c c c w i t h r e f l e x i v e o b j e c t U ( v i a t h e maps F,G).
these data determine a s y n t a c t i c a l a p p l i c a t i v e s t r u c t u r e
m(e)
( = W(O,U,F,G))
Then as
follows: ( i ) The domain o f
m(C)
is
IUI.
( i i ) L e t Ap : U 2 + U be t h e map evU,Uo F x i d For f , g : A + U d e f i n e f a A g = Apo ( f , g ) . X.Y
= x .T y = Apo
(
IUI
x,Y).
As a p p l i c a t i v e s t r u c t u r e I ( Q ) ( i i i ) Uo = T, Un”
U’ In particular for x,yE
is (IUI,.)
= UnxU. L e t A = x l ,
...,x
. be a sequence o f d i s t i n c t v a r i 
a b l e s . W r i t e UA = U”. (iv)
nXA
: UA+U
i
i s t h e c a n o n i c a l p r o j e c t i o n on t h e i  t h c o o r d i n a t e .
226
H. BARENDREGT
,..., f n : A  t U ,
(v) I f fl
then ( f l
= '' A
0
(fl,...,fn+l)
= ((f,,
,..., f n ) : A + U n
i s d e f i n e d by
. . . ,f n ) , fn+l).
Clearly 7rA
0
(f,,
..., f n ) =
fi
xi ( v i ) Let
r
. . ,Y,
= yl,.
with
{q}~{z}.
Define
rlA = ( 7 r A
) :
,...,AA
y1
ua+ur.
ym
This i s the canonical "thinning",
IM(x,y)ll X?Y
[XIA
,. .. , x n ) .
b e i n g "X(x,y).[M(cx,c
(
y1
,.. . ,y,
)".
U A + U ( w i t h intended i n t e r p r e t Y
) l " ) as
follows.
;
= ITA
[canA = a o upon,
"h(xl
d e f i n e i n d u c t i v e l y [Ml,:
( v i i ) For AZFV(I1) a t i o n o f e.g.
i.e.
'
,
= upnA.ua
(for a €
tul) ;
nuA;
I A x . P I A = G O A ( [ P I I ~ , ~ )where , by t h e v a r i a b l e c o n v e n t i o n we assume x!€{Al. ( v i i i ) For a v a l u a t i o n p i n I U I l e t
[MI
P
=
[MILO
pA w i t h A=FV(M).
EM1 E I U I . P ( i x ) Finally%V(G) i s the s t r u c t u r e
Clearly
[ M I X :=N]lr
Proof. ( i ) , ( i i ) (i) [xy.Plro
(lUl,.,[l).
=
I n d u c t i o n on t h e s t r u c t u r e o f M. We o n l y t r e a t M=Xy.P. A = GoA(UPlr,y) o A
rfr
= GoA([P]r,yo = G~A([PI,,~~
= GoA(UPIA,y)
IH = niy.pnA.
n f x i d U ) , by 4 . 2 ( 1 ) ,
221
Lambda Calculus and its Models Here I H denotes “by
the induction hypothesis”.
+
+
3
= [~y.~[;,y
( i i ) u(A~.P)[x:=NII,
:=~,yIn, 3
= GoA(UP[;,y
:= N , ~ l l , , ~ )
+
(uN1,,y,
= GoA(IIPIA,yo IH
z
GoA(UPIA,ya(Exlr)x
= GoA(l[PIA,y)
0
*
i d U ) , see below,
([xl,)
= [[Xy.PD,o(U$), where
[yl,,y)),
;
i s shown as follows
(~ih,,~, u ~
+
I ~ , ~= ) ( ~NI,
0
n:.Y,
ni’y
(i) 3
p,,
= (UNIr,o
+
i d U o ( p1 ,p 2 )
= ([Nl,)x =
( i i i ) Apply ( i i ) t o A’=A,x
4.5
3
((“1,)
and
idUO pg
x idU.
+
r,
w r i t i n g A=y
PROPOSITION. L e t M,NEA(!V?(O)) and { A l z F V ( M N ) .
Proof. 
+
and M [ x : = N ] G M [ y , x
Then
I n d u c t i o n on t h e l e n g t h o f p r o o f of M = N . We t r e a t t h e e s s e n t i a l axiom
and r u l e . Axiom (Xx.P)Q = P [ x :=Ql. (hx.P)QIA = (GoA(uPIA,x))
0
UA
Rule P = Q
=
BPI,
. *
*
Xx.P =
[ a , , by
[PlA0
uAx.PnA
=
[aA
AX.Q.
the induction hypothesis,
rIp
upnA,x =
= nQl,o
n p
an,,^. nxx.~,.
4.6
THEOREM. Every c c c
mc)
=
Proof. 
+
:=y,N].
t
0
w i t h r e f l e x i v e o b j e c t U determines a Aalgebra
(iui,.,un). Immediate f r o m
4.5 and t h e d e f i n i t i o n of 111
P’
0
0
H.BARENDREGT
228
4.7
PROPOSITION. L e t
I=IIR($,U,F,G).
( i ) Let {A}?FV(M).
Then
( i i ) U has enough p o i n t s U ( i i i ) U Z U v i a F,G I (iv) Proof.
u . U z U via
( i ) [lM],
inm.
[ l M ] I A = GoFo[M]I,
k 1
IIR
i s a Amodel.
= I.
F,G and U has enough p o i n t s

Sn i s e x t e n s i o n a l .
= UAy.MyjA = GoA(evo ( Fo [ M I
UylA,y))
A,Y’
= GoA(evo ( F o [ M ] l A o IIA”, nA”))
A
Y
= GoA(evo(FoIIMDAopl,pz)) = GoA(evo (Fo [MI,)
x id)
= GoA(ev) o Fo [ M j A = GoFo [MI,.
(ii)
(1
for a , b € I
Vx€Iax=bx
evo(Foa,x) = evo(Fob,x)
*
evo (Foa)x i d o ( id,x)
9
e v o (Foa)
* Hence

1 L e t U have enough p o i n t s . Then t h e same i s t r u e f o r U ( g U ) .
IIR
(1 Then
= evo (Fob)x i d o ( id,x)
i d = e v o (Fob) x i d , s i n c e U’ has enough p o i n t s ,
X
l a = l b , s i n c e by ( i ) l c = GoFoc = GoA(evo ( F o c ) x i d ) . i s a Amodel. Suppose

Ii s
a Amodel
VxE I U l f O X = gox
and l e t f , g : U + U .
*
Vx i . x = g . x , where
1.f =
=9
7
= GoA(fopZ) and s a t i s f i e s f . x = f o x ,
1.9
G ~ F ~= TG
~
F
~
~
A ( f o p 2 ) = A ( g o p z ) , s i n c e FOG = i d , fop* = SOP2 f=g.
9
T h e r e f o r e U has enough p o i n t s . U ( i i i ) (*) I f U E U v i a F,G, t h e n GoF = i d U , hence by ( i )
UIMIA =
UMII,
i n p a r t i c u l a r 11x1,
[in, i.e.
II.
(1
=
= [xi,.
[Ax.ixnA
Then as i n t h e p r o o f o f = U A ~ . ~ =~ 11111, ,
1 = I. Assume
II.
1 = I . Now
8 1 1 = [Axy.xyB
= GoA(llXy.xyJx)
= GoA(GoFo [ x I x ) ,
= G O A ( G O F ~ P; ~ )
by ( i ) ,
,
4.5
i t follows that
Now
229
Lambda Calculus and its Models
1 ID
and
= 60A(p2). Therefore
G O A ( G O F O=~ G ~ )O A ( ~ ~ )
*
A(GoFop2) = A(p2)
=s
GoFop2 =
(use F) (Ah u n i q u e l y d e t e r m i n e s h )
P2 GoF = idU
*
,
(use ( !
( i v ) By 2.10 and ( i i ) ,
idU)).
0
(iii).
L e t B be a A  a l g e b r a t h a t a r i s e s f r o m a c a t e g o r y t h a t i s " c o n c r e t e " , r o u g h l y one t h a t i s based on s e t s . Then
B
B
i.e.
i s a Amodel and t h e i n t e r p r e t a t i o n i n
has a s i m p l e form.
4.8
DEFINITION. A c c c Q i s s t r i c t l y c o n c r e t e i f t h e r e i s a f u n c t o r 0 :@ + S e t
such t h a t 1. 0 i s f a i t h f u l 2. 0 i s f u l l
3.
(i.e.
(i.e.
i n j e c t i v e on a r r o w s ) .
s u r j e c t i v e ) o n Hom (T,A)
Q.
for AEC.
# p r e s e r v e s t h e t e r m i n a l o b j e c t , p r o d u c t s and p r o j e c t i o n s .
4. For a l l A,BE Q
Note t h a t t h i s i m p l i e s t h a t e v e r y o b j e c t i n
where ASet
,...,a n )
g(al
=Ad. g(al
,..., an,d)
C
has enough p o i n t s . Moreover
f o r g : Xn+l+Xn
i n Set. W r i t e
f o r t h e t e r m i n a l o b j e c t i n Set. Complete p a r t i a l o r d e r s o r c o m p l e t e l a t t i c e s w i t h c o n t i n u o u s maps a r e s t r i c t l y concrete c c c ' s .
4.9
DEFINITION. L e t Q be a s t r i c t l y c o n c r e t e c c c w i t h r e f l e x i v e o b j e c t U. ( i ) cp: ( i i)
IUI
+
: #(Uu)
#(U) +
i s t h e b i j e c t i o n cp(x) = # ( x ) ( * ) .
#(U)
( i i i ) a.b = #(F)(a)(b)
i s t h e map
=
#(GI.
f o r a,bE#(U).
= V ( U ~  ' ( M ) ~ ~  ~f ,o~r ) MEA(O(U)).
( i v ) [MI:
. ,u
( v ) XQ# = ( ~ ( u ) ,
no).
4.10 THEOREM. (Koymand [ 1 9 8 3 ] ) .
( i ) The map
# 1. uxn, = P W ,
2. ucaj:
3.
= a,
f o r aE#(U);
UPQP = uPi#.ud'; P P P
4. U A X . P I ~= 0 6 d . P
:=d)).
'1
i n 4.9 s a t i s f i e s
{*I
H. BARENDREGT
230 ( i i ) a @ i s a Amodel
isomorphic t o a ( c ) .
P r o o f . ( i ) As an example we show b . L e t p o = cplop and A = FV(Ax.P).
A
(*), where f o r s i m p l i c i t y we assume cp'(P) = P ,
UAx.PIQ' = @ ( G O A ~ ( ~ P ~ , o, ~p,) ) P = ~ ( A s e p D ( ~ P I A , x )()P
A
1)
= o ( A d . O(UPJA,x)(p(x : = d ) A ' x ) ) = o()Sd. Q'(UPl
op0(x : = ~  ' ( d ) ) ~ ' ~(*I) )
A.x
= o ( X d * (P(Up1 cplop(x : = d ) 1) = o(Xld. I P l O p ( x : = d ) ) .
+ : I i' s by 4 . 9 ( i v ) ( i i ) The map @::I($) enough p o i n t s , t h e s t r u c t u r e : I ( $ )
an isomorphism. S i n c e U i n i t f o l l o w s that!#
i s a Amodel;
C
has
i s a Amodel.
0 0
i s c a l l e d t h e c o n c r e t e v e r s i o n of:I(t)
v i a t h e f u n c t o r 0.
Now i t w i l l be p r o v e d t h a t e v e r y A  a l g e b r a can be o b t a i n e d f r o m a ccc w i t h a reflexive object.
4.11 DEFINITION.
L e t B be a A  a l g e b r a .
The Karoubi e n v e l o p o f 8, n o t a t i o n
i s t h e c a t e g o r y d e f i n e d as f o l l o w $ . L e t aob = A x . a ( b x ) ,
Objects: { a € $
I ao a
= a}.
Arrows: Hom(a,b) = I f € 8 Identity: i d Composition:
$(a),
for a,b€8.
1
bofoa = f } .
= a.
fog.
I t i s easy t o v e r i f y t h a t t ( 8 ) i s indeed a c a t e g o r y . Karoubi [1978] d e f i n e d t h e e n v e l o p f o r a d d i t i v e c a t e g o r i e s u n d e r t h e name " d e r i v e d pseudo a b e l i a n c a t e g o r y " .
T h i s can be g e n e r a l i z e d t o a r b i t r a r y categories.
The n o t i o n t h e n a p p l i e s t o a A  a l g e b r a
w(a)
= ({aes
Ia
F
by i n t r o d u c i n g t h e monoid
=la},o,~)
c o n s i d e r e d as c a t e g o r y w i t h one o b j e c t and as arrows t h e a E w ( 8 ) w i t h c o m p o s i t i o n . We need some n o t a t i o n f r o m t h e A  c a l c u l u s .
L e t [M,N]
= Az.zMN be p a i r i n g i n
a .a.), f o r i = 1 ,Z. Let 1 2 I w i t h n? t h e c a n o n i c a l Aterms such [MI] = M1, EM1 ,...,Mn+ll = "M1,.. *~MnI;Mn+lI n+l n+l n that n7[M1 Mn] = Mi f o r l < i < n . [ T I , = I , nn+l = TI^, ni = TI. o n l f o r t h e A  c a l c u l u s w i t h p r o j e c t i o n s ni = Ay.y(Aa
'
,...,
1S iSn.
1
4.12 PROPOSITION. ( S c o t t 119803). ( i i ) I i s a reflexive object Proof.
( i ) t ( 8 ) i s a ccc. n
t(n)
v i a the arrows F = G = l .
( i ) 1. T e r m i n a l o b j e c t . Th s i s t = A x y . y .
Note t h a t f : a + t
f=t.
23 1
Lambda Calculus and its Models
2 . P r o d u c t s . L e t a ,a
1
2
E C ( 8 ) . Then al X a 2 = X Z . [ ~ ~ ( T ~ aZ2)( ,n 2 z ) ] i s t h e
Cartesian product w i t h projections a a pi1 = a . o 71
i '
I
(f,g)
.
= Xz. [ f z , g z l .
3. Exponents. L e t a , b E C ( 8 ) .
Then
ba = Xz.b o z o a ev
a,b
= Xz.t(Tlz(a(T2z)))
A(f) = Xxy.f[x,y]. The c a l c u l a t i o n s t h a t show t h a t e v e r y t h i n g works a r e s t r a i g h t f o r w a r d and a r e l e f t t o t h e reader. ( i i ) Note t h a t 1 1 = 1 , 1 : l + l ,
1 : 1 + 1 and l o l = l = i d
4.13 THEOREM. (Koymans [ 1 9 8 3 1 ) . %V(U(%),l,l,l)
1'
0
=fl.
P r o o f . L e t I =YX(Q(8)). By i n d u c t i o n on t h e s t r u c t u r e o f M E A one can show
m
[MJ3 =
(+)
Xz. M
[x,
,... ,xn
:=
7172
,...,71:zl.
As an example we t r e a t M E 1 y . P . [Xy.PDz = GoA(lIPng
?Y
= loXpq.(Az.P[x,
,...,xn,y
n+ 1
:= ITl
z,..
IH = Xpq.P[x l,...,xn,y
hp. (Xy.P) [x,,
=
:= 71p ;
. . . ,x
... ,xn
hz.M[xl,
L e t 0 be a p p l i c a t i o n i n
,...,r;p,q
:= nyp,.
:= 7172,
I.Note
...,i
. . ,lr; T
p
that
a Q b = Ap o ( a , b )
= ev
o ( 1 o a,b) 1 9 1
= Xz.ev
[ l (az) ,bz] 1 3 1
= Xz.az(bz)
= Sab. Now d e f i n e c p : fl+m by cp(a) = Ka. Then cp i s c l e a r l y i n j e c t i v e .
x E I1
I =
t+
If
I , t h e n x i s c o n s t a n t s o x = K ( x l ) = c p ( x l ) ; t h e r e f o r e cp i s s u r j e c t i v e .
F i n a l l y cp i s a homomorphism: 1 . cp(xy) = K ( x y )
= S(Kx) ( K y ) , s i n c e 8 i s a X  a l g e b r a ,
H. BARENDREGT
232 = cp(x) o c o ( y ) .
2.
w(K)
=
KK = [K]"
by (+) and s i m i l a r l y f o r 5.
T h e r e f o r e cp i s an isomorphism and I
Zm.
C
I t f o l l o w s t h a t e v e r y A  a l g e b r a (Amodel) can be o b t a i n e d f r o m a c c c w i t h r e f l e x i v e o b j e c t U ( h a v i n g enough p o i n t s ) . Remarks.
( i ) I t i s n o t h a r d t o show t h a t i f 91 i s a Amodel,
then e v e r y o b j e c t o f
$ ( ? I ) has enough p o i n t s , see Koymans [ 1 9 8 3 ] . ( i i ) I t i s n o t t r u e t h a t C(%V(t,U))
t.
The c a t e g o r y
0:
may have many more
objects. The n o t i o n o f A  a l g e b r a homomorphism can be c h a r a c t e r i z e d i n a c a t e g o r i c a l way.
4.14 DEFINITION. A f u n c t o r @ between two
i s C a r t e s i a n i f Q, p r e s e r v e s t h e
CCC'S
t e r m i n a l o b j e c t , p r o d u c t s and e x p o n e n t s .
4.15 PROPOSITION. ( i ) F o r i = 1 , 2 t h e maps F. ,Gi.
L e t 0 :t 1 + C 2
l e t It. be a ccc w i t h r e f l e x i v e o b j e c t s U. v i a
be a C a r t e s i a n f u n c t o r w i t h Q(U ) = U 2 , Q ( F 1 ) = F2, 1
Q,(G1) =G2. Then @ i n d u c e s a homomorphism @ : %V(d,) ( i i ) I f cp:%+912
cp+ : C ( ? I 1 )
+
+* = c p
o v e r cp
i s a homomorphism,
then
(0
2 up t o isomorphism.
( i ) For x € I U
( i i ) For a an o b j e c t o f $ ( ? I , ) d e f i n e cp+(f) = cp(f). S i n c e
(0
T h i s i s a homomorphism s i n c e
d e f i n e cp+(a) = cp(a) and f o r f€Homd(al)(a,b)
p r e s e r v e s a l l c l o s e d Aterms,
f u n c t o r p r e s e r v i n g I and 1 . C l e a r l y
55.
m(C,).
Q(91 ) p r e s e r v i n g t h e r e f l e x i v e e l e m e n t s I and r e t r a c t i o n map 1 . More
I d e f i n e @ * ( x ) = Q ( X ) € IU21. 1 p r e s e r v e s F,G and t h e C a r t e s i a n s t r u c t u r e .
Proof. 
+
induces a C a r t e s i a n f u n c t o r
+* = c p
(0
t h i s i s a Cartesian
o n m(C(21)) 2 8.
0
OTHER MODEL DESCRIPTIONS; CATEGORICAL MODELS. Lambda models were d e f i n e d as lambda a l g e b r a s s a t i s f y i n g t h e M e y e r  S c o t t
axiom. S i n c e t h e c o m b i n a t o r y axioms d e s c r i b i n g A  a l g e b r a s a r e n o t memorable, on'e may wonder w h e t h e r t h e s e c a n be s i m p l i f i e d i n p r e s e n c e o f t h e new axiom. T h i s i s indeed t h e case;
5.1
t h e r e s u l t i s due i n d e p e n d e n t l y t o Meyer and S c o t t .
DEFINITION. D e f i n e t h e f o l l o w i n g c o m b i n a t o r y terms.
l 1 = 1 = S(KI);
ln+l = S(Kl)(S(Kln)).
= 1 , In+, = S ( K I n ) 5.2, Remark. U s i n g t h e s i m p l e r d e f i n i t i o n l 1
5 . 3 and 5 . 6 ( i )
233
Lambda Calculus and its Models remain v a l i d .
5.2
LEMMA. ( i ) I f !D? i s a c o m b i n a t o r y a l g e b r a , t h e n
LY?C
( i i ) If!D?i s a A  a l g e b r a ,
then
. ab l...bn.
A k In= Aab l . . . b Proof. 
(i) (ii). 1,
5.3
.
= ab l . . . b
lnabl . . . b
I n d u c t i o n on n, e.g.
i n a Aalgebra
= S(KI) = (Axyz.xz(yz))KI
= Ayz.Klz(yz)
THEOREM. (Meyer [19801; S c o t t [ 1 9 8 0 ] ) .
0
= Xyz.yz.
L e t A = (X,.,k,s).
Then!D? i s a
A
model i f f !D? s a t i s f i e s 1 . Kxy = x ,
2 . sxyz = xz(yz),
3. Vx ax
*
= bx
l a = lb,
4. 1 2 K = K , 5. 1 Proof. 
(*)
s=
5.
I f A i s a Amodel,
then by d e f i n i t i o n 1,2,3
hold. MoreoverA i s a
hence s a t i s f i e s 4,s s i n c e t h e s e e q u a t i o n s a r e p r o v a b l e i n
Aalgebra,
(*)
3
A.
F i r s t show t h a t f o r a l l a , b E A l ( K a ) = Ka and l ( S a b ) = Sab.
Indeed, Ka = l Z K a = S(K1)Ka = l ( K a ) and s i m i l a r l y f o r S. S i n c e 1x.A i s always o f t h e f o r m KP o r SPQ i t f o l l o w s t h a t
(*)
l(Ax.A)
= Ax.A.
T h e r e f o r e A i s weakly e x t e n s i o n a l : Vx
* * *
A=B
Vx(Ax.A)x = (Ax.B)x l(Ax.A) = l(Ax.B), Ax.A = Ax.B,
by
by 3,
(*I.
I t remains t o show t h a t !D? i s a A  a l g e b r a ,
f o l l o w s by i n d u c t i o n on t h e p r o o f o f M = N , rule P=Q
9
Xx.P
i.e.
A I
M=N
*
(JR
C M = N . This
weak e x t e n s i o n a l i t y t a k i n g c a r e o f t h e
= Ax.Q.
The f o l l o w i n g , d e f i n i t i o n o f Meyer [1980]
s i m p l i f i e s even more t h e d e s c r i p 
t i o n o f t h e essence o f a Amodel.
5.4 fying
DEFINITION.
( i ) A c o m b i n a t o r y model i s a s t r u c t u r e A = (X,,k,S,E)
(1)
Kxy = x ,
(2)
sxyz = xz(yz),
(3)
EXY
=
(4)
Vx
ax = b x
XY,
+
Ea = Eb.
satis
H.BARENDRECT
234
( i i ) A c o m b i n a t o r y model i s s t a b l e i f moreover
(5) (6)
EE = E ,
E ~ K= K,
(7)
5.5
s,
E3S =
Here, o f c o u r s e ,
E
1
= E and E

n+l
=S(KE)(S(KE~)).
LEMMA. L e t 9 R = ( X , . , ~ , S , E >
(i) ~ (ii)
E
~ = +a a = a
be a c o m b i n a t o r y model.
~ A V X E(ax) =
vx l . . . x i
~9
(iii) l i s stable
Proof.
Ea a =
~
...x . )
€(axl
ax. = ax l . . . x . ,
O
~ , k , k a , s , s a , s a b a r e f o r a l l a,b
c9
f i x e d p o i n t s o f E.
( i ) (a)By a s s u m p t i o n
a = S(KE)(S(KEn))a
(1)
= E(S(KEn)a).
Hence by 5.4(3) ax = S ( K E ) a x = E (ax) ; t h e r e f o r e by 5 . 4 ( 4 )
and (1)
Ea = E(S(KEn)a) = a. (e)I n a c o m b i n a t o r y model one has cab = ab,
therefore
(2)
E(Ea) = Ea
Now ~ ~ + E~( Sa( K=E ~ ) ~ )t,h e r e f o r e by (2)
(3)
ax
Hence by 5.4(4)
= ~
E(En+la)
but also
=
E
~
,+
~
a
(ax) = ax.
and assumption E ( E ~ + ~ = ~Ea) = a .
Together w i t h
(3) t h i s i m p l i e s ~
~ = +a .
~
a
( i i ) By i n d u c t i o n on n .
( i i i ) (=+) As t o Sab: (4)
Sab =
E
3
Sab = E(Sab).
As t o Sa: Sa =
and i t f o l l o w s by (4)
E
3
Sa = S(KE) (sa)
,
and ( 2 ) t h a t Sa i s a f i x e d p o i n t of
E.
S i m i l a r l y i t f o l l o w s t h a t S, Ka and K a r e f i x e d p o i n t s . By assumption E i s a f i x e d p o i n t o f E. (e)By a s s u m p t i o n and ( i i ) .
5.6
~
0
PROPOSITION. ( i ) L e t B = ( X , . , ~ , S , E )
= and l( X , . , k , s )
i s a Amodel.
(ii) I f l = (X,.,~,S,E)
be a s t a b l e c o m b i n a t o r y model. Then
Moreover k , s a r e u n i q u e l y d e t e r m i n e d by E .
i s a c o m b i n a t o r y model, t h e n W = ( X , * , k ' , s ' , ~ ' ) i s
a s t a b l e cornbinatory model, where k ' = e g k , s ' = E k and E '
3
=EE.
235
Lambda Calculus and its Models
Proof. 
( i ) Note t h a t
*
xy = l x y ; by 5.4(4),
EE = € 1
, , ,
E = 1
,
by s t a b i l i t y and 5 . 5 ( i i i ) .
Ex = E(1X)

= lx
* T h e r e f o r e (X,.,k,s)
by 5 . 5 ( i i i )
since l x = S ( K l ) x ,
by 5 . 4 ( 4 ) ,
i s by 5.3 a Amodel.
As t o u n i c i t y ,
l e t ( X , . , k o , s 0 , ~ ) be a l s o a s t a b l e c o m b i n a t o r y model,
in
o r d e r t o show k = k o, s = s o . Then kxy = x = k 0x y ss
E(kx) = E(kOx)
+
s(kE)kx = s(kE)kox
+
E(s(kE)k) = E(s(kE)ko)
+
~
=+ Similarly s = s
k
2
=
~
2 0
k = k
k
0'
0'
( i i ) F i r s t note
(1)
EYZ
= yz, therefore
E(EY) = EY.
Now l e t x E { ~ ' , k ' , k ' a , s ' , s ' a , s ' a b } . By 5 . 5 ( i i i ) since then
E'X=EEX=EX=X.
i t s u f f i c e s t o show t h a t E X = X
S i m p l e c a l c u l a t i o n s show t h a t x = ~ yf o r some y ( e . 9 .
x = k ' = ~k = s ( k e ) ( s ( k E ) ) k = E ( s ( k s ) k ) ) . 2 0 E X = E ( E Y ) = EY = x.
Then by ( 1 ) i t f o l l o w s t h a t
A l t h o u g h i n Amodels k,s a r e u n i q u e l y d e t e r m i n e d by 1 = s ( k i ) , p r e s e r v e s a p p l i c a t i o n and c o n s t a n t map Ax. 1 ' : I +
5.7
DEFINITION.
a map t h a t
1 i s n o t n e c e s s a r i l y a homomorphism: t a k e e.g.
the
I' .
( i ) L e t m = (X,.)
be a c o m b i n a t o r y c o m p l e t e a p p l i c a t i v e s t r u c 
t u r e . An e x p a n s i o n o f I i s o f t h e f o r m (?V,k,s)
= (X,.,k,s)
which i s a combinatory
a1 gebra. (ii)
I= (X,.)
i s a c a t e g o r i c a l Amodel
t h e r e i s a u n i q u e e x p a n s i o n (9?,k,s)
making
( A  a l g e b r a , cornbinatory a l g e b r a ) i f
I into
a Amodel
( A  a l g e b r a , combi
natory algebra). ( i i i ) An element E o f I i s c a l l e d a s t a b l e E i f E E = E Vxax=bx
5.8
A
cab=ab
A
Ea=Eb.
+
THEOREM.@Let !Dl = (X;)
be c o m b i n a t o r y complete.
f r e p r e s e n t a b l e } and d e f i n e F : X + [ X + X l t o a Amodel
i f f there exists a G : [ X + X ] + X
1.
FOG = i d
2.
GoFE [ X + X ] .
Let [ X + X ]
= {f : X+X
such t h a t
[x+xl;
( i i ) The G ' s s a t i s f y i n g 1,2
I
by F ( x ) ( y ) = x y . Then !Dl can be expanded
i n ( i ) correspond e x a c t l y t o s t a b l e
E'S.
H.BARENDREGT
236
( i i i ) VJt i s a c a t e g o r i c a l Amodel
i f f the G i n ( i ) i s unique i f f there i s a unique s t a b l e Proof.
(i)
(1
E.
L e t (m,k,s) be a Amodel.
Oef i ne G(f) = l a f f o r some af r e p r e s e n t i n g f . G i s w e l l  d e f i n e d :
i f a x = f ( x ) = a ' x f o r a l l x, then
l a = l a ' by t h e M e y e r  S c o t t axiom. C l e a r l y F ( l a ) = F ( a f ) = f , so FOG= i d . Moreover f GoF(a) = l a , s i n c e a r e p r e s e n t s F ( a ) ; hence GoF i s r e p r e s e n t a b l e . (e)L e t k O , s O E X s a t i s f y t h e k,s
axioms. D e f i n e E,,=G(GoF).
Then
( X , . ,kO,sO,EO) i s a c o m b i n a t o r y model:
EOab = G(GoF)ab
= F(FoG(GoF) (a)) (b) = F(a)(b),
since FoG=id,
= ab.
Vxax=bx
=e
F ( a ) = F(b)
=e
~
GoF(a) = GoF(b) a
0
0
=
~
b
s i n c e E~ r e p r e s e n t s GoF. I t f o l l o w s by 5 . 6 ( i i )
t h a t R can be expanded t o a Amodel.
( i i ) As i n ( i ) d e f i n e G E ( f ) = E a f actually stable:
~ =E
E
and E ~ = G ( G o F ) .F i r s t n o t e t h a t
F (~ E ~ ) ( E ~= ) F(G(GoF)) (G(GoF))
E~
is
= E ~ .
Moreover AG.E
and XE.G a r e i n v e r s e o f each o t h e r : G ( f ) = EGaf = F O G ( G O F ) ( ~ ~ )
G EG
= GoF(af) = G ( f ) ; = GE(GEoF) = Ea
E
GE
= EE,
since
E
GEoF
represents G O F :
GEoF(b) = EaF(b) = Eb, = E.

( i i i ) R i s a c a t e g o r i c a l Amodel
c)
t h e r e a r e u n i q u e k , s m a k i n g W i n t o a Amodel t h e r e a r e u n i q u e k , s , ~m a k i n g W i n t o a s t a b l e c o m b i n a t o r y model there i s a unique s t a b l e E t h e r e i s a u n i q u e G s a t i s f y i n g 1,2
in (i).
0
F i n a l l y some r e s u l t s a b o u t t h e c a t e g o r i c i t y o f two Amodels. The argument is taken from Longo C19831, where the result i s shown in a general setting.
5.9
THEOREM. ( i ) (Bruce, Longo).
Pw i s a c a t e g o r i c a l Amodel.
231
Lambda Calculus and its Models ( i i ) Pw i s n o t a c a t e g o r i c a l c o m b i n a t o r y a l g e b r a . Proof. 
( i ) D e f i n e x E P w t o be s a t u r a t e d i f ( n , m ) E x
A
enEen,
=)
(n',rn)Ex.
Step 1 . Assume x , x '
a r e s a t u r a t e d and Vy xy = x ' y , Then x = x ' .
Proof.
*
mExen=x'e
*
3en, c e (n'm) E x '
*
(n,m) E x ' , by s a t u r a t i o n .
(n,m)Ex
T h e r e f o r e x c x ' ; hence x = x ' by symmetry. Now l e t G s a t i s f y 1,2 o f 5.8 f o r Pw. Step 2.
( i ) a={(n,m)}.)aEiG(F(a)).
(ii) Vxf(x)zg(x)
Proof.
01
*
G(f)zG(g).
( i ) G(F(a))en = aen = { m l
*
3en, s e n ( " ' ,m) E G(F(a))
*
m E G ( F ( a ) ) e n ' = ae
*
n = n ' , o t h e r w i s e aen,
*
(n,m) E G(F(a))
n'
=0,
.
( i i ) Assume Vx f ( x ) _ c g ( x ) . Note t h a t
G(f)
= lG graph(f).
I mEf(en)}
where g r a p h ( f ) = {(n,m)
i s t h e s t a n d a r d g r a p h f o r Pw.
Therefore G ( f ) = lG graph(f)
5
lG graph(g)
uz
= G(g). Step 3. G ( f )
i s saturated.
P r o o f . Assume Then (n',m)
( n , m ) E G ( f ) and e n c e n I .
E GoF({(n',m)}),

c GoF({(n,m)}),
5
GoF(G(f)),
= G(f). F i n a l l y , l e t G,G'
by 2 ( i ) , by 2 ( i i ) ,
by 2 ( i i ) ,
0,
s a t i s f y 1,2 o f 5.8.
Then
G ( f ) ( x ) = f ( x ) = G ' ( f ) (x)
*
G(f) = G ' ( f ) ,
=)
G = G'.
T h e r e f o r e by 5 . 8 ( i i i )
by 3,1
,
Pw i s a c a t e g o r i c a l Amodel.
3 , k = [ K J P w i s s a t u r a t e d . C l e a r l y k f 0 , s a y (n,m) E k . L e t e n 9 e n l . Then ( n ' , m ) E k and k ' = k  { ( n ' , m ) } a c t s e x t e n s i o n a l l y t h e same as k . ( i i ) By ( i ) s t e p
Hence k i s n o t u n i q u e .
0
H. BARENDREGT
238
5.10
[19831). ( i ) DA i s n o t a c a t e g o r i c a l Amodel.
(Longo
THEOREM.
( i i ) As c m b i n a t o r y a l g e b r a s Pw and D Proof.
A
a r e n o t isomorphic.
I bEf(B)}.
( i ) I n DA t h e map G i s d e f i n e d by G ( f ) = { ( B , b )
But
G ' ( f ) = G ( f ) U A a l s o works. ( i i ) By ( i ) and
5.9(i).
119833
I n Longo
0
i t i s a l s o shown t h a t as a p p l i c a t i v e s t r u c t u r e s Pw and D A
a r e f o r c o u n t a b l e A m u t u a l l y embeddable i n t o each o t h e r .
REFERENCES Barendregt, H.P.,
119811
The Lambda c d c u l ~ t b ,k2 6yntaV and 6emaM.tic6, N o r t h H o l l a n d , Amstei.dam.
B a r e n d r e g t , H.P.
and Koymans, C . P . J . ,
[19801 Compaihing dome [ 19803, 287302. Barendregt, H.P.,
dU4deA 06
Coppo, M.,
h b d a cdclLeus mod&,
DezaniCiancaglini,
i n H i n d l e y and S e l d i n
M.,
[19831 A d a e h Lambda model and Rhe compLeeteneA6
06
t y p e a6ignment, J. Symbolic
L o g i c , t o appear. Engeler, E.,
[1981]' Aegebhab and combinatom, A l g e b r a U n i v e r s a l i s , 13, 389392. H i n d l e y , J.R.
[1980]
and Longo, G.,
Lambda cdcu&6 B d 2 6 , 289310.
mod&
and e.xtenbionu&ty,
H i n d l e y , J.R. and S e l d i n , J.P. (Eds.). [19801 To H.B. C u r r y : EdsUyb on combinatoty Academic P r e s s , New Y o r k and London.
2. Math. L o g i k Grundlag. Math.,
Logic, LambdacdcLLeus and d o h m d h m ,
Jacopini, G.,
[19751
P h i n d p i o di M t e n b i o n W n e l cafkolo d e i combinatohi, C a l c o l o 1 1 , no. 4 , 465471.
K a r o u b i , M.,
[1978]
Ktheoty,
Koymans, C.P.J.,
[1983] Mode&
WI
in&odu&on,
06 .the
S p r i n g e r , B e r l i n and New York.
Lambda CaecLLeUd, I n f o r m a t i o n and C o n t r o l , t o appear.
Longo, G . ,
[19831 setRhtheoh&d mod& 06 h b d a c d c u R u s : theohicb, expavlbionb, homo%pkism, An. Math. L o g i c , t o appear. Meyer, A., i.4 a model 06 the h b d a C a e c l l h P, p r e p r i n t , L a b o r a t o r y f o r Comp u t e r Science, 545 Technology Square, Cambridge, Massachusetts 02139, USA.
[19801 What [1981]
Expanded v e r s i o n o f Meyer
[1980], Information and Control 52, 87122.
Statman, R . ,
[I9801 On Xhe & f e n c e
d o s e d .tmi n the t y p e d AcdclLeud I , i n : [ l 9 8 0 ] , 511534. 119821 CompLetenedd , i n v h n c e and Ade~inabLLiAy, J S y m b o l i c L o g i c , v o l . 47, 1 726. 04
H i n d l e y and S e l d i n
.
S c o t t , D.,
[1980]
R e l a t i n g theohie~06 t h e h b d a Caecu&~, i n : H i n d l e y and S e l d i n t19801, 403450.
Lambda Calculus and its Models Visser, A . , [19801 Numehatiom, Xcdcu&h 259284.
and ahithmetic, in: Hindley and Seldin [19801,
239
LOGIC COLLOQUIUM'82 G. Lolli, G . Long0 and A. Marcia (editors) 0 Elsevier Science Publishers B. V. (NorthHolland). 1984
24 1
EXTENDED TYPE STRUCTURES AND FILTER LAMBDA MODELS M. COPPO (1) F. HDNSELL ( 2 )
M. DEZANICIANCAGLINI (1) G. LONGD ( 3 )
D i p a r t i m e n t o d i Scienze d e l l ' I n f o r m a z i o n e , C. M. D ' A z e g l i o 42, 10125 T o r i n o . (2) Scuola Normale S u p e r i o r e , Pisa. (3) D i p a r t i m e n t o d i Scienze d e l l ' I n f o r m a z i o n e , U n i v e r s i t a d i Pisa. ITALY Research p a r t i a l l y s u p p o r t e d b y M.P. I . (Comitato p e r l a Matematica, f o n d i 40%). (1)
An Extended A b s t r a c t Type S t r u c t u r e i s a p r e  o r d e r e d s e t X which i n c l u d e s a l a r g e s t element and a + b, a A b, whenever a, b a r e i n X. Extended Type S t r u c t u r e s (ETS) may be g i v e n o v e r a p p l i c a t i v e s t r u c t u r e s , b y i n t e r p r e t i n g t h e p r e  o r d e r and " A " by s e t i n c l u s i o n and i n t e r s e c t i o n , r e s p e c t i v e l y . F o r any ,D model o f 1  c a l c u l u s , t h e c l a s s o f b a s i c open s e t s , w i t h r e s p e c t t o t h e S c o t t t o p o l o g y , forms an ETS. The s e t o f f i l t e r s o f an ETS ( f i l t e r domain) i s an a l g e b r a i c complete l a t t i c e and may be t u r n e d i n t o a continuous a p p l i c a t i v e s t r u c t u r e . Domain w h i c h a r e models o f Acalculus ( f i l t e r Amodels) a r e c h a r a c t e r i z e d . A c h a r a c t e r i z a t i o n i s a l s o g i m Amodels which a r e r e f l e x i v e domains, t h a t i s which a r e domains where t h e s e t o f t h e c o n t i n u o u s f u n c t i o n s i s a r e t r a c t i o n . As a m a t t e r of f a c t , n o t any f i l t e r ?model t u r n s o u t t o be a r e f l e x i v e domain. I n any f i l t e r h o d e l t h e i n t e r p r e t a t i o n o f a t e r m i s an element o f a t y p e ( s e t ) o f d a t a , as usual, as w e l l as a c o l l e c t i o n o f types; namely t h e f i l t e r o f t y p e s assigned t o i t by t h e t y p e assignment t h e o r y determined by t h e a s s o c i a t e d ETS. Moreover the f i l t e r &model i n [ 2 1 i s shown t o be i s o m o r p h i c , as a f i l t e r model. Also, domain, t o an e x p l i c i t l y g i v e n s u b s t r u c t u r e o f a D, any ( c o u n t a b l e ) a p p l i c a t i v e s t r u c t u r e may be i s o m o r p h i c a l l y embedded i n t o t h i s f i l t e r domain.
0.
INTRODUCTION
Type symbols a r e used i n v a r i o u s areas of Mathematical L o g i c and Computer Science as a f o r m a l r e p r e s e n t a t i o n o f c o l l e c t i o n s o f f u n c t i o n s , o f f u n c t i o n a l 5 o v e r f u n c t i o n s etc... I n Recursion Theory i n h i g h e r t y p e s one u s u a l l y b e g i n s w i t h j u s t one atomic t y p e symbol, 0 say, t o be i n t e r p r e t e d as t h e s e t o f n a t u r a l numbers N , and t h e n d e f i n e s t h e s e t o f t y p e symbols as t h e s m a l l e s t s e t c l o s e d under " + ' I . The meaning o f 0 0' i s t h e s e t o f f u n c t i o n s f r o m N t o N (see, f o r example, [131). Thus t h e i n t e r p r e t a t i o n o f t y p e s i s f i x e d and i t g i v e s t h e Type S t r u c t u r e as c o l l e c t i o n s o f f u n c t i o n s o v e r N e t c . i n any f i n i t e type. The use o f t y p e s f o r s t u d y i n g f u n c t i o n a l p r o p e r t i e s o f terms o f (untyped) 1  c a l c u l u s i s due t o C u r r y (+) [51. C u r r y ' s t y p e s (which s h o u l d be b e t t e r c a l l e d " t y p e schemes") a r e j u s t s y n t a c t i c o b j e c t s , b u i l t f r o m a s e t o f v a r i a b l e s by an o p e r a t o r " + I ' o f t y p e f o r m a t i o n . They a r e assigned t o 1  t e r m s
..,
(+)
While we were w r i t i n g t h e f i n a l v e r s i o n o f t h i s paper, t h e was saddened by t h e d e a t h o f P r o f e s s o r H a s k e l l B. Curry. P r o f e s s o r C u r r y s e v e r a l times. Besides h i s c o n t r i b u i t i o n s , a landmark f o r L o g i c , we c o u l d a l l admire h i s l i v e l y presence, h i s enthus iasm.
l o g i c community Most o f u s met which have been and encouraging
2 42
M.COPPO ET AL.
by formal assignment r u l e s . In t h i s way d i f f e r e n t types can be assigned t o t h e same 1term. This i s the main d i f f e r e n c e between Curry's approach and t h e typed Acalculus 1111, where types a r e b u i l d a s in Recursion Theory and each term has a uni ue type. In [ 21 a n d ? 3 I a conservative extension of Curry's system i s g i v e n . This i s done by allowing e x p l i c i t l y t h a t t o a l  t e r m i s assigned more than one type. More formally a new o p e r a t o r " 4 ' ' of type formation and t h e " u n i v e r s a l " type w a r e introduced. The main f e a t u r e of t h i s extension i s t h a t types c h a r a c t e r i z e completely t h e functional behaviour of Aterms, see [ 3 I ( n o t i c e t h a t any 1term has a t l e a s t one t y p e ) . S c o t t [181 gave a mathematical semantics f o r Curry's types which can be n a t u r a l l y extended t o t h e new types in [ 2 1 . Given an a p p l i c a t i v e s t r u c t u r e, types a r e i n t e r p r e t e d a s s u b s e t s of 0 , where, f o r A,BcD, A + B = { dcD PdecA d .eeB 1. Moreover " A " i s i n t e r p r e t e d a s set i n t e r s e c t i o n and I, w" a s t h e whole s e t D. By t h i s i n t e r p r e t a t i o n one has i n c l u s i o n r e l a t i o n s between t y p e s , which a r e represented in [ 2 1 by t h e (formal) r e l a t i o n " L " . A t y p i c a l case i s t h a t i f 0'50and T(T' then U+TLO' + T I ( c f r . 1.1.7): i f 0' i s smaller than u and r i s s m a l l e r than T', then t h e "functions" from u t o T a r e l e s s than those from u ' t o T I . On t h e o t h e r hand one may think i n a dual way. S t a r t i n g from t h e f a c t t h a t a type r e p r e s e n t s , in some sense, a domainrange information about a Aterm, one may c o n s t r u c t a 1model in which t h e i n t e r p r e t a t i o n of a term i s t h e s e t of i t s types. Since t h e s e s e t s a r e closed under " A " and " 5'' (upward), they turn o u t t o be f i l t e r s . This c o n s t r u c t i o n i s e x p l i c i t l y done in [ 2 1 , but i t i s , t o some extend, a common f e a t u r e of a l l 1models defined by P l o t k i n ' s technique [141. Take, say, S c o t t ' s Pw model [191 o r Engeler's D A model [ 6 1 . There any d c DA i s a s e t of " i n s t r u c t i o n s " such a s ( B e b ) , where B E D A i s a f i n i t e s e t . I f ( g * b ) ~ d , then BLecD implies b f d  e , t h a t i s d c & + { " b } & D A , A where =Id cDAl c c _ d > . Thus ( B * b ) i s c l e a r l y r e l a t e d t o t h e type k + { i ~ l .This w i l l be used in several places i n t h i s paper and more formally s t u d i e d , p a r t i c u l a r l y when dealing with D, models. In t h e present paper t h e notion of types and t h e i r i n c l u s i o n p r o p e r t i e s a r e a b s t r a c t l y formalized i n t h e d e f i n i t i o n of Extended Abstract Type S t r u c t u r e (EATS). Actually EATS a r e information systems i n t h e sense of S c o t t [211 ( c f . 1 . 9 ) . I t i s i n t e r e s t i n s t o consider EATS given over a p p l i c a t i v e s t r u c t u r e s (they a r e c a l l e d ETS). ETS can be viewed a s i n t e r p r e t a t i o n s ( i n t h e sense of S c o t t ) of formal types. Continuous a p p l i c a t i v e s t r u c t u r e s over EATS ( f i l t e r domains) a r e defined. We then i n v e s t i g a t e embeddings and isomorphisms between f i l t e r domains. In p a r t i c u l a r , t h e f i l t e r domain defined in [21 i s shown t o be " u n i v e r s a l " in t h a t any f i l t e r domain i s isomorphic t o the range of a c l o s u r e operation which i s an element of IFI. Moreover we o b t a i n simple r e l a t i o n s between t h e p r o p e r t i e s of "2 of an EATS and the c l a s s of r e p r e s e n t a b l e f u n c t i o n s over the a s s o c i a t e d f i l t e r domain ( c f . 2.13). Some f i l t e r domains can a c t u a l l y be turned i n t o models of type f r e e 1calculus ( f i l t e r Amodels). An i n t e r e s t i n g c l a s s of them (which has a simple c h a r a c t e r i z a t i o n in terms of " < " ) i s t h e c l a s s of f i l t e r domains i n which a l l continuous functions a r e representable. However t h e r e e x i s t a l s o f i l t e r Amodels i n which not a l l continuous f u n c t i o n s a r e r e p r e s e n t a b l e ( s e e 4.11). Embeddings and isomorphisms between r e f l e x i v e domains ( c f . [ l n ) and f i l t e r h o d e l s a r e s t u d i e d : i n p a r t i c u l a r any Dmspace i s isomorphic t o a ( s u i t a b l e ) f i l t e r h o d e l . L a s t l y j u s t using ( e a s i l y axiomatizable) a b s t r a c t 'I<'' r e l a t i o n s between formal type, we c o n s t r u c t some f i l t e r 1models isomorphic toD, spaces. In one of these D ,spaces t h e " u n i v e r s a l " f i l t e r isomorphically embedded.
1model
can be
2 43
Extended Type Structures and Filter Lambda Models 1.
EXTENDED TYPE STRUCTURES
1.1. D e f i n i t i o n . An Extended A b s t r a c t Type S t r u c t u r e (EATS) S i s a s t r u c t u r e < X , ~ , A . + , ~ > , where X i s a set,weX,"A"and 14diare t o t a l f u n c t i o n s f r o m X x X t o X and k " i s a p r e o r d e r r e l a t i o n on X s a t i s f y i n g : 1. a L W 2. w 5 w+w 3. a L a A a 4. h b l a aAb5b 5. (a;b) A +c) L a +(bhf) 6. a s . , b d * a h b c a ' h b 7. a ' z a , & ' * a + b c a ' + b ' e X. where a,b,c,a',b'
(7
L e t a ?.b i f f a5bQ. Observe t h a t w w + w
and a
A
(bAC)
(ahb)hc.
...,in}, then
NOTATION: I and J w i l l always be f i n i t e s e t s o f i n d i c e s . I f I= {il, ,A a . means a . A a
I
'
11
i2/\
i;
1.2. EXAMPLES. ( i ) L e t T be t h e s e t o f f o r m a l t y p e s b u i l t f r o m w a n d a ( c o u n t a b l e ) s e t At= {$o, 4 l,. 1 o f t y p e v a r i a b l e s by t h e ( s y n t a c t i c ) o p e r a t o r s
..
" +'I
and
"A"
o f 1.1 t h e n
o f type formation. F=is
I f ''zO" i s t h e minimal p r e o r d e r s a t i s f y i n g 17 t h e f r e e EATS o v e r g e n e r a t o r s
...
4 0 . 4 ~ ~
defined i n [ 21. (ii)Consider $ =where P i s t h e s e t o f w . f . f . of ( p r o p o s i t i o n a l ) d e r i v a t i v e + A  l o g i c [16, p. 2851 , aeP and "(I' i s d e f i n e d by " p l q i f f p t q " . 9' i s an EATS s i n c e 17 t r i v i a l l y h o l d . and
Our " c o n c r e t e " EATS w i l l always be g i v e n o v e r an a p p l i c a t i v e s t r u c t u r e and " A " w i l l be ( i n t e r p r e t e d b y ) s e t t h e o r e t i c i n c l u s i o n and i n t e r s e c t i o n .
"2'
1.3. D e f i n i t i o n . ( i ) L e tbe a ( p a r t i a l ) a p p l i c a t i v e s t r u c t u r e and A , B g . Define then A+B = { d r 01 VeeA d.eeB 1 €PO (if i s a p a r t i a l o p e r a t i o n , b y d.eeB we mean: I'de i s d e f i n e d and d e B " ) . ( i i ) L e t be an a p p l i c a t i v e s t r u c t u r e . An Extended T e S t r u c t u r e (ETS) ( o v e r D ) i s an EATS S = < p, 5 , n, +,D>, where PSPD, C_ a n d K p a r e s e t i n c l u s i o n and s e t i n t e r s e c t i o n and B t P .
"."
I n o t h e r words an ETS i s a s e t o f subsets o f an a p p l i c a t i v e s t r u c t u r e , n o t c o n t a i n i n g t h e empty s e t , and c l o s e d under " n " and ' I + ' ' . I t i s t h e n easy t o check t h a t t h e c o n d i t i o n s o f 1.1 a r e s a t i s f i e d (indeed, "L'" i s a p a r t i a l order). 1.4. EXAMPLE. ( i ) L e t
i n [ 1 9 1 ) and P = I A ' d d e A } . T h e n < P , E , n , + , D > i s an ETS. (ii)Each c o l l e c t i o n o f a l g o r i t h m s as d e f i n e d i n [ 1 2 1 i s an ETS c l o s e d under i n f i n i t e intersection. L e t T be t h e s e t o f f o r m a l t y p e s as d e f i n e d i n 1 . 2 ( i ) . Given any EATS S o v e r a c o u n t a b l e s e t X we can d e f i n e t h e f o r m a l t h e o r y o f S b y a ( s u i t a b l e ) o r d e r r e l a t i o n on T. 1.5. then
Definition.
( i ) L e t T be t h e s e t o f t y p e s o f 1 . 2 ( i ) . If u ,TET, i s a formula ( 1 ) . (ii)A t y p e t h e o r y T i s any s e t o f f o r m u l a s c l o s e d under 17 o f 1.1 p l u s t r a n s i t i v i t y and r e f l e x i v i t y . u c T 'I s t a n d s f o r ULTET. UZT
M.COPPO ET AL.
2 44
( i i i ) I f 1 i s any s e t o f f o r m u l a s t h e n T ( z ) i s t h e t h e o r y generated by z . < z s h o r t f o r i T [ 2)(iv) S (T)
.
.
For any t y p e t h e o r y 1.2(i). 1.6.
is
T, S ( T ) i s t r i v i a l l y an EATS. L e t To be t h e f r e e t h e o r y o f
A , + ,w> Definition. ( i ) L e t Ss < X , ( , i s a f u n c t i o n V: T + X such t h a t :
be
an
EATS.
Then
a
2. V( O A T ) = V ( U) A V ( T) 3. v(U*T)= v ( U)* V(T). We say t h a t < S,V> i s a t e model (a c o n c r e t e t y p e model when S i s an ETS). (ii)If<S.V> i s a t y p h t s t h e o r y TV i s g i v e n by TV = { O T I V ( O ) G v(T)}*
E.VT
stands f o r
U ~ T
Tv.
N o t i c e t h a t g i v e n any EATS S, we can always f i n d many V:T+X such t h a t <S,V> i s a t y p e model. C l e a r l y , i f X i s c o u n t a b l e , V can be made s u r j e c t i v e . O f course, t h e c o n d i t i o n on c o u n t a b i l i t y may be dropped i f one t a k e s t h e s e t A t o f atoms o f the desired cardinality. F i.n a l l y , i f V i s o n t o , Obviously Tv= T (Tv) and ToETV ( i . e . ''2; extends " 3 " ) one c l e a r l y has
z v , , + A
,w>
< X,L,
I
A
+ ,
@>
.
Some more work can be done w i t h EATS, l o o k i n g a t c o l l e c t i o n s o f t h e i r subsets. be an EATS. D e f i n i t i o n . L e t S= <X, < A , *,w> An a b s t r a c t f i l t e r x o f 3 i s a non empty subset o f X such t h a t : 1. a.be x =. h b e x 2. a; x, a 3 * k x . ( i i ) If AGX, t A i s t h e a b s t r a c t f i l t e r generated by A. I f A= {a), f o r fIa1. ( i i i ) IS1 i s t h e s e t o f a b s t r a c t f i l t e r s o f S ( f i l t e r domain o f S ) . 1.7. (i)
I f S i s an ETS, IS1
i s clearly the set o f f i l t e r s o f
f a stands
S i n t h e u s u a l sense.
1.8. LEMMA. < IS1 & > i s a complete a l g e b r a i c l a t t i c e , where f w and X a r e t h e l e a s t and t h e l a r g e s t elements ( r e s p e c t i v e l y ) . Moreover i f x,ye I S I : (i) x w = t (xL5) ( i i ) xny = X ~ Y ( i i i ) IfA G l S l i s a d i r e c t e d s e t , t h e n U A = U A . ( i v ) The finite elements are exactly the principal filters, i.e. x = u { f a 1 facx 1 Proof. Easy. 0 1.9. REMARKS. (i) EATS a r e i n f o r m a t i o n systems i n t h e sense o f S c o t t 1211. I n f a c t , an EATS s<X,< , A , *, w> i s an i n f o r m a t i o n system (X,u,Con,+) where Con c o n s i s t s o f a l l  f i n i t e subsets o f X and, i f A = {al, a n } , Atb iff a,&
... ~a~
...,
5 b (and +Ib i f f
w~
b ) . Moreover
IS1 i s t h e s e t o f elements o f
t h e c o r r e s p o n d i n g i n f o r m a t i o n system. ( i i ) Any ETS < p , c,n , *,D > i s a neighbourhood system i n t h e sense o f [ Z O I . Moreover i f we d e f i n e : AfdB * deA+B (where deD and A , W P ) t h e n f d i s an approximable mapping, as d e f i n e d i n [201.
245
Extended Type Structures and Filter Lambda Models FILTER DOMAINS
2.
This s e c t i o n mainly deals w i t h p r o p e r t i e s o f f i l t e r domains ( o f EATS), viewed as a p p l i c a t i v e s t r u c t u r e s . I n t h e sequel complete l a t t i c e s w i l l always be considered w i t h t h e S c o t t topology ( c f . [17]). D e f i n i t i o n . ( i ) I f D i s a complete l a t t i c e ( w i t h respect t o "I")and " ' " : DxD+D i s continuous, thenis a continuous a p p l i c a t i v e s t r u c t u r e . ( i i ) A continuous a p p l i c a t i v e s t r u c t u r e i s a l g e b r a i c i f f D i s algebraic.
2.1.
Given any EATS structure. 2.2.
S, one may t u r n Is1 i n t o an a l g e b r a i c continuous a p p l i c a t i v e
D e f i n i t i o n . L e t S be an EATS. Define: x IS1 + IS1 by xy = { b 13aEy a + bexl.
"'":ISI
2.3. (ii)
LEMMA. ( i ) x , y r I S I * x ' y ~Is!i s an a l g e b r a i c continuous a p p l i c a t i v e s t r u c t u r e .
Proof. Routine (cf.Lemma 1.8.).
0
REMARKS. ( i ) L e t T be a type theory and S ( T ) as i n 1.5. Using2.4. one can e a s i l y show t h a t T i s t h e theory o f a concrete t y p e model. J u s t d e f i n e vT (oi)= I X E I S ( T ) I 1 oiex 1 S * ( 1) = <;(T) , C , n ,+,IS ( T ) I > Then an easy i n d u c t i o n shows t h a t V T ( 0 ) = I x E [S(T)[I x 1
(2).
and t h a t <S* (T), VT > i s a type model whose theory i s e x a c t l y T ( c f . Theorem 1.10 o f [ 201). as defined i n [ 61). Given a s e t A, l e t XA be t h e ( i i ) (Connections w i t h, c l o s u r e o f A " h ) ( w h e r e o s A ) under ' ' + I ' and "n."Then, i f SA =<XA, 5, A , +,w >, i t can be e a s i l y proved t h a t < I S A ~ : ,E>~
,C > ( 3 ) .
Define v : XA +DA by v (W)' P v (a)= {a} f o r a l l aeA v (b +c)= { v ( b ) + d I d c V ( C ) 1 v (br, c ) = V(b)uv(C) and v* : I S A ~ DA by +
v*(x)="
cc x
v(c).
A r o u t i n e c a l c u l a t i o n shows t h a t v* i s an embedding. As usual, i fi s an a p p l i c a t i v e s t r u c t u r e t h e s e t o f representable f u n c t i o n s over i s given by: .(DiD)= { f : D'+D 13x'D VyeD xy = f ( y ) } . Clearly, i f < D 1  , L > i s a continuous a p p l i c a t i v e s t r u c t u r e , then (D+D) C C(D,D), t h e s e t o f continuous functions from D t o D. I f we d e f i n e F ( x ) ( y ) = x  y then F i s a continuous map o f D i n t o C(D,D) (onto (D+D)). Notice t h a t ( D + D ) i s a complete l a t t i c e by t h e c o n t i n u i t y o f F.
2.5.
D e f i n i t i o n . (i) A r e  r e f l e x i v e domain i s a t r i p l eD i s a complete*lattice (2) FE C(D,C(D ,O)) and GcC((D+D), 0 ) (where (D+D)=F(D)) ( 3 ) F O G = i d (4).
(1)
such t h a t
M.COPPO ET AL.
246
!
i s algebraic i f f D i s algebraic.
domain 
REMARK. I fi s a p r e  r e f l e x i v e domain t h e n G O F i s a r e t r a c t whose 2.6. range i s i s o m o r p h i c t o ( D + D ) . I f i s a d d i t t i v e ( c o a d d i t i v e ) t h e n G O F i s a closure (projection). < I S I , * > can be t u r n e d ( i n more t h a n one way i n g e n e r a l ) i n t o a p r e  r e f l e x i v e domain. However, i t i s u s e f u l t o c o n s i d e r a p a r t i c u l a r c h o i c e o f G. B u t we f i r s t need a lemma.
2.7. (1) (2) (3)
LEMMA. L e t S be an EATS and X E I S I . Then t h e f o l l o w i n g a r e e q u i v a l e n t : a+bex bex  t a a+ b e t { c 4 I dex. t c }.
Proof. (1) * ( 2 ) . By d e f i n i t i o n o f ' I  ' ' . c + bex * a+ bex ( s i n c e c + b j a + b ) . (2) * ( 1 ) . box  ? a 3ccta ( 3 ) * ( 1 ) . By assumption f o r some I ( 5 ) A c i + d i i a + b =)
a n d V i € 1 di (2)
=.
EX
.fci
. Thus,
I by ( 2 ) * ( l ) ,V i
E
I
ci +diex
and t h e n a+ bex.
(3). T r i v i a l . 0
The lemma suggests how t o o b t a i n , g i v e n an EATS, a c a n o n i c a l G. 2.8.
THEOREM. L e t S be an EATS. D e f i n e
feC(ISI
, Isl),
G,(f)=tIa+b
L e t Go be t h e r e s t r i c t i o n o f ,G
I
".'I
bef(ta)}
( a n d F ) as above a n d s e t , f o r
. Then
FoG,:id.
t o (IS1 +IS1 ) ,i s a c o a d d i t i v e
p r e  r e f l e x i v e domain. Proof. L e t s = < X , < , A , + , W > and f e C ( I S I , I S I ) . Since { t a l a e X } i s t h e s e t o f f i n i t e elements oPIS1 by 1 . 8 ( i i i ) one has f ( x ) = u { f ( t a ) ] . Thus, f o r a l l x e IS1 aex f ( x ) ={bl 3aex bef(fa)} c{bl 3aax a+be G (f)) G*(f).x. That i s fLF,G,
(f).
Note t h a t i f f e ( I S 1 + I S I ) , f = F ( z ) say, then, by Lemma 2.7, b e f ( t a ) = z * t a . Thus, i n t h i s case, one a c t u a l l y has f = FaG,,(f). Moreover, t a k e bi
E X
?ai
CE
, ieI.
Eo
F ( x ) = ?{a+ b
By Lemma 2.7
I bexfal.
Vie1 ai+bi
I t i s a r o u t i n e c a l c u l a t i o n t o show t h a t the Scott topology. n
Then, f o r some 1 , A a . +b I
E X
&
a + b e G o ( f ) =.
which implies
CEX.
i
Thus
< c with
i$0
FLid.
and F a r e c o n t i n u o u s w i t h r e s p e c t t o
I f S i s such t h a t , f o r some G ' ,i s a non t r i v i a l r e f l e x i v e domain t h e n we would o b t a i n a A  c a l c u l u s model ( c f . [ 1, 18.11 ). We w i l l say t h a t i s a f i l t e r Amodel i f f i t i s a &model. A f i l t e r Xmodel i s n o t n e c e s s a r i l y a r e f l e x i v e domain, c f . 4 . 1 1 ( i i ) . EXAMPLE. L e t F= be as i n 1 . 2 ( i ) , t h e n i s a c o a d d i t i v e r e f l e x i v e domain (see t h e remark a f t e r Lemma 2.13). I f we d e f i n e G'(f)= G o ( f ) u A t where f e ( l FI +I FI ) and A t i s d e f i n e d i n 1 . 2 ( i ) we can e a s i l y p r o v e
2.9.
t h a t
i s an a d d i t t i v e r e f l e x i v e domain ( j u s t mimic [ 9 ] f o r a p r o o f . ) .
247
Extended Type Structures and Filter Lambda Models
2.10.
REMARKS.
( i ) Theorem 2.8 a c t u a l l y proves t h a t
i s a continuous
r e p r e s e n t a t i o n between C ( ISl,ISI) and IS1 a c c o r d i n g t o t h e d e f i n i t i o n o f 1151, i . e . F o G& i d and 0 F 5 id.
(ii)F o & i s a c l o s u r e o f C(ISI ,IS1 ) whose range i s (IS1 +IS1 ) (use f L F o C, ( f ) f o r feC(IS1 ,I 9 ) and f = F 0 G, ( f ) f o r f e (IS1 + IS I)). ( i i i ) L e t Dk be as i n 2 . 1 1 ( i ) .
I t i s easy t o show t h a t i fis
i( Dk )s( D +Dlk. (G^(
(coaddi t t i v e ) a1 g e b r a i c p r e  r e f 1 e x i ve domain t h e n
i0((
an a d d i t i v e
(D
+
D l k )GDk 1.
Therefore I SI+lSI ) ) c Is f o r a l l EATS S. Moreover t a k e F as i n 2.9, t h e n k f( I Flk)cC( IF1 , IF1 ) k y s i n c ei s an a d d i t t i v e r e f l e x i v e domain. I t i s
k
easy t o see t h a t t h i s i s n o t t r u e f o r a l l EATS S . There a r e some s i m p l e c o n d i t i o n s on EATS which correspond t o t h e d e f i n a b i l i t y o f c l a s s e s o f c o n t i n u o u s f u n c t i o n s (among them, t h e c l a s s o f a l l continuous functions). 2.11. D e f i n i t i o n . (i) L e t D be an a l g e b r a i c complete l a t t i c e . D e f i n e Dk'ICeDI c i s f i n i t e I .
(ii)L e t D and D ' be a l g e b r a i c complete l a t t i c e s . A s t e p f u n c t i o n f a b : D + D ' d e f i n e d by
1
fab(c)'
is
b i f a& I' o t h e r w i s e
where aeDk, I X D l k and
1'
i s t h e l e a s t element o f D ' .
The f i n i t e elements o f C ( D , D ' ) a r e e x a c t l y g i v e n by t h e f u n c t i o n s Uf,.,, where a.eD b. E D ' i e I . Note t h a t faibrc)=Y {bil ai cl. I ii 1 k' k' Thus (*) iff J = { i / a i r c l # m a n d dL Ubi. J 2.12. D e f i n i t i o n . L e t S an EATS. We d e f i n e t h e f o l l o w i n g c o n d i t i o n s on S : C1) ai +bi z c + d h bi 'd
y
fcdbyfaibi
+
C2) C3)
C
I
a +b_u td and dl;w=r c a and b y A ai+bi 5 c+d=dl;w*J={i Iczai
I
l#S@3
biId.
C o n d i t i o n (*) i s c l e a r l y e q u i v a l e n t t o C3, where we t a k e ISlk and ,G
as d e f i n e d
i n 2.8. A d i f f e r e n t f o r m u l a t i o n o f C3 w h i c h w i l l be used i n many p r o o f s i s : a 1. + b i z c +d afidl;w93J#@ C I c z f ai and A bi(d. J C l e a r l y C3*C2*Cl.
9
A t y p e t h e o r y T s a t i s f i e s Cl(C2 o r C3) i f f S ( T ) s a t i s f i e s Cl(C2 o r C3). 2.13.THEOREM.Let S be an EATS. Then ( i ) satisfies C I * ( 1.~1) contains a l l constant functions. ( i i ) S s a t i s f i e s c2 9 ( I S [ + ISI) c o n t a i n s a l l s t e p f u n c t i o n s . ( i i i ) S s a t i s f i e s C3 0 (1Slt I S l ) = C ( ISI, 1st) ( i . e . < IS1 ,F,Go> i s a r e f l e x i v e domain and, t h u s , a f i l t e r Amodel). P r o o f . We p r o v e o n l y ( i i i ) . The p r o o f s o f (i) and (ii) a r e s i m i l a r and e a s i e r . Let ai+bi(c+d. Take f e C ( I S I , 6 I) d e f i n e d by f ( x ) = $ K t b i I t a i s x l = L e t now t { b i l a + x , i e I } . I t i s t h e n easy t o show t h a t G o ( f ) = t / t a i + b i .
+
1
J = { i Ic Lai
c +de Go(f).
(*).
1 ( t h u s C 5 9 a i ).
Then
L a s t l y , d7.w i m p l i e s J # a.
Go(f)* tc= t A b J i
and
3 9Id,
since
I t i s enough t o p r o v e t h a t a l l sups o f f i n i t e s e t s o f s t e p f u n c t i o n s a r e
M. COPPO ET AL.
248
r e p r e s e n t a b l e . Then t h e p r o p e r t y f o l l o w s f r o m t h e f a c t t h a t ( IS I + I S I ) i s a complete l a t t i c e . ) . We p r o v e t h a t L e t f be d e f i n e d as above (observe t h a t f = & I taifbi x f = k ( f ) = f f a i * bi r e p r e s e n t s f, i . e . t h a t 'dye IS1 xf * y = f $ bi, where
J =Iila.ey,
In
'ieI}.
1
fact
dexf'y
* 3cey
;aTbi:c+d.
Now,
if
J ' = {iI c i a . 1,' we have t h a t J'c_ J , s i n c e c L a i by C3. T h e r e f o r e
aie y. Thus J ' P O and biid d. Moreover i t i s easy t o p r o v e t h a t A b. < x * y and t h e J I f
3 bi
result follows.0
C o n d i t i o n C3 i s t h e c o n d i t i o n o f Lemma 2 . 4 ( i i ) o f [2]. Thus t h e r e p r e s e n t a b l e f u n c t i o n s o v e r < I F I , . > a r e e x a c t l y t h e c o n t i n u o u s ones. We can now g i v e some examples o f s t r u c t u r e s which s a t i s f y o n l y C 1 ( o r C2). B u t we f i r s t need a lemma. 2.14. LEMMA. L e t Sbe an ETS. I f t h e r e e x i s t A,BieP t h a t ASyB and Vie I A 5Z B i, t h e n S does n o t s a t i s f y C3.
( i e I ) such
(y
Bi does n o t
Proof. Observe t h a t , g i v e n any Ce p, n e e d t o belong t o P ) . n
Bi+
C=("I B i) + C G A + C
2.15. EXAMPLES. ( i ) L e tbe an a p p l i c a t i v e s t r u c t u r e such t h a t Vd,eeD d  e = e. Then any (non t r i v i a l ) ETS S o v e r does n o t s a t i s f y C1. One a c t u a l l y has t h a t VA,BcD: A + A = B + B = D . (ii)L e t be such t h a t de=d. Then any (non t r i v i a l ) ETS S o v e r does n o t s a t i s f y C2, s i n c e vA,BED A+ B = D +B. C l e a r l y S s a t i s f i e s C1. ( i i i ) L a s t l y , we show an ETS which s a t i s f i e s C2 b u t n o t C3. L e t < x ,  >be t h e Kleene a p p l i c a t i v e s t r u c t u r e d e f i n e d by nm = { n 1 (m)
s a t i s f i e s C2. L e t A,B,C,E
. Actually
be non empty subsets o f 2 and EP 2. Then,
i f A + B G C + E , c l e a r l y BGE. Moreover l e t p e g \ E and q E " I k l ( x ) = i f x= r t h e n p e l s e q " i s such t h a t ke A+B b u t i s g i v e n by t h e C2 i s s a t i c f i e d . A X p l e T S o v e r < & , a > 1.4. Namely, by t h e I 1 Recursion Theorem t a k e no such t h a t
m e 2 . Then
{ A G yl
any ETS o v e r
B. I f r e C \ A, t h e n k t C+E. Thus G A and same argument used i n { n o X m ) = no, f o r a l l
no E A } i s an ETS and does n o t s a t i s f y C3 by Lemma 2.14.
A l s o t h e e x t e n s i o n a l i t y p r o p e r t y o f qISI,.> has an easy c h a r a c t e r i z a t i o n i n terms o f t h e p r o p e r t i e s o f S As u s u a l , an a p p l i c a t i v e s t r u c t u r ei s e x t e n s i o n a l i f f VG D a. c = b  c * a=b f o r a,beD.
.
2.16. THEOREM. L e t S*X, 2, A , +,u> be an EATS. ( i ) V'ze I S l x . z = y  z 0 (Va,beX a + b e x * a + b E y ) , f o r x , y l S I . ( i i ) . ISl,*>is e x t e n s i o n a l i f f VaeX 31 a * A b + ci.
I
Proof. ( i )
()
a+bex
i
b e x  f a (by Lemma 2.7) * be . f a * a+iey. Immediate f r o m t k e d e f i n i t i o n o f ''.''. ( * ) Let x a = f { b + c l a
(r).
(ii)
since and
< I S I , * > is
9 bi+
( c )Easy
3.
extensional.
Thus
ae xa,
that
is
By ( i ) x a = f a ,
3 1 Vi € 1 bi+ ci e xa
c. 5 a. 1
from ( i ) . 0
EMBEDDINGS AND ISOMORPHISMS
D e f i n i t i o n . L e t D be an a l g e b r a i c complete l a t t i c e . D e f i n e (i) 'c= {x I c L x } f o r c EDk, t h e cone o v e r a ( f i n i t e ) element.
3.1.
2 49
Extended Type Structures and Filter Lambda Models (ii)
K ( D ) = { z l ceD2.
( i i i ) C ( D ) as t h e c l o s u r e o f K ( D ) under f i n i t e union. As w e l l known, K(D) i s a b a s i s f o r t h e S c o t t t o p o l o g y on D. 3.2. REMARK. C(D) i s i f f C ( D ) i s c l o s e d under
c l o s e d under 'In''.,
, n ,+
,D >
is
an
ETS
'I+
THEOREM. L e tbe an a l g e b r a i c c o n t i n u o u s a p p l i c a t i v e s t r u c t u r e . IfS = i s an ETS t h e n < D , * ,&>c+< I s c [ , ,c_> C < I S I,.,g>. ( i i ) If S K = < K ( D ) , E , n , + , D > i s an ETS t h e n < D , * , G K Proof. ( i ) L e t Emb : D + I S C I be Emb(d)= { X E C(D) I dsX}. The embedding o f D 3.3. (i)
i n t o I S c [ as
lattices
is
trivially
verified.
To
prove t h a t
Emb p r e s e r v e s
a p p l i c a t i o n observe t h a t : X E Emb(a'b) * a*beX
* 3ceO d b
a.ceX,
f o r { a * c I ceDk
ccb} i s d i r e c t e d
w D k d c ae*c+ x k =. X E Emb(a) Emb(b). =)

The r e v e r s e i n t r i v i a l . (ii) L e t Iso: D+lSKI be I s o ( d ) = & K ( D ) I The isomorphism as l a t t i c e s i s a p p l i c a t i o n goes as i n case (i). 3.4.
crd}.
immediate.
The
REMARKS. ( i ) I t can be e a s i l y proved t h a t
defined
as
i n 161,
N o t i c e t h a t K(P,)
are
closed
under
"+
"
that
C(Pu) and
and
and K(DA) a r e n o t c l o s e d under
proof
so
Is0
preserves
C(DA), where D A i s
Theorem 3 . 3 ( i )
applies.
'I+''.
.
(ii) K(I S ( T ) I ) i s c l o s e d under ' I + " f o r a l l t y p e t h e o r i e s T I n t h i s case Theorem 3 . 3 ( i i ) j u s t amounts t o say t h a t i f we b u i l d f i l t e r s o f f i l t e r s we do n o t change t h e c o n t i n u o u s a p p l i c a t i v e s t r u c t u r e (modulo isomorphisms). ( i i i ) I f. K There i s a s i m p l e c o n n e c t i o n between t h e c o a d d i t t i v i t y o f and t h e c l o s u r e o f K(D) under ' I + ' ' . 3.5.
THEOREM. L e tbe an a l g e b r a i c r e f l e x i v e dom;in. i s c o a d d i t t i v e , , * K ( D ) i s c l o s e d under I' K(D) i s c l o s e d under + ' I * 3 6 ' such t h a t r e f l e x i v e domain.
(1) (11)
Proof. de
+
( i ) We p r o v e t h a t Va,trDK
*a +b*=)F(d):
".
.
i s a coaddittive "
I
a + b = G(fab). O b v i o u s l y G ( f a d s a + b .
fab
*G(F(d)) !G(fab) (by c o a d d i t t i v i t y ) . *dlG(f
,d
(ii)D e f i n e a r b by : a L b = >+E, Notice t h a t b d e f i n i t i o n (1) f,bfF(d) * deZ+"b * arbgd
where a , b D
K'
and G ' ( f )  C H a r b l f a , C f ) .
Moreover
M.COPPO ET AL.
250 and
f abC f * a c b LG'(f) (2) Thus F(G' ( f ) ) c K f a'b I f a b & F(G' ( f ) ) 1 = U [ f a b l a + b L G ' ( f ) l , by ( 1 ) CKfabl fabEf} f.
by ( 2 )
=
G'(F(d))=Uabb I f
CF(d) } abU a r b I a+b E d } , b y (1) d.
3.6.
REMARK. Since any
o f Theorem 3.5 i t
i s an e x t e n s i o n a l r e f l e x i v e domain, f r o m t h e p r o o f
f o l l o w s t h a t Va,b e (D,
)k a c b = f a E
From t h e p r e v i o u s r e s u l t s we o b t a i n t h e isomorphism o f any a l g e b r a i c c o a d d i t t i v e r e f l e x i v e domain w i t h t h e f i l t e r Amodel, b u i l t on i t s compact cones. 3.7. D e f i n i t i o n . [ 151 An isomorphism between t h e r e f l e x i v e domainsand i s a p a i r , where w,w> i s an isomorphism between D and D ' such t h a t (1) F'(d)= v 0 F (w(d)) o w (2) G ' ( f ) = v(G(w0 f a v ) ) where d E D ' and f e C ( D ' , D ' ) . (1) i m p l i e s t h a t v and w p r e s e r v e " * ' I . By ( 1 ) and ( 2 ) V d e D v ( G o F ( d ) ) = G ' o F ' ( v ( d ) ) ( c f . t h e n o t i o n o f isomorphism i n [ l , 5.3.21). 3.8. THEOREM. L e t be an a l g e b r a i c c o a d d i t t i v e r e f l e x i v e domain and S = . Then d),F',G'> = < I S I , F,G,>. K K Proof. S Ki s an ETS b y Theorem 3.5. Since < D , F ' >  < I S K I , F > (by Theorem 3 . 3 ( i i ) ) t h e range o f F i s C(IS 1,l Sd ) I i . e . < I S I,F,GO> i s a r e f l e x i v e domain. K K I n 1151 Sanchis n o t i c e s t h a t g i v e n two c o a d d i t t i v e r e f l e x i v e domains, o n l y one o f t h e c o n d i t i o n s o f 3.7 s u f f i c i e s t o have t h e isomorphism. So we a r e done, s i n c e < I SKls,F,Go> i s c o a d d i t t i v e by Theorem 2.8 and c o n d i t i o n ( 1 ) o f 3.7 h o l d s , b y Theorem 3 . 3 ( i i ) . 0 Another i n t e r e s t i n g c l a s s o f embeddings i s d e f i n e d c o n s i d e r i n g f i l t e r s o f EATS b u i l t from tvoe theories. F o l l o w i n g [ f i ' l an element U E I FI i s a c l o s u r e o e r a t i o n i f f i t s a t i s f i e s : G , ( i d ) L u = u 0 u ( where u 0 u =G,(Az. u.(u.z)T). 3.9. (i (ii)
THEOREM. L e t T be a t y p e t h e o r y . Then one has % < I S ( T ) I ,  , c _ > i s i s o m o r p h i c t o t h e range o f a c l o s u r e o p e r a t i o n u e I F I .
Proof.
.
.
( i ) Observe t h a t a b s t r a c t f i l t e r s o f
S ( T ) are abstract f i l t e r s o f
then, a r e c l o s e d u n d e r a p p l i c a t i o n . Thus < I FI , * ,g >.
F and,
i s a substructure o f
25 1
Extended Type Structures and Filter Lambda Models
( i i ) As p o i n t e d o u t i n 1.6, z T extends 2 0 . D e f i n e u = t Io+TI O < T T I E I FI a n d + A d S ( T ) I as t h e f i l t e r generated by t h e s e t o f t y p e s A ( n o t i c e t h a t u i s c l o s e d under lo w h i l e + A i n c l o s e d under
3.We p r o v e t h a t u*A=+A.
+Ac_ u . A i s t r i v i a l . F o r t h e r e v e r s e observe t h a t T E
u.A3Ue
A
*
u
s
O+T
* 30s A * 30 E A * 30 E A =. 3 0 c A
31 V i e 1 u ~ ~ ~ T o ~ ~& + 31 V i e 1 0 ~ i ~ ~ a n d 3 J uz0 c _ I3 u i
33 ui
U L ~ /Jui T
T ~ A .
,
OT+ T
~
~
~
,I~T~ , f (o r~F T satisfies
C3
zT$~iiO~
s i n c e < extends lo T
Obviously u _ > i = t { o + . r l u i o T I
and u = u

3
u. C l e a r l y S ( T ) i s t h e range o f u. 0
Theorem 3.9 proves t h a t i s " u n i v e r s a l " ( i n t h e sense o f [ 191) f o r a l l f i l t e r domains. R e c a l l i n f a c t t h a t each such domain i s t r i v i a l l y isomorphic t o an EATS g i v e n by a s u i t a b l e t h e o r y (see what p o i n t e d o u t a f t e r 1.6). We can a c t u a l l y p r o v e t h a t any a p p l i c a t i v e s t r u c t u r e can be embedded i n t o . 3.10. THEOREM. L e t % .
b e a ( c o u n t a b l e ) a p p l i c a t i v e s t r u c t u r e . Then
Proof. L e t A = { a . I i L l } and x A = { $ i i $ j + $ d a i * a j = Emb: A + I S ( T z
)I by
Emb(ai)=
t$i
a h } . Define
(iL1).
A
Emb(ai)* Emb(a.)= Emb(ai. a .). _> i s t r i v i a l and C_ i s g i v e n J J by t h e m i n i m a l i t y o f iZA, as d e r i v e d f r o m x A ( f o r t h i s some b o r i n g c a l c u l a t i o n s
We c l a i m t h a t
a r e needed. We l e a v e them as an e x e r c i s e ) . Moreover < I S(Tz ) I ; > k < l A
FI
>;
by Theorem 3 . 9 ( i ) .
0
The c o n d i t i o n on t h e c a r d i n a l i t y o f A may be dropped j u s t t a k i n g enough atoms, i . e . t a k i n g A t l a r g e enough and c o n s t r u c t i n g T f r o m i t as i n 1 . 2 ( i ) . 4.
FILTER AMODELS
As a l r e a d y p o i n t e d o u t , any EATS s a t i s f y i n g C3 y i e l d s a f i l t e r Amodel. A c t u a l l y any such Amodel i s g i v e n by a r e f l e x i v e domain, i.e. i t has t h e s t r o n g p r o p e r t y t h a t any c o n t i n u o u s f u n c t i o n i s r e p r e s e n t a b l e . T h i s i s more t h a n what i s r e q u i r e d by an a p p l i c a t i v e s t r u c t u r e t o y i e l d a Amodel. Theorem 4.8 c h a r a c t e r i z e s EATS S such t h a t < S,F,G > i s a f i l t e r Amodel. Theorem 4.11 g i v e s a f i l t e r Amodel, which i s n o t a r e f l e x i v e domain. F o r t h e n o t i o n o f (expanded) combinatory a l g e b r a and Amodel we m o s t l y r e f e r t o [ I 1 s [ 9 1 9 [I01
.
4.1. D e f i n i t i o n . L e t S= <X,(,A,+,W> be an EATS ( r e c a l l t h a t a +b+ c stands f o r a + ( b + c ) ) . D e f i n e K = t { a + b +c I CE t a 1 S = t { a + b +c+d I dcta.tc(tb.tc)} E * t { a  . b +c I CE t a  t b l .

Note t h a t
K,
S a n d 5 have been d e f i n e d j u s t u s i n g G*
o f 2.8.
M. COPPO ET AL.
252
4.2. LEMMA. L e t S be an EATS. Then x  z (y.z) ~ _ S  x . y  z and x.y@x*y. ( i ) Vx,y,ze I S I x g : x . y , i s a combinatory a l g e b r a , t h e n KSK and S C S . Moreover, f o r ( i i ) I f < lSI,,S,K> I=SKK, a+ b € 1 0 a 2 b. Proof. ( i ) By FOG*,
i d (see 2.8).
( i i ) Observe t h a t K . t a  t b = t a i m p l i e s by 2.7 a+b+ccK f o r S. Moreover a + b e I 0 b E I  t a = t a . 0

f o r a l l ceta. S i m i l a r l y

4.3. THEOREM. L e t s be an EATS. I f t h e r e a r e S, K such t h a t < I S I , ,S,K > i s an expanded combinatory a l g e b r a t h e n a l s o d s l , , $,E > i s an expanded combinatory algebra.
Proof. Immediate f r o m 4.2. THEOREM. L e t S = < X , L , A , + , w > 4.4. ( i e 1 ) o n e has(*) /\(bi+ci)+bi+ciza+b*3J
be an EATS. Assume t h a t f o r any a,b,bi,ci a 2 A d . e .&I. J J 3I Then, i fi s a combinatory a l g e b r a , < l S I , . , E > i s a 1model.
EX
P r o o f . F o l l o w i n g [ l o ] , we j u s t need t o show t h a t ( 1 )g.x.y = x  y ( 2 ) vz x.2 = y.z =$ g. x =g.y (3) E.4 = E . As f o r (17, n o t e t h a t g=S(K(S&)). Then use 4.3. As f o r ( 2 ) , observe f i r s t t h a t a + b e E 31 ? a i + b i  + c i ( a + b f i V i E I a 1. < b . 1+ c i
,3I
A(bi+ci) +bi+ci5a+b I =, 35 a L A d .+e by ( * ) J J JTake now begax, t h e n 3 a ~ xa + b E s Use t h e p r e v i o u s argument t o o b t a i n J such
.+,
t h a t a z A d .e.:h.
J J J
d j +e j e y . NOW,
Since a
E
x,
by 2 . 1 6 ( i )
and t h e assumption i n ( 2 ) , V j E J
A ( d j  t e j ) + d j + e j E E , by d e f i n i t i o n .
J
Thus VjEJ dj+ e . E E  Y and b EE_.Y. J ( 3 ) follows from t h e d e f i n i t i o n s . n Note t h a t , i f < IS I . . , E > i s a xmodel, t h e n E =S(K(SKK)) ( c f . 4.1) and, by d o e s n ' t need t o 4.2,s C _ E . However, a l s o i f < l S I ,  , ~ > i s a xm<delSbe a m o d e l , f o r < IS I , ,K,S_> may j u s t be a combinatory a l g e b r x ( o r a x  a l g e b r a ) . Each t y p e t h e o r y T induces a system o f t y p e assignment, i n t h e sense o f [ 2 1 , f o r t h e s e t A o f xterms. By t h i s , Theorem 4.8 c h a r a c t e r i z e s t h e t y p e t h e o r i e s which y i e l d 1models. N o t a t i o n and concepts a r e m o s t l y f r o m [ 21. I n p a r t i c u l a r i f CJ E T and M E A , then UM i s a statement, where u is t h e p r e d i c a t e and M t h e s u b j e c t . A basis i s a s e t o f statements w i t h o n l y v a r i a b l e s as s u b j e c t s . 4.5. D e f i n i t i o n . L e t 7 be a t y p e t h e o r y . The (extended) t y p e induced by T i s d e f i n e d by t h e f o l l o w i n g n a t u r a l d e d u c t i o n system
assignment
253
Extended Type Structures and Filter Lambda Models
( + ) i f x i s n o t f r e e i n assumptions on which T M depends o t h e r t h a n T
W r i t e B t
oM i f
OX.
oM i s d e r i v a b l e f r o m t h e b a s i s B i n t h i s system.
4.6. D e f i n i t i o n . L e t T be a t y p e t h e o r y and S ( T ) be t h e EATS d e f i n e d i n 1 . 5 ( i v ) . F o r any map 5 f r o m v a r i a b l e s o f A t o IS( ~ ) and 1 M E A. define: (i) B =Ioxloe s(x)}
5
(ii) (II b y i n d u c t i o n on t h e s t r u c t u r e o f M) u x f =
UPQ 1 = F ( EPII [Ax. P
T
( UQII
T
I T = ( ,G he E 1 S( T ) I .[ P 1 ) 5 c [ x/el
(see 2.8).
( T h i s i s w e l l d e f i n e d , by t h e c o n t i n u i t y o f F and Note t h a t i f
I f B i s a basis, l e t B h = { o y 4.7.
> i s a xmodel,
I oy
Thus V i
C_
. By i n d u c t i o n on M.
I
I
, by
induction
T
, by
rule ( < ) T
Bd x / t a . l ~ BiP 1
* BPUIaix}t~;P T
* B ta.+Bi 5 1
Xx.P
, by (
The r e s u l t f o l l o w s b y u s i n g (
II J
i s the
[MI
T
5
= Io l B
T
5
toM}.
The o n l y non t r i v i a l case i s M  X x . P .
r
E
then
E B and y z x l .
THEOREM. L e t T be a t y p e t h e o r y . Then
Proof.
).
A
+I).
I ) and ( z T )
(standard)
M. COPPO ET AL.
254
.
3
T By i n d u c t i o n on t h e d e d u c t i o n B g t  oM. We j u s t check when ( + I ) i s used.
The r e s t i s t r i v i a l . Note t h a t i f
t ax1
t h e n we have, by a s h o r t e r d e d u c t i o n , gives the result. 0
BF[x/+alg 8p. The
i n d u c t i v e hypothesis
L e t G o be as i n 2.8. THEOREM. L e t T be a t y p e t h e o r y . Theni s a xmodel * T T lB/X t U+T xx.M * B / x ~ (UXI F TM). Proof. =*. R e c a l l t h a t G o i s t h e r e s t r i c t i o n o f G, t o ( I S ( T ) I + I S( T ) I ) , t h e
4.8.
IS( T ) I t o
IS( T ) I,
i s representable.
Thus t h e
representable functions.
By assumption,
which i s d e f i n e d by a
xterm
semantics o f xterms i n
IS( T ) I i s d e f i n e d e x a c t l y as i n 4 . 6 ( i i ) ,
.
use G o i n s t e a d o f G, Let 4.7
g ( x ) = f { u J u z ~o r B
r
B/x+u+T
T
oxeB}. O b v i o u s l y B F p N * B I\x.M]
XX.MU+TE
any f u n c t i o n f r o m
(using constants),
T
I~N gB
Te
F(Go(f))(fu) f o r T
I
f=AeeIS
(T)
T
* B
/X U { U X } t T M . gB =. The p r o o f o f Theorem 3.5 i n [ 2 ] remains v a l i d , c which r e q u i r e s t h e g i v e n c o n d i t i o n .
4.9. Oefine
except f o r
point
(iii),
we can now g i v e a c l a s s o f f i l t e r M o d e l s , which a r e n o t
D e f i n i t i o n . ( i ) Choose 0
c*=
b y Theorem
r
q x / t u 1‘:’?ince
Using Theorem 4.8, r e f l e x i v e domains.
. Then,
where one may
{ ULU[ $/p]
IUE
A t and PcT such t h a t
T}.
( i i ) I * a n d ? i s short f o r (iii)u ~ i Tf f e i t h e r (1) u 3 T
zz* and &(E*),respectively.
0 does n o t o c c u r i n
p
.
Extended Type Structures and Filter Lambda Models 4.10. (ii)
ui*
LEMMA. ( i )
T
=)
o ~ A a + 6< * A y
. ,...,
d O / p I ~ * T[ $/PI + 6 = ~ * 3pl
pn
255
LT.
o b p l z . . . < p
I i i J i in and each p (1 I h I n ) i s an i n t e r s e c t i o n o f arrows. Fi B&M~=) BL +/PI P T [ 4/01 M . ( i v ) vie1 B/XU{a.X} 6 f3.M and ai+Bi & A y . 4 . * VjeJ B / x U { y . x } p 6 . M . 1 1 J J J J J
.
.
.
(2
2.4(ii) i n [2]). Bi
I n f a c t v j ~ J f a ~ + By.+a.implies ~ i ~ J J
i t j . Therefore
a.X 1
:
from t h e derivations
3K?@SIy
j 3
pi and
f o r a l l i e K one may o b t a i n :
6 fM
ujx a.x
(lo1
1
K
i t
6iM

tBiM
(A1)
6.M
(50)
J
I f case (2.1) YiX
:
obtain
a.X 1
i n 4.9 a p p l i e s , t h e n f r o m t h e d e r i v a t i o n YiX BiM j u s t u s i n g (i*)i n and
a.x
6;M
one may
.
1
I f case (2.2) i n 4.9 a p p l i e s t h e n yi+6i(ai+Bi) QyPdO/Ply f o r a l l 0.y. B h U{ai[ $/PI X } P Bi[ 4/01 4.11. (ii)
[$/p
1.
By d e f i n i t i o n
T h e r e f o r e f r o m ( i i i ) B/xu(aixl@BiM
implies
M.
THEOREM. (i)B~*u+TAx.M BUIUX}PTM. L e t =S , < T , i * , A , +, w > . Theni s
a f i l t e r Amodel.
Proof.
(i)L e t D be t h e d e d u c t i o n showing B ~ U + T A X . M . Assume t h a t , f o r some I,ai+BiAx.M,icI, a r e a l l t h e statements i n D on which u+TXX.M depends and which
a r e c o n c l u s i o n s o f (+I). I t i s t h e n easy t o p r o v e t h a t +ai+6izu+~ i s d e r i v e d f r o m t h e ai+Bi 2.7(iii)
i n [ 21).
,Since u+TAx.M
Ax.M u s i n g o n l y r u l e s ( A I ) , ( A E ) and (2,). (Cf. Lemma
By 4 . 1 0 ( i i )
above,
each uh i s an i n t e r s e c t i o n o f arrows.
3p1
,..., 11, + a i + .1%. <~l~...~p n
,
where
The r e s u l t f o l l o w s by i t e r a t e d a p p l i c a 
tions o f 4.10(iv).
(ii)By (i) and 4.8.
0
S+ does n o t s a t i s f y C2 (and hence C3), w h i l e i t i s easy t o check t h a t p i* $.
f o r 4+ 6
*I
P
+
P
and
0
I* P ,
M. COPPO ET AL.
256
4.12. REMARK. ( i ) Theorem 4.7 g e n e r a l i z e s 3.5 i n 121. By t h i s a l s o t h e completeness r e s u l t i n [2,3.1Dl can be g e n e r a l i z e d . Given a 1model nl =, an environment 5 and a t y p e i n t e r p r e t a t i o n V: T +PD, l e t m,5, VCoM and m,E, V I  B be as i n 1.2 o f 1 2 1 . Now, g i v e n a t y p e theorVy,= , d e f i n e BI=ToM i f f f o r a l l m , s,V, such t h a t T c T one has m, 5, m,E,V k oM. V By t h e same t e c h n i q u e as i n 121, u s i n g 4.7 and 2 . 4 ( i ) one can e a s i l y p r o v e
;
r
r
B I= UM * B F uM. (ii)(Comparing t h e completeness r e s u l t s i n [ 7 1 , [a1 and 121). The p r o o f i n [ 71 uses a t e r m model o f k a l c u l u s , namely t h e s e t o f 1terms up t o 16convertib i l i t y , i.e. m ( A g ) = { [ M l I M i s a Aterm 1 where [ M ] = { N l x p . F M = N }. Then, f o r a g i v e n b a s i s 8, ty es a r e i n t e r p r e t e d by (1)
vH={
mn 5 0 IB+@P
c_
O M }
rn(xe1
where c 0 i s t h e t r i v i a l environment d e f i n e d by c 0 ( x ) = [ o f B, see [ 6 1 . I n 121, t y p e s a r e i n t e r p r e t e d by Vo(o) = I d / o e d } E I F [ . However, g i v e n a b a s i s B, Theorem 4.7 see t h e p r o o f o f 4.8) Vo(o) = ( [ M I
%+
IB'F
implies t h a t ( f o r the d e f i n i t i o n o f
5B'
UM 1.
As a m a t t e r of f a c t , n o t e t h a t V o 5.
and B + i s a v a r i a n t
To
To (2)
XI
and V
EXTENDED TYPE STRUCTURES AND D
H
a r e v e r y much a l i k e .
AMODELS
I n t h i s s e c t i o n we p r e s e n t some i n t e r e s t i n g t y p e t h e o r i e s o b t a i n e d by i n t e r p r e t i n g t y p e s i n wellknown a p p l i c a t i v e s t r u c t u r e s , i . e . D, 1 models o f S c o t t [ 171. F i r s t , we show how t o o b t a i n e x a c t l y t h e t h e o r y To by i n t e r p r e t i n g t y p e s i n an i n v e r s e l i m i t space,
DZ
,
c o n s t r u c t e d by s t a r t i n g w i t h t h e l a t t i c e P,
.
As immediate consequence, we have t h a t 'the completeness theorem f o r C u r r y ' s t y p e assignment system (and f o r i t s c o n s e r v a t i v e e x t e n s i o n i n 1 2 1 ) i s proved by t h e use o f a "mathematical" Amodel. Dl
T h a t i s , by a model (which i s a s u b s t r u c t u r e o f
) c o n s t r u c t e d by means which a r e n o t s y n t a c t i c i n n a t u r e ,
f i l t e r Amodel
i n 1 2 1 o r t h e t e r m model i n 171.
assignment f o r t n e model P, by Theorems 3.9 and 3.10, 0
i n t o Dm
has been proved,
such as t h e
(Completeness o f C u r r y ' s t y p e
i n d e p e n d e n t l y , i n [ 41). Moreover,
any ( c o u n t a b l e ) a p p l i c a t i v e s t r u c t u r e can be embedded
.
I n t h e second p a r t t h e r e l a t i o n s between t y p e t h e o r i e s and some 0,
Amodels
are studied. We use s t a n d a r d n o t a t i o n on i n v e r s e l i m i t spaces ( c f . [ 17 ] o r [ 11). An i n v e r s e l i m i t D:
(i,j) of
i s c o m p l e t e l y determined by a complete l a t t i c e D D 1
=C(Do,Do)
on Do. As u s u a l we i d e n t i f y dt.0
t h a t , i f D o i s a l g e b r a i c , a l s o D, 0
L e t D,
i s a l g e b r a i c and (D,)
be t h e i n v e r s e l i m i t space determined by
p r o j e c t i o n (il,j,),
where i,(d)=AeeD
0
0
and a p r o j e c t i o n
w i t h an,
n
k
=
u (Dn) ne
(d)eDm. R e c a l l
k.
Do= P, and t h e s t a n d a r d
.d and j l ( f ) = f ( l ) .
Extended Type Structures and Filter Lambda Models 5.1.
Definition. ( i )
(ii)
v!
&=
D:>. "
T+K(D:)
s a t i s f i e s C3.
LEMMA. (i)V u e T V n V o ( u ) c _ { n l .
(ii) vC~(D:)
Proof. deD,.
I
i s t h e t y p e i n t e r p r e t a t i o n g i v e n by Vo($n)={nl+{Ol.
Note t h a t , by 2 . 1 3 ( i i i ) and 3 . 4 ( i i i ) , S m 5.2.
257
3 p e V ~
(vo(~)~vo(T)G*
(i)
Observe
that,
{%}=f
Then
with
l e t u'(flah+Bh)A(a$
i, ( d ) = f
t h e standard projection,
= $ + { n'}=D:+{
i } ( c f . 3.6).
@{n} The p r o o f i s by i n d u c t i o n on u . I f Else,
O ( ~ )~ v o ( T ) + v o ( ~ ) fi v o ( p ) & ) .
U ~ Ut h e
for all
proof i s t r i v i a l .
~ ' ( ~ ) s { " n > . Then,
k ) and assume
i d
by ~ 3 , f o r some HIGH
and
D:G(H",Vo(ah))n(n,IkI) (n,Vo(Bh))n(;7, [ O l ) ~ { % } . K H which i s i m p o s s i b l e by t h e i n d u c t i o n hypothesis.
and K ' S K such t h a t K ' U H ' P 0 ,
I f K ' = 0, t h e n n, V '(6 ) C { n ' l H h  , If K'#@, then V F K ' DoE{kl, which i s impossible. m
(ii)
.
Again by i n d u c t i o n on u
Thus t a k e
If
UEU
t h e assumption may h o l d o n l y f o r c=@.
PEW.
I f u? ( $ a h + B h ? A ( A $ ) , by C3, f o r some H'GH and K'EK, H ' U K ' f Q , V'(T)C_ K k ((H",V "(a h) )"($ [k I) and ( V O(8 h) In( I f K'=@, take p A B H' I f K ' # @ , t h e n VkeK V O( T)c_Ik 1, which i s i m p o s s i b l e by ( i ) .
pl{61)s;.
5.3.
THEOREM. (i) T o :T V o .
(ii)c, l.
m
,,E>
.
Proof. (i)By d e f i n i t i o n T o G T V O . Conversely, we p r o v e t h a t TSU. T r i v i a l . ~4,.
I f u'u
U ~ ~ T * U % T
, t h e n u~ VO $ n
L e t t h e n u:(fiah+Bh)
A
;
O(
.
does n o t h o l d .
and
,; 611S{b I,
bh) 1
T
(2 $ k ) 
05 V ~ $ n * 3 H ' c H 3K'CK H ' U K ' # $ ( ,V
b y i n d u c t i o n on
{ n k ( f l , V o ( a h ) ) n ( n , [ i } ) and K by C3. I
",
By Lemma 5.2 we have K ' = 0 ( e l s e
H and
"
V o ( ~ h ) C { O I ) . Thus V k e K ' { n } g { k } ,
i.e.
VkeK' n=k. T h e r e f o r e A , $ =$ OU'A$ f o r some U ' E T. T h i s g i v e s u%$ K k n T:T~+T~. The p r o o f goes as i n p r e v i o u s case a p p l y i n g i n d u c t i o n . T C T ~ A T ~ Use .
(ii)
We
U ~ V ~ T ~ A TU 2L *V ~ T i
have t o
show
that
and
ULVoT2
there
Define
io:IFI+IK(D_) I ^vO
by i o ( d ) = ? { V o ( u ) l u e d }
an embedding
as
lattices
.
i s i n j e c t i v e . It i s c l e a r l y i n c l u s i o n preserving.
We p r o v e t h a t ? ' ( d ) * i o ( e ) L i o ( d  e ) .
.
.
exists
a p p l i c a t i v e s t r u c t u r e s . The isomorphism h o l d s by 3 . 4 ( i i i ) . BY ( i ) ,
n
The r e v e r s e i n c l u s i o n i s e a s i e r .
and as
M. COPPO ET AL.
258
u)g.
N o t i c e f i  r s t t h a t E E V q d ) =+ 3ue d V,"( * , . ( d ) * V o ( e ) =+ 31be\j0(e) b + C E V o ( d ) *CE V * " * 3Tee 3 u Ed V0(r)E b a&vVo(~)E b+c * 3 ~ e e3 0 E d VO(u)c VO(,)+ c ;+ 3 r e e 3 0 Ed 3peT Vo(u) c_ VO(,)+ Vo(p) V 0 ( p ) c c, by 5 . 2 ( i i ) * 3 p t T 3 ~ ee T+$d V o ( p ) C c, b y (i) =+ 'ceVo(de). n
Then
and
T h i s theorem has an immediate consequence. SOROLLARY. 5.4. i n t o D, .
Any ( c o u n t a b l e ) a p p l i c a t i v e s t r u c t u r e
Proof. By Theorems 3.10 and 5 . 3 ( i i ) .
can be embedded
0
As a l r e a d y mentioned, t h e h y p o t h e s i s an t h e c o u n t a b i l i t y o f A can be dropped by t a k i n g enough t y p e v a r i a b l e s i n t h e d e f i n i t i o n o f T i n 1 . 2 ( i ) and u s i n g Pa i n s t e a d o f Pw , f o r a l a r g e enough c a r d i n a l a , i n the construction o f Dana S c o t t ( p e r s o n a l communication)
.D:
has a d i r e c t argument f o r a s i m i l a r
embedding r e s u l t . By Theorem 3.5, 0m and type
IK(D_)[
any D,
a r e i s o m o r p h i c as r e f l e x i v e domains.
interpretations
V:
domains. < I S ( T V ) I ,F,G t h e c o n s i d e r e d 0,
space y i e l d s an ETS o v e r K ( D _ ) . Moreover, by 3.8, T+K(Dm ) ,
Thus f o r a l l s u r j e c t i v e
and D,are
IS ( T V ) I
isomorphic r e f l e x i v e
> can be used as a t o o l f o r i n v e s t i g a t i n g p r o p e r t i e s o f
. For t h i s
purpose, however, we need a t h e o r y TV easy t o
handle (we ask, f o r example, t h a t T V i s r e c u r s i v e l y a x i o m a t i z a b l e ) . I n t h e r e s t of t h i s s e c t i o n we d e f i n e some s i m p l e t y p e t h e o r i e s w h i c h y i e l d f i l t e r ?,models i s o m o r p h i c t o i n t e r e s t i n g i n v e r s e l i m i t spaces. We f i r s t need a r e s u l t which i s w o r t h d i s p l a y i n g by i t s own i n t e r e s t f o r t h e axiomatization o f type theories. 5.5.
THEOREM. L e t T be a t y p e t h e o r y s a t i s f y i n g C3. Assume t h a t
and 3J $.+ A $ . + $ . where VjeJ V@ieAt "i.;L? 1 T J J J Define Z T [ $ $ j ~ @ i l ,"~j~~i}Ut"+13'"ij+"jI, QT Then T=T(Zl. Proof. We o n l y need t o p r o v e t h a t TC_T (
x
$.EAtU[w},$.
3 J$.+Q. J J 1
), i.e.
that
0
€At
u and
T
.
TEW.
Trivial.
TZ@ E
A t . I f uzw
or u
E
. .
u ~ ~ T ; + u ( ~ T
The p r o o f i s by i n d u c t i o n on t h e number o f arrows i n cases on
J
We work by
A t t h e r e s u l t i s obvious.
Otherwise, l e t u = (Aah+Bh )A((?k $1 ' s a r e as i n t h e assumption.
) , w i t h @k+T
t$pl, where t h e
$
1
' s and
Extended Type Structures and Filter Lambda Models 0 5T T

uLT
34 .++j
I'A
f o r some J, by h y p o t h e s i s L ~ ++j* ~
h ) ~$1+( t+1)
vje
J
=. V j
J 3H'L H 3 L ' c L
E
259
ah+3
( A , 6 )A(;,
H'UL'#
0
and $ .<
 Jr'
fl, ah)A(t, $1) and
$1 IT+ by c 3
 V j e J 3 H ' zHH 3 hL 1 & L '
,9
H ' U L ~ #o
and
$ ;~filah)A(~,+l)and
($,~~)h((!,+~)5~ +jj. b y t h e assumption i f H ' i s empty and by t h e i n d u c t i o n hypothesis otherwise
* The case
, by
5< T
z
T E T ~4 T~
i s t r i v i a l w h i l e t h e case
the induction hypothesis.
5.6.
LEMMA.
Let
1.1.
D m be any
t h a t V C ~ ( D 31 ~ )V(+$l)=c ~
T S T ~ + T ~e a s i l y
f o l l o w s f r o m C3 and
n inverse
limit
space.
Let
V:T+
K ( D m ) be such
(where $ i e A t f o r a l l i d ) . Then V i s s u r j e c t i v e .
P r o o f . R e c a l l t h a t i f ce(Dm ) k , t h e n c ~ ( D , ) ~ f o rsome n. The p r o o f i s by i n d u c t i o n on n. n=O. By t h e assumption.

.
L e t c ~ ( D ~ + ~Then ) ~ . f o r some I and ai,b;Dn, c = y fa,b. Now, =; ii+6 i, by Remark 3.6 1 1 c=nf I a.b. *;= ?V(ui)+V(ri) f o r some I, by t h e i n d u c t i o n h y p o t h e s i s JE=v(Au.+T
1 1
). i
(i)( P r o j e c t i o n s ) Given a complete l a t t i c e D o , d e f i n e t h e f o l l o w i n g p r o j e c t i o n s o f C(Do,Do) on Do : ie(d)= f je(f)=f(e) where eeDo ed i f Do= Pw,set i*(d)uf{nl{nll nedl j*(f)U(dl i*(d)Lfl. , and N L i s t h e f l a t (ii) ( I n v e r s e l i m i t spaces) L e t 6 = {1 , T ) , where I(T l a t t i c e o f i n t e g e r s ( i . e . x y i f f x = l o r Y=T o r x=y) Do = and p r o j e c t i o n s Then se:, w i t h 5.7.
Definition.
P
w
pw
NL NL 4 Dm
5 Dm
( iii) ( Formul a s )
6 0
M. COPPO ET AL.
260
(iv)
(Type I n t e r p r e t a t i o n s ) D e f i n e , f o r q=O,
... ,5,
Vq:l
vo(Qn)=v'(@n)=Iil V 2 ( Qn)=V3(Qn)=il
.
v4(Qn)=v5(@n)= i
The p r o j e c t i o n s ( i j ) have been i n t r o d u c e d by Park e' e It i s easy t o check t h a t ( i * , j * ) i s a c t u a l l y a p r o j e c t i o n o f C(Pw,Pw) on Pw
.
It
has been d e f i n e d u s i n g P r o p o s i t i o n 3.10 o f [ 17 1. 5.8.
THEOREM. L e t q= 0
(i)
V'
(ii)
T(Zq) = T v q
,... ,5.
Then
i s onto K ( D ~ ) .
.
P r o o f . By 5.5 and 5.6. 5.9.
REMARK. The s e t o f t y p e s i n T(Cq) and T ( z , ) a c t u a l l y c o n t a i n s j u s t one t y p e
v a r i a b l e (up t o " ^ . " ) , w h i l e t h e o t h e r t h e o r i e s above d e f i n e d , c o n t a i n i n f i n i t e t y p e v a r i a b l e s . T h i s corresponds t o t h e f a c t t h a t 6 has o n l y one compact element, T,
d i f f e r e n t from
elements.
I,
while NL
and Po have c o u n t a b l y many incomparable compact
I n general a complete a l g e b r a i c l a t t i c e , by t h e same t e c h n i q u e ,
would
g i v e a t y p e t h e o r y w i t h as many non e q u i v a l e n t t y p e v a r i a b l e s as t h e c a r d i n a l i t y o f t h e compact elements i n i t ( e x c e p t f o r The isomorphism between f i l t e r
I
).
Amodels
i n i n v e s t i g a t i n g t h e t h e o r i e s o f D,
spaces.
and D _ s p a c e s may be o f some h e l p
I n p a r t i c u l a r we c o n j e c t u r e t h a t
n o t any D, Amodel has a maximal t h e o r y . A n a t u r a l g e n e r a l i z a t i o n o f EATS i s o b t a i n e d by a l l o w i n g " A " t o be a p a r t i a l f u n c t i o n which s a t i s f i e s some ( n a t u r a l ) c o n d i t i o n s ( f o r example, i f U / \ T i s d e f i n e d and U < ( J ' , TLT' t h e n a l s o d AT' must be d e f i n e d , c f . t h e s e t Con o f r 2 1 1 ) . I n t h T s case t h e f i l t e r domains t u r n o u t t o be c.p.0.s. This i s also t h e approach o f S c o t t ' s i n f o r m a t i o n systems [ 2 1 ] . We c l a i m t h a t most o f t h e p r o p e r t i e s o f t h i s paper s t i l l h o l d . FOOTNOTES
"I" i s
(1)
Note t h a t i n D e f i n i t i o n 1.1 r e d as a s y n t a c t i c o b j e c t .
a r e l a t i o n w h i l e , here,
(2)
As u s u a l , i f f: A+B and CCA, f ( C ) = I f ( a ) I aeC1.
(3)
As u s u a l ,L, ( t h e r e i s an i n j e c t i v e ( b i j e c t i v e ) homomorphism v:
"I" i s
conside
means t h a t D';,E'
>.
Extended Type Structures and Filter Lambda Models
261
(4)
I d denotes always t h e i d e n t i t y f u n c t i o n on some s e t which i s c l e a r from the context.
(5)
We keep u s i n g t h i s abuse o f language: 3 I.. stands f o r " t h e r e e x i s t f i n i t e l y many elements o f t h e i n t e n d e d s e t , indexed i n I , such t h a t . . . " .
.
"2' between t y p e s i s i n t e n d e d modulo p e r m u t a t i o n s and and EUAW ).
(6)
" A W "
( f o r example
BATETAU
ACKNOWLEDGEMENTS A few d i s c u s s i o n s we had w i t h Henk Barendregt and K a r s t Koymans, i n t h e b e a u t i f u l s u r r o u n d i n g s o f Mount Gran Paradiso, were v e r y h e l p f u l i n c l e a r i f y i n g o u r view p o i n t and some o f o u r r e s u l t s . REFERENCES
[l]
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[2 ]
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[3 ]
Coppo,M., D e z a n i  C i a n c a g l i n i , M. and Venneri, B., F u n c t i o n a l Characters o f S o l v a b l e Terms, Z. Math. L o g i k Grundlag. Math. 27 (1981) 4558.
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Coppo, M., Completeness o f Type Assignment i n Continuous Lambda Models, Theor. Comput. S c i . ( t o appear).
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and Feys,R.
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263
D E C I S I O N PROBLEMS I N PREDICATE LOGIC
Egon B o r g e r ( + ) Lehrstuhl Informatik I 1 U n i v e r s i t a t Dortmund P o s t f a c h 500 500 D4600 Dortmund 50
ABSTRACT I n t h i s paper we s u r v e y fundamental methods and r e s u l t s a b o u t t h e d e c i s i o n problem f o r c l a s s e s o f f i r s t o r d e r l o g i c a l f o r m u l a e . We beg i n w i t h a kintotocal. account which t e l l s t h e main s t e p s i n t h e deve: lopment o f t h e f i e l d f r o m H i l b e r t ' s f o r m u l a t i o n o f t h e Entscheidungsproblem t o today. We t h e n d i s c u s s i n more d e t a i l n meMod due t o Aanderaa and m y s e l f which b u i l d s upon and extends i d e a s o f T u r i n g and Buchi and i s p a r t i c u l a r l y w e l l s u i t e d 60. logical. &chipLLvMo 0 6 compuRnLLvml. pmblem,; we e x p l a i n how b y t h i s method (and var i a n t s t h e r e o f ) s t r u c t u r a l p r o p e r t i e s o f c o m p u t a t i o n f o r m a l i s m s and o f t h e i r d e s c r i b i n g f o r m u l a e a r e i n t i m a t e l y c o r r e l a t e d i n such a way t h a t many r e c u r s i o n and c o m p l e x i t y t h e o r e t i c a l p r o p e r t i e s by t h i s r e d u c t i o n a r e e a s i l y c a r r i e d o v e r f r o m t h e c o m b i n a t o r i a l d e c i s i o n problems t o t h e c o r r e s p o n d i n g l o g i c a l d e c i s i o n problems. As example we produce by s l i g h t and n a t u r a l v a r i a t i o n s o f t h a t method u ~ d o m(and e a s y ) pmo@ f o r : NPresp. x 1 r e s p . n 1 r e s p . n 2  c o m p e t e n u n 0 6
t h e d e c d i o n pmblem f o r p r o p o s i t i o n a l (Cook) resp. f r i s t o r d e r l o g i c (Church, T u r i n g ) r e s p . o f t h e emptiness ( T r a c h t e n b r o t , B u c h i ) r e s p . t h e i n f i n i t y problem f o r f i r s t o r d e r npectm, t h e chamctehizuLLon o f t h e l a t t e r ( S c h o l z ' s problem) as t h e NEXPTIMEacceptable s e t s ( B e n n e t t , Rodding, Schwichtenberg, Jones, Selman) r e s p . o f t h e gener a l i z e d s p e c t r a as t h e NPsets ( F a g i n ) , s i m p l e axioms f o r e s s e n t i a l l y u n d e c i d a b l e and i w o m p l & t e t h e o h i u resp. n a  t h @ a b l e d o m u h e w i t h o u t fiecumive mock& d e s c r i b i n g enumeration programs f o r xounseparab l e r . e . s e t s , Loweh compeexity bounds and i n d e e d completeness r e s u l t s f o r many n a t u r a l s o l v a b l e cases o f f i r s t o r d e r l o g i c a l d e c i s i o n problems as s u b r e c u r s i v e analogues t o t h e u n d e c i d a b l e r e d u c t i o n c l a s s e s , and o t h e r c o m p l e x i t y r e s u l t s f o r f i r s t o r d e r o r p r o p o s i t i o n a l l o g i c problems l i k e a n a t u r a l l o g i c a l c h a r a c t e r i z a t i o n o f n e t work o r T u r i n g machine campLexity o BooLmn duncLLo~owhich i s s t r o n g l y r e l a t e d t o t h e P = NPprob em. Our main concern i s t o r e v e a l t h e deep s t r u c t u r a l and c o m b i n a t o r i a l s i m u l a r i t i e s between computat i o n s and l o g i c a l d e d u c t i o n s , which b r i n g o u t e x p l i c i t e l y t h e fundamental and u n i f o r m reason f o r many u n d e c i d a b i l i t y and c o m p l e x i t y r e s u l t s f o r c o m b i n a t o r i a l and f o r l o g i c a l d e c i s i o n problems (see t h e above c i t e d examples). HISTORY OF IDEAS From t h e v e r y b e g i n n i n g of mathematics a g r e a t amount o f mathematical r e s e a r c h has been d e v o t e d t o f i n d i n g a l g o r i t h m i c s o l u t i o n s t o g i v e n problems. An i m p o r t a n t subc l a s s o f such problems a r e t h e so c a l l e d d e c i s i o n problems c o n s t i t u t e d b y a c l a s s X of o b j e c t s t o g e t h e r w i t h a p r o p e r t y ( o r r e l a t i o n ) P on X; such d e c i s i o n problems a r e c a l l e d s o l v a b l e if t h e r e e x i s t s an a l g o r i t h m which e n a b l e s t o d e c i d e f o r e v e r y o b j e c t i n X whether i t shares t h e p r o p e r t y P o r n o t . The e f f o r t t o s o l v e problems +) P r e s e n t l y a t I s t i t u t o d i M a t e m a t i c a , I n f o r m a t i c a e S i s t e m i s t i c a o f U n i v e r s i t y o f U D I N E I I t a l y , on l e a v e f r o m U n i v e r s i t y o f Dortmund.
4
E. BORGER
264
i n an a l g o r i t h m i c manner on t h e one hand r e p l i e s t o t h e p r a c t i c a l need o f mechanizing a p p l i c a t i o n s o f mathematical reasoning, o f breaking down complicated mathematical processes i n t o a succession o f elementary steps which can be performed i n a p u r e l y mechanical way w i t h o u t deeper mathematical understanding and w i t h o u t any need f o r i n t e l l i g e n t e x t e r n a l c o n t r o l o r i n t e r v e n t i o n . On the o t h e r hand t h i s e f f o r t o f a l g o r i t h m i s i n g mathematical t h i n k i n g i s c o r r e l a t e d t o and has been s t i m u l a t e d by an o l d dream o f humanity t o f i n d a general r e l i a b l e method by which a l l p h i l o s o p h i c a l and s c i e n t i f i c problems and a l l d i s p u t a t i o n s on them could be s e t t l e d i n an e f f e c t i v e and d e f i n i t e way. This Ldidea 06 a u n i v e m d pmbLem n o t u h g a.tgo&LZhn, present f o r ex. i n Raimundus L u l l u s ' reasoning about an ars magna, has been made more p r e c i s e by L e i b n i z f i r s t l y i n h i s d i s t i n c t i o n between an ars inveniendi  an a l g o r i t h m f o r searching and l i s t i n g s y s t e m a t i c a l l y s o l u t i o n s t o a l l problems  and an ars i u d i c a n d i  an a l g o r i t h m f o r deciding f o r every p a r t i c u l a r problem posed whether the answer t o i t i s yes o r no ; secondly by r e a l i s i n g t h a t such a u n i v e r s a l search o r d e c i s i o n method presupposes a mathem a t i c a l l y precise u n i v e r s a l language i n which a l l problems can be unambigously expressed. This Leibnizean p r o j e c t o f a chamc.tehid.tiCa u n i v e a ~ a L ihas ~ been f u l l y r e a l i z e d o n l y by Frege's f o r m u l a t i o n o f what we c a l l today c l a s s i c a l f i r s t order l o g i c ( p r e d i c a t e c a l c u l u s ) and Godel's (1930) completeness theorem t h a t t h e l a t t e r i n deed describes e x a c t l y t h e n o t i o n o f u n i v e r s a l l o g i c a l v a l i d i t y . This r a t i o n a l calculus, t o say i t i n Leibnizean terms, enabled H i l b e r t t o t u r n the o l d human dream of an ars magna i n t o a s p e c i f i c mathematical problem, namely H i l b e r t ' s program. (1) H i l b e r t asked i n p a r t i c u l a r t o c o d i f y the various branches o f mathematics by f i r s t order axiom systems so t h a t t h e p r o o f o f a n y p a r t i c u l a r mathem a t i c a l statement comes up t o d e r i v i n g i t from t h e axiomsby the i n d i c a t e d p u r e l y l o g i c a l means; t h a t would t u r n d e r i v a t i o n s o f mathemacical r e s u l t s a t l e a s t i n p r i n c i p l e i n t o a mechanical game w i t h concrete o b j e c t s , namely s t r i n g s o f symbols representing f i r s t order l o g i c a l d e r i v a t i o n s . This i s the reason why the d e c i s i o n problem f o r c l a s s i c a l p r e d i c a t e l o g i c , i . e . t h e problem t o know i f t h e r e e x i s t s (and e v e n t u a l l y t o e x h i b i t ) an a l g o r i t h m by which f o r any w e l l formed statement o f p r e d i c a t e l o g i c i t can be decided i n a f i n i t e number o f steps whether i t i s l o g i c a l l y t r u e o r not, has been c a l l e d by H i l b e r t "WS ENTSCHEIDUNGSPROBLEM" t o u t court, considered one i f n o t the main mathematical problem o f t h a t period.
A l l attempts i n the Twenties and e a r l y T h i r t i e s t o solve the Entscheidungsproblem f a i l e d and came up w i t h n o l d o v l b o f the d e c i s i o n problem o n l y 6012 phticduh nubcasu. To formulate the most s i g n i f i c a n t o f these e a r l y p a r t i a l s o l u t i o n s ...) t h e c l a s s o f a l l closed prenex formulae o f r e s t r i c t e d denote by n(ml,m2,m3, predicate l o g i c ( i . e . w i t h o u t f u n c t i o n symbols o r i d e n t i t y s i g n ) having a p r e f i x o f form and c o n t a i n i n g a t most m. p r e d i c a t e symbols o f rank i. The f o l l o w i n g subcases o f p r e d i c a t e l o g i c have a'solvable d e c i s i o n problem w i t h respect t o s a t i s f i a b i l i t y ( 2 ) and a r e optimal t h e r e f n i n a sense t o be made p r e c i s e l a t e r : Monadic p r e d i c a t e l o g i c V. VAV V V... VMV V VA A V...
..
... ... ...
(Lowenheim (1915)) (Ackermann (1928)) (Godel (1932),Kalmar(1933),SchUtte (1934)) (Bernays fi Schonfinkel (1928))
(l) I t i s e p i s t e m o l o g i c a l l y i n t e r e s t i n g t h a t H i l b e r t ' s program was formulated w i t h t h e i n t e n t i o n t o defend mathematics against t h e a t t a c k o f t h e foundational c r i s i s due t o the discovery o f various paradoxes w i t h i n systems o f s e t theory, by p u t t i n g mathematical reasoning on a safe epistemological basis. ( * ) Mostly f o r t e c h n i c a l convenience we s h a l l speak o f l o g i c a l d e c i s i o n problems always i n terms o f s a t i s f i a b i l i t y instead of l o g i c a l v a l i d i t y ; t h i s i s wothout l o s s o f g e n e r a l i t y since a formula i s s a t i s f i a b l e i f f i t s negation i s n o t l o g i c a l l y v a l i d . For a c l a s s o f formulae we consider t h e r e f o r e i f {FIFEC,F i s s a t i s f i a b l e ) i s r e c u r s i v e o r not.
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I n 1936/7 i t has been p r o v e d t h a t H d b e m 2 EnXAcheidu~1gop4obbe.mas a whole noR nabuabbe. abgohithmicaUy. Indeed Church (1936) showed t h a t any p a r t i a l r e c u r s i v e f u n c t i o n can be r e p r e s e n t e d i n a f i n i t e e x t e n s i o n o f f i r s t o r d e r p r e d i c a t e l o g i c ; as a c o n c l u s i o n t h i s e x t e n s i o n and t h e r e b y a l s o D r e d i c a t e l o g i c cannot be a d e c i d a b l e t h e o r y . T u r i n g (1937) o b t a i n e d t h e r e s u l t i n d e p e n d e n t l y by an e x p l i c i t d e s c r i p t i o n of T u r i n g machine computations by l o g i c a l f i r s t o r d e r formulae t h e r e b y r e d u c i n g e f f e c t i v e l y an u n s o l v a b l e c l a s s o f p a r t i c u l a r word problems f o r T u r i n g machines t o t h e d e c i s i o n problem f o r corresponding f i r s t o r d e r formulae, c o n c l u d i n g t h e u n d e c i d a b i l i t y o f t h e l a t t e r f r o m t h e undecidab i l i t y o f t h e former. Since a t t h a t t i m e a l a r g e number o f s p e c i a l c l a s s e s o f f i r s t o r d e r formulae was known a l r e a d y t o have a s o l v a b l e Entscheidungsproblem, T u r i n g found i t i n t e r e s t i n g t o observe i n o p . c i t . t h a t f o r h i s r e d u c t i o n he r e a l l y needed o n l y a small p o r t i o n o f p r e d i c a t e l o g i c , namely t h e subclass V A V A6(0,). By T u r i n g ' s c o n s t r u c t i o n , Godel I s completeness theorem t e l l i n g t h a t f i r s t o r d e r l o g i c a l v a l i d i t y i s r e c u r s i v e l y enumerable and t h e u n i v e r s a l i t y o f T u r i n g machines t h e above c l a s s i s even a "ke&LLon claoo", i . e . a c l a s s X o f f o r m u l a e w i t h a d e c i s i o n problem t o which t h e whole Entscheidungsproblem i s manyone r e d u c i b l e , i n o t h e r words f o r which a procedure e x i s t s a s s o c i a t i n g t o e v e r y f i r s t o r d e r formulaF a f o r m u l a F i n X which i s e q u i v a l e n t t o i t i n t h e sense t h a t F i s s a t i s f i a b l e i f f F i s s a t i s f i a b l e . A f t e r t h e appearence o f T u r i n g ' s paper, many e f f o r t s were spent t o i m prove t h i s r e s u l t and t o 4e@%2H d b s 4  f ~EnXAcheidcngsp&obbem, on t h e one hand by p r o d u c i n g " s m a l l e r " such " r e d u c t i o n c l a s s e s "  t h e r d o r e w i t h undecidable, even manyone complete Entscheidungsproblem, see Suranyi (1959)  on t h e o t h e r hand by e x h i b i t i n g d e c i s i o n procedures f o r l a r g e r and l a r g e r subclasses, see Ackermann (1954) f o r t h e s t a t e o f t h e a r t i n t h e F i f t i e s . I n measuring t h e l o g i c a l c o m p l e x i t y o f formulae m a i n l y t h e above mentioned two c l a s s i f i c a t i o n p r i n c i p l e s were pursued c o n s i d e r i n g t h e s t r u c t u r e o f t h e p r e f i x  l e n g t h and number (and k i n d ) o f q u a n t i f i e r changes and/or number and a r i t y o f o c c u r i n g p r e d i c a t e symb o l s . Most r e d u c t i o n s proceeded by s k i l f u l and o f t e n v e r y c o m p l i c a t e d f i r s t o r d e r f o r m a l d e s c r i p t i o n s o f d i r e c t t r a n s f o r m a t i o n s o f models f o r g i v e n formulae F ( o f w e l l known l o g i c a l s t r u c t u r e ) i n t o models o f a " c o d i n g " F o f F such t h a t f r o m any model f o r F a model f o r F can be e x t r a c t e d . These a x i o m a t i z a t i o n s o f d i r e c t model t r a n s f o r m a t i o n s o f t e n y i e l d e d much more t h a n what was r e q u i r e d by t h e v e r y n o t i o n o f r e d u c t i o n c l a s s , namely secondorder l o g i c a l equivalences; and indeed i t i s a w i d e l y open, i n t e r e s t i n g problem t o know what a r e t h e b e s t r e d u c t i o n c l a s s e s w i t h r e s p e c t t o secondorder d e d u c i b l e e q u i v a l e n c e between ( t h e e x i s t e n t i a l c l o s u r e s o f ) F and i t s r e d u c t i o n f o r m u l a F ( i n s t e a d o f a s k i n g m e r e l y t h a t F i s s a t i s f i a b l e ( f i r s t o r d e r d e d u c i b l e ) i f f F i s ) . B u t i n s p i t e o f many new r e d u c t i o n s which were found up t o t h e end o f t h e F i f t i e s , n o t even t h e d e c i s i o n p r o blem o f a l l f o r m a l l y s p e c i f i e d c l a s s e s o f formulae l i k e t h e p r e f i x c l a s s e s (I o r t h e p r e f i x  s i m i l a r i t y c l a s s e s n(ml,m *,...) c o u l d be s e t t l e d .

S t r a n g e l y e n o u g h T u r i n g ' s i d e a t o l o o k f o r smooth and d i r e c t l o g i c a l d e s c r i p t i o n o f machine computations o r s i m i l a r processes was n o t r e a l l y pursued up t o 1962. Only t h e n Buchi (1962) t o o k up a g a i n T u r i n g ' s approach and combined i t w i t h s k i l f u l use o f *Lea theohems &e Ro Shabern. These theorems t e l l t h a t a prenex f o r m u l a o f r e s t r i c t e d p r e d i c a t e l o g i c ( i . e . w i t h o u t f u n c t i o n symbols and w i t h o u t i d e n t i t y s i g n ) i s s a t i s f i a b l e i f f i t s Skolem normal f o r m i s , and t h a t i n models f o r such Skolem normal forms one can r e s t r i c t a t t e n t i o n t o t h e domain o f terms b u i l t up f r o m t h e i n d i v i d u a l c o n s t a n t s and f u n c t i o n symbols o c c u r i n g i n t h e f o r m u l a and t o i n t e r p r e t a t i o n o f t h e terms by themselves. T h e r e f o r e i n t h e l o g i c a l d e s c r i p t i o n o f computation processes l i k e t h e one g i v e n i n T u r i n g ' s (1937) paper one has n o t t o c a r e any more about t h e f o r m a l r e p r e s e n t a t i o n o f t h e o b j e c t s o f computation  l i k e numbers, words, sequences, domino p o s i t i o n s and l i k e ; these d a t a a r e r e p r e s e n t e d j u s t as i n d i v i d u a l terms appearing i n f o r mulae i n Skolem normal form. A f o r m u l a F i n VAVA o f form VAVAG f o r example i s uxvy s a t i s f i a b l e i f f i t s Skolem normal f o r m M G U , V ( O , ~ ' )  w i t h Hx 1'. . ,xn ( tl 3. . . I t n )
w
.
E. BORGER
266
d e n o t i n g t h e r e s u l t o f simultaneous s u b s t i t u t i o n o f xi by ti  i s s a t i s f i a b l e o v e r the naturals IO,O',O",O'", ...1 w i t h t h e i n d i c a t e d i n t e r p r e t a t i o n o f t h e zerop l a c e resp. oneplace f u n c t i o n as t h e number 0 r e s p t h e n a t u r a l successor VA o f f o r m f u n c t i o n x + 1. S i m i l a r l y a f o r m u l a F i n AV
...
AV
xv1
. ..
VA G VnY
i s s a t i s f i a b l e i f f i t s Skolem normal f o r m G ; v.,l
. .,vn
(al,.
. . ,an)
i s s a t i s f i a b l e o v e r t h e domain of a l l words o v e r t h e a l p h a b e t
...,an)
{al.
w i t h t h e oneplace f u n c t i o n symbols i n t e r p r e t e d as word successor f u n c t i o n s . S i m i l a r l y formulae o f p r e f i x f o r m M V t a l k about t h e F i t c h domain (Dyck language D1) o f a l l c o r r e c t p a r e n t h e s i s expressions b u i l t up f r o m t h e 2ary p a r e n t h e s i s (,); V n M V corresponds t o t h e s t r u c t u r e o f b i n a r y t r e e s w i t h leaves l a b e l e d by symbols al,. . ,a ecc.
.
Buchi's s i m p l e b u t f u n d a m e n t a l o b s e r v a t i o n c o n s t i t u t e d a breakthrough. I t became c l e a r t h a t i n o r d e r t o show a c l a s s X t o be a r e d u c t i o n c l a s s one had t o l o o k above a l l f o r an a p p r o p r i a t e t y p e o f c o m b i n a t o r i a l system  l i k e T u r i n g machines, Thue systems, P o s t correspondence problems, domino games  where a p p r o p r i a t e n e s s means t h a t t h e d a t a s t r u c t u r e o f t h e c o m b i n a t o r i a l system can be c o n v e n i e n t l y encoded i n t o t h e t e r m s t r u c t u r e o f t h e (Skolem normal f o r m o f t h e ) f o r m u l a e i n X, and then d e s c r i b e an u n s o l v a b l e d e c i s i o n problem f o r t h e c o m b i n a t o r i a l system by a s a t i s f i a b i l i t y ( o r d e d u c i b i l i t y ) problem o f a f o r m u l a i n X.(1) By t h i s m t h o d Biichi (1962) showed t h e u n s e t t l e d p r e f i x c l a s s t o be a (even c o n s e r v a t i v e T 2 ) ) r e d u c t i o n c l a s s . Indeed f o l l o w i n g t h i s l i n e o f a t t a c k w i t h i n 4 y e a r s t h e d e c i s i o n problem o f a l l p r e f i x  s i m i l a r i t y c l a s s e s n(ml,m 2,...) could be s e t t l e d by proving the f o l l o w i n g t o be c o n s e r v a t i v e r e d u c t i o n c l a s s e s : AVA(,l)
Kahr (1962), i m p r o v i n g t h e r e d u c t i o n c l a s s AVA (0,m) Kahr, Moore, Wang (1962)
in
A V ~ A ( O , ~ K) o s t y r k o (1964), Genenz (1965) ; s t r e n g t h e n e d i n Deutsch (1981) AVAVm(O,l) Gurevich (1966)
(1) I t i s i n t e r e s t i n g t o n o t e i n t h i s c o n t e x t t h a t v a r i o u s small u n i v e r s a l combinat o r i a l systems o r s t r o n g c o m b i n a t o r i a l t o o l s have been developed i n t h e a t t e m p t t o d e c i d e a g i v e n l o g i c a l d e c i s i o n problem. Good example a r e : Ratmey'n (1928) theanem developed i n t h e course o f t h e s t u d y o f t h e d e c i s i o n problem f o r t h e SchGnf i n k e l  B e r n a y s c l a s s V" f i m , a n d o t h e r s ; R a d d i t t g ' o (1969) p t h pmbLem i n t h e f i r s t (Gaussian) q u a d r a n t w i t h n a t u r a l c o o r d i n a t e s developed f o r an e l e g a n t p r o o f o f t h e u n d e c i d a b i l i t y o f t h e AVA (,1) case  t h i s u n s o l v a b l e problem, a k i n d o f g e o m e t r i c a l model encoding a r b i t r a r y machine computations, t u r n e d o u t t o be extreml y u s e f u l f o r c o n s t r u c t i o n o f s m a l l u n i v e r s a l T u r i n g machines (see K l e i n e Buning & Ottmann (1977) and f o r d e c i s i o n problems i n g e n e r a l i z e d v e c t o r a d d i t i o n systems (see K l e i n e Buning (1980)); t h e fineah nampfing pnobLem d e v i c e d by Aanderaa d u r i n g t h e s t u d y o f subcases o f t h e AVA  d e c i s i o n problem (Aanderaa (1966) see below t h e AVA Subclass Theorem) which prompted Lewis (1979) t o g i v e a new p r o o f f o r B e r g e r ' s (1966) theorem of t h e u n s o l v a b i l i t y o f t h e u n c o n s t r a i n e d domino problem. (2) A reduction class X i s c a l l e d conservative i f also f i n i t e s a t i s f i a b i l i t y i s preseri f t h e r e d u c t i o n procedure a l s o f u l f i l l s t h a t F i s s a t i s f i a b l e i f f F i s . ved,i.e.
267
Decision Problems in Predicate Logic
These t h r e e fundamental r e s u l t s t o g e t h e r w i t h t h e d e c i d a b l e cases mentioned a t t h e b e g i n n i n g (and w i t h some a d d i t i o n a l easy r e d u c t i o n s , see f o r ex. Gurevich (1966) o r K o s t y r k o (1966) y i e l d t h e f o l l o w i n g p r e f i x  s i m i l a r i t y theorem, f o r which we g e n e r a l i z e s l i g h t l y o u r
Not~~tition.F o r any c l a s s n o f p r e f i x e s and any c l a s s u o f p r e d i c a t e symbols l e t n ( u ) be t h e c l a s s o f c l o s e d prenex f o r m u l a e o f r e s t r i c t e d p r e d i c a t e l o g i c w i t h p r e f i x i n n and p r e d i c a t e symbols i n U. L e t V"A" : = I V ~ Am,n~ ~= 0,1, ... I mn) be a c l a s s o f ml monadic,m2 and s i m i l a r l y f o r V m ~ V m e t c . L e t (ml,m2, b i n a r y ,...,mn nary and no o t h e r p r e d i c a t e symbol.
...,
Phe6ixSimiJhhity Theohem doh k e n h i c t e d ptedicute
F o r any c l a s s p r e f i x e s and any c l a s s u o f p r e d i c a t e symbols t h e f o l l o w i n g h o l d s :
n
of
1. e i t h e r n(~) i s a c o n s e r v a t i v e r e d u c t i o n c l a s s o r has a s o l v a b l e d e c i s i o n problem 2. n(o) has a s o l v a b l e d e c i s i o n problem i f f e i t h e r u c o n t a i n s o n l y monadic p r e d i c a t e symbols o r II 5 V"A" or
u V"AV"
u V"MV"
and ~  ( v " A " u V"AV" u V"MV")
are f i n i t e
( i . e . e s s e n t i a l l y o n l y t h e subcases mentioned a t t h e b e g i n n i n g have s o l v a b l e d e c i s i o n problem). 3. Any p r e f i x  s i m i l a r i t y r e d u c t i o n c l a s s i s more c o m p l i c a t e d o r equal t o one o f the f o l l o w i n g nine minimal conservative reduction classes: f i n i t e p r e f i x : AvA(m,l)
A3V (", 1
i n f i n i t e prefix: A"V(0,l)
A 3 ~ " ( ~1),
v"A3v( 0,1)
V"AVA(O,~)
q u a n t i f i e r changes :1
AVA"(O, 1), AV"A (0, 1
AVAV"
(o,~)
:2 :3
where n ( u ) i s n o t more c o m p l i c a t e d t h a n ( i . e . < ) n ' ( u ' ) i f f t h e elements o f n can be o b t a i n e d f r o m t h e elements o f n' by d e T e t i o n o f q u a n t i f i e r s and t h e r e i s a 11mapping f r o m u t o U ' which does n o t decrease t h e a r i t y o f p r e d i c a t e symbols i n u . An analogous theorem h o l d s when a l s o f u n c t i o n symbols and t h e e q u a l i t y s i g n a r e allowed, denote by ( ~ ; u ; T ) t h e c l a s s o f c l o s e d prenex f o r m u l a e w i t h p r e f i x i n n
(2)
A good source f o r a p r o o f a r e f o r p a r t 2 t h e papers by Lowenheim, Godel, Bernays E S c h o n f i n k e l c i t e d a t t h e beginning, f o r p a r t 3 t h e papers Gurevich (1966), Kostyrko(1964) and f o r t h e AVA(m,l) case e i t h e r K o s t y r k o ( l 9 6 6 ) o r Rodding (1969).
E. BORGER
268
and p r e d i c a t e s resp. f u n c t i o n s as i n d i c a t e d by 0 r e s p . T. We t h e n can f o r m u l a t e t h e f o l l o w i n g v a r i a n t o f t h e above theorem which a g a i n has been e s t a b l i s h e d piecemeal by v a r i o u s a u t h o r s :
P m & i x  S U n i t y Theoaiem doh. @&?. ptredicate L o g i c . 1. F o r i d e n t i t y f r e e p r e d i c a t e l o g i c a c l a s s ( 7 ; O ; T )
w i t h a t l e a s t one f u n c t i o n symbol e i t h e r has a s o l v a b l e d e c i s i o n problem and i s i n c l u d e d i n a t l e a s t one o f t h e c l a s s e s o f a l l c l o s e d prenex f o r m u l a e w i t h  o n l y monadic p r e d i c a t e and f u n c t i o n symbols or  a t most one u n i v e r s a l q u a n t i f i e r o r t h e c l a s s i s u n s o l v a b l e and i n d e e d a c o n s e r v a t i v e r e d u c t i o n c l a s s and i n c l u d e s a t l e a s t one o f t h e c l a s s e s o f a l l c l o s e d prenex f o r m u l a e w i t h p r e f i x M a n d one b i n a r y p r e d i c a t e and one monadic f u n c t i o n symbol o r v i c e versa one b i n a r y f u n c t i o n and one monadic p r e d i c a t e symbol.
2. F o r f u l l p r e d i c a t e l o g i c w i t h = a c l a s s ( T ; O , T ) w i t h a t l e a s t one f u n c t i o n symbol e i t h e r has a s o l v a b l e d e c i s i o n problem and i s i n c l u d e d i n a t l e a s t one o f t h e c l a s s e s o f a l l c l o s e d prenex f o r m u l a e w i t h


o n l y monadic p r e d i c a t e and a t most one, a m o n a d i c > f u n c t i o n symbols or o n l y one u n i v e r s a l q u a n t i f i e r and a t most one, a monadic, f u n c t i o n symbol or only existential quantifiers
o r i t i s a c o n s e r v a t i v e r e d u c t i o n c l a s s and i n c l u d e s a t l e a s t one o f t h e classes w i t h

prefix A
,2

prefix A
,
p r e f i x A\
monadic f u n c t i o n s and no p r e d i c a t e besides = or 1 b i n a r y f u n c t i o n and no p r e d i c a t e besides = or 1 monadic f u n c t i o n , 1 b i n a r y p r e d i c a t e besides =
The same theorem h I d s i f f u n c t i o n s may be i n t e r p r e t e d n o t as t o t a l b u t as p a r t i a1 f u n c t i o n s .?I) A t t h e b e g i n n i n g o f t h e S e v e n t i e s Krom (1970), Aanderaa (1971) and m y s e l f i n 8 o r g e r (1971) i n d e p e n d e n t l y came up w i t h a t h i r d i d e a which pursued B u c h i ' s approach t o i t s l a s t consequences. Whereas Buchi and h i s f o l l o w e r s i n t h e 6 0  i e s d e s c r i b e d c o m p u t a t i o n processes w i t h e x p l i c i t r e f e r e n c e t o t h e t i m e component, we r e a l i z e d t h a t t h i s i s n o t necessary i f one aims a t a d e s c r i p t i o n o f p r o p e r t i e s o f computations where t h e t i m e needed a t a d e s c r i p t i o n o f p r o p e r t i e s o f computations where t h e t i m e needed t o reach t h i s p r o p e r t y i s i r r e l e v a n t . Such a p r o p e r t y i s f o r ex. t h a t a computation j u s t h a l t s , w i t h o u t w o r r i n g a b o u t how many s t e p s t h i s may take. T h e r e f o r e we t r i e d and succeeded i n b c ~ i b i n c jcompu*a.tion p o c e s n e s &thou2 ae6eaencing .time. T h i s method a l l o w e d enormous s i m p l i f i c a t i o n s o f r e d u c t i o n f o r m u l a e d e s c r i b i n q machine and l i k e problems and r e s u l 
(1)
F o r p r o o f s see Gurevich (1969),(1973),(i976)  f o r a s i m p l i f i c a t i o n o f t h e main r e d u c t i o n i n t h e l a s t paper a l s o Borger (1978) o r W i r s i n g (1977 b ) ; t h e decidab i l i t y o f t h e o n e  q u a n t i f i e r c l a s s w i t h = and o n l y one, a monadic, f u n c t i o n symb o l i s due t o Shelah (1977); t h e case f o r p a r t i a l f u n c t i o n s i s t r e a t e d i n Abramsky (1980).
Decision Problems in Predicate Logic
269
ted i n almost t r i v i a l i z i n g many proofs(')and i n much s h a r p e r reduction c l a s s e s as before, defined by imposing r e s t r i c t i o n s not only on p r e f i x and s i m i l a r i t y , b u t a l s o on t h e t r u t h  f u n c t i o n a l s t r u c t u r e of reduction formulae, on the s t r u c ture of atomic subformulae i n them and on t h e number of occurences of atomic subformulae. F u r t h e r a n a l y s i s using s t r o n g l y the idea of describing computations without bothering about t h e time d e s c r i p t i o n and based on these new s t r o n g e r c l a s s i f i c a t i o n p r i n c i p l e s i n some cases again brought t o t h e border l i n e between decidable and undecidable cases, although not in such a complete and natural way a s i n the case of t h e p r e f i x  s i m i l a r i t y problem. For a systematic ( b u t not complete) account we r e f e r t o t h e two r e c e n t books Dreben & Goldfarb (1979) and Lewis (1979) and l i m i t ourselves here t o mention only f o u r t y p i c a l and outstanding examples f o r such (almost) minimal undecidable cases (indeed reduction c l a s s e s ) . The f i r s t example i s about Khom ,pmunLLeae, i.e. formulae in prenex conjunctive normal form with a matrix containing only binary d i s j u n c t i o n s . The i n t e r e s t f o r t h i s c l a s s of formulae comes from two f a c t s : a ) the proof by Herbrand (1930: pg. 118), (1931: pg. 33 sq.) t h a t t h e decision problem f o r formulae in prenex conjunctive normal form where the matrix i s a conjunction of atomic o r negated atomic formulae i s s o l v a b l e ; b) Chang's & K e i s l e r ' s (1962) normal form theorem showing t h a t any i d e n t i t y  f r e e f i r s t order formula can be p u t i n t o prenex conj u n c t i v e normal form with a l t e r n a t i o n s of length a t most 3. Krom (1964), (1966), (1967), (1967 a ) , (1968) studied then formulae with binary d i s j u n c t i o n s from various points of view and obtained in Krom (1970) the r e s u l t t h a t t h e i r decision problem i s unsolvable. Thourough research based on the method developed in Aanderaa (1971) and Borger (1971) r e s u l t e d i n the following &om and H o m ~Pnehix Theohem. R e s t r i c t e d t o K m m (prenex conjunctive normal form w i t h matrices containing only binary d i s j u n c t i o n s ) and t o Hohn formlilae ( i . e . no d i s j u n c t i o n contains more than one nonnegated atomic subformula) a l l p r e f i x c l a s s e s except f o r t h e c l a s s e s AVAVn f o r n = 1,2, ....m  whose decision problems a r e s t i l l open, but we c o n j e c t u r e them t o be solvable  a r e e i t h e r conservative reduction c l a s s e s o r have a s o l v a b l e decision problem. In p a r t i c u l a r the following a r e minimal undecidable c l a s s e s : VAVA AVVA AVM M V A : r e s t r i c t e d t o Krom & Horn AVAV A3V : ( r e s t r i c t e d t o Krom allowing = ) o r ( r e s t r i c t e d t o Horn) whereas t h e following c l a s s e s have a s o l v a b l e decision problem: AVA (even with = allowed) VmAmVm : r e s t r i c t e d t o Krom AVA : r e s t r i c t e d t o Horn The two Krom and Horn c l a s s e s MV nand A V V A a r e h i s t o r i c a l l y t h e f i r s t minimal ones which have been proved t o posses an unsolvable decision problem by the method developed in Aanderaa (1971) and Borger (1971); f o r t h e sake of exempli
(2) Recently Jones & Matijasevich (1982) applied t h e same idea t o a d i r e c t desc r i p t i o n of r e g i s t e r machine h a l t i n g problems by exponential diophantine equat i o n s ; t h i s r e s u l t e d in a tremendous s i m p l i f i c a t i o n of t h e proof f o r the DavisPutnamRobinson theorem t h a t every r e c u r s i v e l y enumerable set i s exponential diophantine avoiding completely use o f the Chinese remainder theorem and t h e t r i c k y number t h e o r e t i c a l c o n s t r u c t i o n s involved.
270
E. BORCER
f i c a t i o n t h i s p r o o f w i l l be reproduced i n t h e n e x t s e c t i o n . o t h e r undecidable cases have been o b t a i n e d by l a t e r r e f i n e m e n t s o f t h i s method. Up t o today t h e p r e f i x  s i m i l a r i t y problem r e s t r i c t e d t o Krom and f o r Horn c l a s s e s i s neverthel e s s s t i l l open; o n l y s c a t t e r e d p a r t i a l r e s u l t s ( a l t h o u g h o b t a i n e d by i n t e r e s t i n g refinements o f t h e above mentioned method) a r e known l i k e t h e u n d e c i d a b i l i t y o f the classes VAVmA(O,l),AVmA( 1,1),AVmA(0,2) r e s t r i c t e d t o Krom and Horn (see Lewis (1976),(1979) based on a d e s c r i p t i o n o f P o s t correspondence problems f o l l o w e d by f u r t h e r r e d u c t i o n s by a technique i n s p i r e d by Sh n o n ’ s (1956) cons t r u c t i o n o f a u n i v e r s a l T u r i n g machine w i t h o n l y 2 s t a t e s . ) ? ? )
“7,
I n t e r e s t i n g l y enough i n t h e Krom case, d i f f e r e n t l y from t h e n o t t r u t h  f u n c t i o n a l l y r e s t r i c t e d c l a s s i c a l case, p r e d i c a t e s o f rank b i g g e r than 2 may p l a y an e s s e n t i a l r o l e f o r t h e ( u n  ) s o l v a b i l i t y o f t h e d e c i s i o n problem o f a c l a s s ; i n f a c t t h e c l a s s AVA(,) r e s t r i c t e d t o Krom has a s o l v a b l e d e c i s i o n problem as proved i n Borger (1973) by r e d u c t i o n t o t h e AVAKrom case, whereas f o r some k t h e c l a s s e s MVA(0,,k) and AVM(0,,k) r e s t r i c t e d t o Krom a r e c o n s e r v a t i v e r e d u c t i o n c l a s s e s . F o r r e a l l y small k l i k e k < 7 n o t h i n g i s known and t h e a c t u a l l y a v a i l a b l e methods do n o t seem t o be s u f f i c i e n t t o s e t t l e these q u e s t i o n s . W i t h o u t c o n s i d e r a t i o n o f t h e p r o p o s i t i o n a l f o r m one has f o r one o f t h e minimal f i n i t e  p r e f i x r e d u c t i o n c l a s s e s t h e f o l l o w i n g i n t e r e s t i n g sharp c l a s s i f i c a t i o n o f subclasses: AVA Subc&eans Theohem. Subclasses o f t h e minimal undecidable f i n i t e  p r e f i x c l a s s AVA(m,l) s p e c i f i e d by any o f t h e 2121 combinations o f a l l o w e d atomic subformulae b u i l t up f o r m t h e v a r i a b l e s i n t h e p r e f i x AVA have an u n s o l v a b l e deXVY c i s i o n problem ( i n d e e d c o n s t i t u t e r e d u c t i o n c l a s s e s ) i f f a t l e a s t t h r e e forms o f atomic sirhfnrmiilae(jrc1uding e i t h e r Rxy t o g e t h e r w i t h Ryv o r Ryx t o g e t h e r w i t h Rvy a r e allowed.
Fw Atomic SubgomLLeae Theomm. ( G o l d f a r b ( 1 9 7 4 ) ) ( 4 ) The subclass o f AVA“ o f a l l formulae w i t h m a t r i c e s o f t h e form (Ao A 1A1) v (A2 A l A 3 ) where A . a r e atomic formulae i s a r e d u c t i o n c l a s s ; t h e c l a s s o f a l l formulae c o n t a i n i n g ( e v e n t u a l l y an a r b i t r a r y number o f d i f f e r e n t occurences o f ) o n l y two d i s t i n c t atomic subformulae has a s o l v a b l e d e c i s i o n problem. The case where t h r e e atomic subformulae a r e a l l o w e d i s open. ( l ) AVAA,AAVA a r e due t o Lewis (see Aanderaa & Lewis (1973)), AVAV and A3V f o r H w n a r e a l s o ( s e e Ph.D. T h e s i s ) and f o r Krom w i t h allowance o f = t o Aanderaa & F r g e r & Gurevich (1982). The d e c i s i o n procedures a r e due r e s p e c t i v e l y t o Aanderaa & Lewis (1973) ( f o r t h e i n c l u s i o n o f = see a g a i n Aanderaa & Borger & Gurevich ( 1 9 8 2 ) ) , Maslov (1964) and G o l d f a r b (1974). F o r c o n s e r v a t i v i t y o f t h e r e d u c t i o n s see Aanderaa & Borger & Lewis (1982). F o r AVA i n Krom w i t h = see G o l d f a r b ’ s Ph.D. Thesis.
(’) An e a r l y r e s u l t o f an u n s o l g a b l e p r e f i x  K r o m c l a s s w i t h a small number o f b i n a r y p r e d i c a t e symbols was AV A (0,4) r e s t r i c t e d t o Krom and Horn and was proved i n Rodding & Borger (1974) by a much s i m p l e r method t h a n t h e one used by H a r r y Lewis f o r h i s s h a r p e r r e s u l t s . Krom (1970) o b t a i n e d t h e u n s o l v a b i l i t y o f t h e Krom c l a s s hVmA(O,k) f o r some ( b i g ? ) k by d e s c r i b i n g d e d u c t i o n s i n P o s t ’ s t a g systems. ( 3 ) F o r t h e d e c i d a b l e cases see Dreben & Kahr & Wang (1962), f o r t h e undecidable cases Aanderaa & Lewis (1974) which i s based on Aanderaa (1966) and t h e i n t e r e s t i n g and v e r y d i f f i c u l t l i n e a r sampling problem e s p e c i a l l y devised f o r t h i s case. T h i s c l a s s i f i c a t i o n was suggested a l r e a d y by Buchi (1962) and a l s o appears i n Wang (1962). K o s t y r k o (1966) proves t h e u n s o l v a b i l i t y w i t h any t h r e e o f t h e f o u r atomic subformulae Rxy,Ryx,Ryv,Rvy b e s i d e s o n l y monadic subformulae. ( 4 ) F o r t h e l o n g t r a d i t i o n t o c l a s s i f y formulae w i t h r e s p e c t t o forms o f t h e i r atomic subformulae see t h e l i s t o f r e f e r e n c e s i n Lewis (1979), pg. 155 i n c l u d i n g among o t h e r s Skolem, Church, Friedman, and Maslov.
271
Decision Problems in Predicate Logic W h i n g ' n T h e o k m . W i r s i n g (1977). The c l a s s o f formulae o f form A
x1
... 9
(S1 = S 2 A S 3 # s 4 )
6
w i t h terms si b u i l t up from a monadic f u n c t i o n symbol f and t h e v a r i a b l e s x ,..., x i s a c o n s e r v a t i v e r e d u c t i o n c l a s s (The cases w i t h 5,4 o r 3 u n i v e r s a l q i a n t i f i h  s a r e s t i l l open problems.) The i n t i m a t e s t r u c t u r a l and c o m b i n a t o r i c a l connections between programs M and the l o g i c a l formulae LX d e s c r i b i n g t h e e f f e c t o f t h e e x e c u t i o n o f M on g i v e n data, once r e v e a l e d by o u r method f o r l o g i c a l d e s c r i p t i o n o f computation processes t o be e x p l a i n e d i n t h e n e x t s e c t i o n , immediately y i e l d as b y  p r o d u c t t h a t v i a t h i s implementation m c k i n e haPLing pmbLemn and Logical? & c h i o n pMbl?emn a m mcwrniueLy inomo?Lpkic. From t h i s isomorphism many c o m p l e x i t y t h e o r e t i c a l consequences can be drawn; some examples w i l l be g i v e n i n t h e t h i r d s e c t i o n below. I n part i c u l a r l e t us mention here two fundamental theorems which w i l l be proved parap h r a s i n g t h e manyone completeness p r o o f g i v e n i n t h e n e x t s e c t i o n f o r H i l b e r t ' s Entscheidungsproblem:
Aandenaa'n Theohem (1971). I f a program M enumerates r e c u r s i v e l y unseparable s e t s then t h e f i r s t o r d e r t h e o r y w i t h t h e program d e s c r i p t i o n a M as n o n l o g i c a l axiom has r e c u r s i v e l y unseparable theoremhood and l o g i c a l f a l s e h o o d and i s t h e r e f o r e e s s e n t i a l l y undecidable and incomplete. (A v a r i a n t o f ) a,,, i s a s a t i s f i a b l e formula
w i t h o u t r e c u r s i v e models.
(We w i l l g i v e t h e p r o o f w i t h t h e s i m p l e r formulae
aM
found i n Borger (1975),
(1982) which y i e l d an analogous statement f o r Enunseparable En+lsets Grzegorczyk's h i e r a r c h y . )
in
Ttiacktenbhot'n Theomm. The c l a s s e s o f c o n t r a d i c t o r y r e s p . f i n i t e l y s a t i s f i a b l e ( r e s p . n o n  c o n t r a d i c t o r y b u t n o n  f i n i t e l y s a t i s f i a b l e ) f i r s t o r d e r formulae a r e r e c u r s i v e l y i n s e p a r a b l e . As a c o n c l u s i o n t h e same h o l d s r e s t r i c t e d t o a r b i t r a r y c o n s e r v a t i v e r e d u c t i o n classes. Note t h a t by t h e u n s o l v a b i l i t y o f t h e Entscheidungsproblem, G o d e l ' s completeness theorem and t h e obvious f a c t t h a t t h e f i n i t e l y s a t i s f i a b l e formulae form a r e c u r s i v e l y enumerable c l a s s ( 1 ) t h e c l a s s o f i n f i n i t y axioms ( i . e . o f noncontrad i c t o r y b u t n o t f i n i t e l y s a t i s f i a b l e formulae) i s t r i v i a l l y n o t r e c u r s i v e l y enumerable. Most known formulae c l a s s e s w i t h s o l v a b l e d e c i s i o n problem have t h e p r o p e r t y t h a t any s a t i s f i a b l e f o r m u b i n them admits a l s o f i n i t e models; t h i s so c a l l e d f i n i t e c o n t r o l l a b i l i t y p r o p e r t y i s s t u d i d e x t e n s i v e l y i n Dreben & Goldf a r b (1979); see a l s o Ash (1975). An i n t e r e s t i n g counterexample i s t h e c l a s s A V A r e s t r i c t e d t o Krom which c o n t a i n s t h e f o l l o w i n g i v f i n i t y axiom (Gxy means: x i s b i g g e r t h a n y; r e d v as successor x o f x ) :
I ::
A V A (Gvx & (GXY
+
Gvy)
A
~GXX)
X V Y
T h i s f i r s t c o n j u n c t asks f o r a " g r e a t e r " element x
~ t +o any ~ g i v e n xn, t h e
second c o n j u n c t l i n k s these elements t o g e t h e r by ( a k i n d o f ) t r a n s i t i v i t y ( 2 ) i n t o a c h a i n (xo,xl,x 2,...) where every xi i s " g r e a t e r " t h a n x . i f i i s b i g g e r than j, J t h e t h i r d c o n j u n c t exludes t h a t an x . may be equal t o some xi f o r i < j . We w i l l J use t h i s s i m p l e i n f i n i t y axiom i n t h e n e x t s e c t i o n t o assure c o n s e r v a t i v i t y o f reductions. ( 1 ) F i n i t e s a t i s f i a b i l i t y i s c o m p l e t e l y a x i o m a t i z e d i n B u l l o c k & Schneider (1973); c f . a l s o H a i l p e r i n (1961) f o r a complete a x i o m a t i z a t i o n o f formulae which a r e i n v a l i d i n some f i n i t e domain. ( 2 ) NB. F u l l t r a n s i t i v i t y cannot be expressed by a Krom formula, see Krom (1966).
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E. BORGER
The o r i g i n a l proofs in Trachtenbrot (1950), (1953) a r e much more involved, they y i e l d i n t e r e s t i n g a p p l i c a t i o n s t o normal form theorems f o r r e c u r s i v e l y enumerable p r e d i c a t e s . Indeed t h e 1950proof i s based o n a construction s h o w i n g t h a t p4ecDeLq t h e gnaphn ad pLhtiaL necuhbive @nc.tiou have a npectmL hepmenah.tivn, i . e . admit a f i r s t o r d e r formula F with =, without function symbols and with cert a i n monadic p r e d i c a t e symbols Pi such t h a t f o r a l l x ~ , . . . , x ~ + ~ : f ( xl , . . . . x ) = x ~ +i f f ~ t h e r e i s a f i n i t e model of F where t h e monadic prea r e i n t e r p r e t e d a s s e t s of d i c a t e symbols P 1 , . . . , P n + l c a r d i n a l i t y x l , ... , x ~ +r e ~s p e c t i v e l y ( ' ) . Related t o these techniques developed by Trachtenbrot and o t h e r s i s the SpectnaLpmblem formulated i n Scholz (1952), i . e . the pmbLem t o chanactehize npectm f o r f i r s t o r d e r formulae including t h e i d e n t i t y symbol, where t h e spectrum ( F ) of a formula F i s defined a s t h e c l a s s of a l l those natural numbers n f o r which F has a f i n i t e model of c a r d i n a l i t y n . (Note t h a t by well known p r o p e r t i e s of f i r s t order l o g i c i t i s reasonable t o formulate t h i s problem a s done above f o r f i n i t e c a r d i n a l i t i e s of models f o r f i n i t e l y axiomatizable l o g i c a l t h e o r i e s with = , where without l o s s of g e n e r a l i t y functions a r e represented by t h e i r g r a p h s . ) Already e a r l y i n v e s t i g a t i o n s i n t o t h i s problem  s e e Asser (1955), Mostowski (1956)  showed t h a t t h e SpekxhzLpmbLem h a ~13 do w L t h compcLta.tivmL p m b L m a n d t h e i 4 cvmpLexity a t lower l e v e l s of t h e Grzegorczykhierarchy ( E n : 0 5 n ) of p r i m i t i v e r e c u r s i v e functions: f o r ex. every GrzegorczykE2 ( r e a d : by a determin i s t i c polynomialtimebounded r e g i s t e r machine a c c e p t a b l e ) set i s a spectrum and the c l a s s of s p e c t r a i s s t r i c t l y included in t h e c l a s s of a l l E3sets ( r e a d : of a l l s e t s acceptable by a d e t e r m i n i s t i c r e g i s t e r machine in exponential t i m e ) , whereas u p t o today i t i s not known whether every spectrum of a f i r s t o r d e r f o r mula i s a l s o an E2set nor what i s t h e answer t o Anne4'n pmbLem whether the complement of every spectrum i s a l s o a spectrum. Considering in the same way f o r any f i n i t e order n the c l a s s SPECTRAn : = {spectrum(F)I F l o g i c a l formula of order n} of s p e c t r a of n  t h o r d e r formulae i t turned o u t t h a t n  t h order s p e c t r a form a s t r i c t hierarchy SPECTRAn+l SPECTRAn+2exhausting e x a c t l y t h e c l a s s of a l l Kalmarelementary s e t s ( E 3  s e t s ) of p o s i t i v e n a t u r a l numbers (Bennett 1962), i n particular: T h e v m (Rodding & Schwichtenberg 1972): y n c SPECTRAn+l 5 y n + l Here y n denotes the c l a s s of a l l s e t s which a r e accepted by a r e g i s t e r machine an(x) within time bound a n ( p ) f o r some polynomial p and a o ( x ) : = x , a n + l ( x ) := 2 B e n n e t t ' s t h e s i s already contained among o t h e r s an autotmah Lhevnetic chamctehiza.tivn ad d i h b t o4dehspecxhz, b u t s i n c e i t was never published t h i s r e s u l t became not known t o t h e s c i e n t i f i c community. Rodding's and Schwichtenberg's above c i t e d paper (which was submitted on february 26, 1971) rediscovered among o t h e r s  without s t a t i n g i t e x p l i c i t e l y a s such  t h i s c h a r a c t e r i z a t i o n as s p e c i a l case obtained by a smooth n = 1 of the proof f o r t h e f i r s t i n c l u s i o n y n 5
( l ) Oeutsch (1975) s t r e n g t h e n s this r e s u l t t o closed prenex formulae with p r e f i x AV ...VA, only one occurence of t h e i d e n t i t y symbol and besides the P . only one, a binary, p r e d i c a t e symbol. His proof i s based on t h e DavisPutnamRabinson (1961) exponential diophantine normal form f o r r e c u r s i v e l y enumerable predic a t e s , s e e f o o t n o t e 2 on page 7 of t h i s paper and c f . Deutsch (1975). See a l s o t h e reduction i n Fagin (1975) of a r b i t r a r y s p e c t r a with only one e x t r a pred i c a t e , a binary one.
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n  t h order l o g i c a l d e s c r i p t i o n of timebounded r e g i s t e r machine computations over f i n i t e domains. Since t h a t paper was w r i t t e n i n german i t needed a t h i r d independent rediscovery of this r e s u l t by N.D.Jones and A.L.Selman  announced i n AMSNotices 19,2(1972) under number *72TE28 and published in Jones & Selman (1974) , and a n a t u r a l extension of i t t o f i r s t o r d e r f i n i t e l y axiomatizable p r o j e c t i v e c l a s s e s of f i n i t e type formulated e x p l i c i t e l y i n Fagin (1974), t o become widely known and c e l e b r a t e d . The fundamental idea underlying a l l these d i f f e r e n t proofs i s simple: t o give an a p p r o p r i a t e f i r s t  o r n  t h order L o g i c a t denchiption 0 4 &m i t e d hnLtLng p m b l m o d nrrckined OWeh 4inite ( n o t any more i n f i n i t e ) danrrim. In f a c t we can e x t r a c t t h i s idea e x p l i c i t e l y by showing t h a t a natural adaption of our l o g i c a l d e s c r i p t i o n of machine computations  namely t o f i n i t e computations t o be described over corresponding f i n i t e domains  y i e l d s almost t r i v i a l l y (and following t h e same proof p a t t e r n a s explained f o r the Church/Turing, the Aanderaa and t h e Trachtenbrot theorem) t h e above mentioned: .btomta RheomaZc chamctehizaaZon 06 & i & t  O h & h specfm : With r e s p e c t t o unary (resp. binary) r e p r e s e n t a t i o n SPECTRAl coincides with the NP ( r e s p . NEXPTIME) s e t s of p o s i t i v e n a t u r a l numbers. As usual P,NP,DEXPTIME,NEXPTIME denote the c l a s s of a l l s e t s which a r e accepted by a d e t e r m i n i s t i c resp. nondeterministic Turing machine within polynomial resp. exponential time in t h e ( b i n a r y ) length of the input. C.A. Christen i n his doctoral d i s s e r t a t i o n "Spektren und Klassen elementarer Funktionen" (ETH Zurich, 1974) has l i f t e d this BennettRoddingSchwichtenbergJonesSelmancharacterization t o higherorder s p e c t r a completing Rodding's and Schwichtenberg's inclusions yn 5 SPECTRAn+l 5 Y ~ t +o ~ SPECTRAn+l = NTIME( an+l)sets of p o s i t i v e numbers. Fagin (1974) observed t h a t t h i s c h a r a c t e r i z a t i o n of s p e c t r a a p p l i e s equally well t o f i n i t e l y axiomatizable c l a s s e s of f i n i t e s t r u c t u r e s : one only needs t o add t o the c o n s t r u c t i o n a n a p p r o p r i a t e encoding of s u b s e t s of f i n i t e s e t s . Remember t h a t a f i n i t e l y ( f i r s t o r d e r ) axiomatizable pmjectiwe c . k s b of f i n i t e type i n the sense of Tarski i s a c l a s s of p r e c i s e l y those f i n i t e s t r u c t u r e s ( i . e . w i t h f i n i t e domain and f i n i t e l y many f i n i t e r e l a t i o n s over t h a t domain) which a r e models of a formula V ... V a without f r e e individual v a r i a b l e s , with t h e bounded p r e d i c a t e '1 'r and some f r e e occuring p r e d i c a t e symbols R1,...,Rd, sometimes v a r i a b l e s P1,...,Pc such a c l a s s i s a l s o c a l l e d R1,...,Rd spectrum of V ... V o r simply genem'1 'r &zed ( f i r s t o r d e r ) specinurn. Assuming t a c i t l y t h a t a l l model c l a s s e s we a r e t a l l i n g about a r e closed under isomorphisms and r e f e r r i n g t o a standard encoding of f i n i t e s t r u c t u r e s i n t o binary words the proof of the above c h a r a c t e r i z a t i o n of f i r s t order s p e c t r a i s e a s i l y extended t o a proof f o r the following:
Chamctehizafion 0 6 genekxfized dih&tohakh s p e c t m : The (encodi ngs o f ) generalized s p e c t r a a r e p r e c i s e l y t h e NPsets (of nonempty words). The method of l o g i c a l d e s c r i p t i o n of f i n i t e computations over f i n i t e domains does not depend on the p a r t i c u l a r machine model. I f we apply i t mutatis mutandis d i r e c t l y t o f o r ex. t h e rudimentary p r e d i c a t e s i n t h e sense of Smullyan we o b t a i n d o h ewehy mckmentahy pmciicate a 6 i ~ O th & h mphedenaiztion i n ,5inite domim and thereby from the e x i s t e n c e of a rudimentary KleeneTpredicate a s c o r o l l a r y the n;completenedn 06 t h e emptineds pmbLem d o h 6imiohdeh &pectm (Buchi 1962) and the ~2compLetenedb 06 .thein: indinity pmblem. The BennettRoddingSchwichtenbergJonesSelma,iFaginChristencharacterization o f ( g e n e r a l i z e d ) s p e c t r a shows a very c l o s e connection between t h e Spektralpro
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blem resp. A s s e r s ' s ( s t i l l unsolved(')) complement problem f o r f i r s t order s p e c t r a and the fundamental and outstanding complexity theone.tical wahiant ad HLLobent'n E n h c h e i d u n g n p ~ ~ b l ewhich m can be formulated a s follows and i s known a s the P = NPproblem o r Cook'n ptroblsm I s t h e r e a d e t e r m i n i s t i c Turing machine wich recognizessatisfiability f o r a r b i t r a r y formulae of propositional l o g i c w i t h i n polynomial time bound ( i n the length of t h e i n p u t formulae)? Indeed t h i s open problem plays a c e n t r a l r o l e f o r the important question about t h e f e a s i b i l i t y o r not of a huge v a r i e t y of algorithms designed f o r combinatorial dec i s i o n problems occuring in almost every area of computer a p p l i c a t i o n s . This i s not the r i g h t place t o d i s c u s s i n more length the importance and the wide range of ram i f i c a t i o n s of Cook's problem i n t o many a r e a s of computer s c i e n c e , operations research, mathematics and l o g i c ; s e e f o r ex. the book by Garey & Johnson (1979) e want t o mention however one important a s p e c t of o r the survey Hartmanis (1982). W t h i s problem which i l l u m i n a t e s t h e analogy between Cook's problem and H i l b e r t ' s Entscheidungsproblem and echoes the analogy between the P = NPquestion and P o s t ' s problem in degree t h e o r y . ( 2 ) The P = NPproblem i s e q u i v a l e n t t o the above given formulation of Cook's problem because the decL5inion pmoblem doh plwpobLtiomf? l o g i c 0 complete d o h NP with r e s p e c t t o polynomialtime computable reductions in t h e same sense a s t h e p r e d i c a t e l o g i c decision problem i s manyone complete f o r t h e r e c u r s i v e l y enumerable s e t s : Cook'n theonem (1971). The s a t i s f i a b i l i t y problem f o r propositional l o g i c i s NPcomplete. The i n t e r e s t i n g p o i n t i n our context i s t h a t n o t only the r e s u l t i s analogous t o the clcompleteness of H i l b e r t ' s Entscheidungsproblem (with r e s p e c t t o deducibil i t y ) , b u t t h a t v i r t u a l l y t h e same proof can be given f o r both theorems: we j u s t eventually i n f i n i t e com u t a t i o n s in r e i n t e r p r e t e our firs_t~crdeyd e s c r i p t i o n ptoponi.tioml Logic terms doh d i n i t e compu*a.tionn. E s s e n ~ i i l T y  W ~  w ~ n T ~  ~ ~ ~ ~ r t amounts t o "look a t " f i r s t o r d e r atomic formulae  representing machine configur a t i o n s  a s propositional v a r i a b l e s and t o use f i n i t e conjunctions i n s t e a d of universal q u a n t i f i c a t i o n s f o r t h e d e s c r i p t i o n of p o s s i b l e machine t r a n s i t i o n s . We w i l l give the d e t a i l s i n the l a s t s e c t i o n . Apart from r e v e a l i n g in a n a t u r a l and s t r i k i n g l y simple way the j u s t one fundamental reason f o r computational completeness of f i r s t order and propositional
for
~
( l ) F o r generalized s p e c t r a where only monadic p r e d i c a t e symbols a r e allowed Fagin (1975 a ) shows t h a t not f o r every spectrum t h e complement i s a l s o a spectrum. See a l s o Fagin (1975 b ) and Yasuhara (1971) where o the c l a s s of number theor e t i c a l f u n c t i o n s d e f i n a b l e from successor and maxfny by composition and t h e folloYing maxQounded p r i m i t i v e recursion: f(x,l) = g(x) f(;,n+l) = max(')(h(;,n,f(;,n)),n+l) i t i s shown t h a t both t h e range and ( i f not empty) a l s o i t s complement a r e firstorder s p e c t r a f o r every element i n t h i s class.Examples of t h u s obtained s p e c t r a a r e the s e t of Fermat resp. o f Mersenne primes and t h e i r complements.Fagin (1975) gives an i n t e r e s t i n g c h a r a c t e r i z a t i o n of those s e t s X of n a t u r a l s where X and i t s complement a r e f i r s t  o r d e r s p e c t r a . (*)For i l l u m i n a t i n g remark on t h i s l a s t analogy s e e f o r ex. Fagin (1974:pp.86 sq.), Hartmanis (1982), Specker & S t r a s s e n (1976). I n t e r e s t i n g degree t h e o r e t i c a l ams i d e r a t i o n s about r e l a t i o n s between t h e complexity of s e t s of n a t u r a l s A,the s e t of formulae v a l i d i n a l l s t r u c t u r e s of c a r d i n a l i t y i n A and r e l a t i v i z e d h a l t i n g problems can be found i n Hay (1973), (1973 a ) , (1975), Selman (1973), (1974).
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l o g i c , f o r Aanderaa's and f o r T r a c h t e n b r o t ' s theorem, f o r complexity t h e o r e t i c a l c h a r a c t e r i z a t i o n s of s p e c t r a , t h i s way t o prove those r e s u l t s permits a l s o t o loc a t e e x p l i c i t e l y and in a p r e c i s e way how the LagLcal h t m a e of program desdeXemnLnhLLc to nondetemnLdLLc comp&c r i p t i o n s .LA addected by pIbAing . t i o ~ n . Cook (1971) proved t h a t the s a t i s f i a b i l i t y problem f o r propositional Krom formulae i s i n P , t h e i r u n s a t i s f i a b i l i t y problem i s even complete f o r nondetermin i s t i c logarithmic space a s shown by Jones & Laaser & Lien (1976) and an analogous completeness r e s u l t holds f o r t h e i r s a t i s f i a b i l i t y problem i n q u a n t i f i e d Boolean l o g i c , s e e Aspvall & Plass & Tarjan (1979). While i n t h e realm of f i r s t order d e s c r i p t i o n s of computations Kmm h ~ c t u mcan in some cases be obtained by an a p p r o p r i a t e choice of computation model and using s u f f i c i e n t l y rich o t h e r l o g i c a l expressive means  p r e f i x s t r u c t u r e , s i m i l a r i t y type e t c . , in some o t h e r cases i t c u t s down t h e complexity of decision problems from undecidable t o decidable a s can be seen from t h e Krom and Horn p r e f i x theorem; i n the propositional case f o r Cook's theorem Krom s t r u c t u r e cannot be obtained unless P = NP: in f a c t a choice between a t l e a s t two possible next s i t u a t i o n s following a given one does not seem t o be d e s c r i b a b l e by a binary d i s j u n c t i o n ( b u t with a t e r n a r y one i t i s ) ; note t h a t t r a n s i t i v i t y cannot be expressed by Krom formulae, c f . Krom 1966). S i m i l a r l y H a m ~ L u c t u ~seems e t o belong t o d e t e r m i n i s t i c computations: the s a t i s f i a b i l i t y problem f o r propositional Horn formulae i s in P and l i k e Krom formulae shares c e r t a i n completeness p r o p e r t i e s , s e e Jones & Laaser (1977) and Aanderaa & Borger (1979). In f a c t t h e Horn s t r u c t u r e can be preserved in going from f i r s t order t o propositional l o g i c d e s c r i p t i o n of the program formulae of d e t e r m i n i s t i c programs, but ( u n l e s s P = NP) t h i s i s impossible f o r t h e i n p u t d e s c r i p t i o n which can however be given by a Krom formula: Cook's theorem can be shown with a conjunction of a program formula which i s not Krom b u t Horn except f o r those conjuncb describing nondeterministic moves and a Krom formula which i s n o t Horn f o r desc r i p t i o n of input and s t o p condition. T h u s i t seems natural t o measure the comp l e x i t y o d Boolean ~unc.titiolzn in terms of minimal length of propositional formulae defining them and having (almost) Horn s t r u c t u r e . This y i e l d s a complexity measure which i s s t r o n g l y r e l a t e d t o Cook's problem and which by a natural adaption of the AanderaaBorger reduction method t o propositional d e s c r i p t i o n s of logical networks has been shown t o be equivalent t o network and Turing machine complexity f o r Boolean f u n c t i o n s , a s w i l l be discussed i n t h e l a s t s e c t i o n . The Spektralproblem and i t s r e l a t i o n t o computational complexity problems i s only one example i n t h e s t i l l growing f i e l d of complexity theory where smooth logical d e s c r i p t i o n s of combinatorial (computational) problems play a d e c i s i v e r o l e . From the examples i n t h e following s e c t i o n s i t should become c l e a r t h a t i f the logical d e s c r i p t i o n i s such a s t o "show" an i n t i m a t e s t r u c t u r a l c o r r e l a t i o n between the combinatorial system described and the l o g i c a l expressive means used, then t h i s link w i l l a l s o c a r r y over complexity phenomena from the computational system t o the corresponding l o g i c a l system; indeed the optimal s i t u a t i o n i s t h a t Rhe l o g i cal nynteni can be u i w e d v i a Rhe tmMnk.tion jut an a naiutuml i.mplemevLta.tion 0 6 the denchibed compu;ting nyhtem. In a way t h i s d e s i r e underlies a l s o t h e many approaches t o d e f i n e semantics of programming languages by l o g i c a l o r a b s t r a c t a l g e b r a i c a l means; i t underlies numerous simulation techniques between various computation models. Success r e s u l t s here not only i n b e t t e r and deeper understanding of t h e s i t u a t i o n , b u t a l s o in b e t t e r technical s o l u t i o n s of given problems. (Take a s a b s t r a c t example t h e simulation techniques developed in Borger 1979.) In any way i t w i l l become c l e a r from t h e next s e c t i o n s t h a t and why the techniques developed f o r e s t a b l i s h i n g lower complexity bounds f o r decision procedures f o r dec i d a b l e l o g i c a l decision problems resemble s t r o n g l y those developed i n t r a d i t i o n a l reduction theory. The reader i s i n v i t e d t o compare t h i s w i t h t h e i l l u m i n a t i n g d i s cussion i n the book of Machtey & Young (1978) where i t i s shown how lower comp l e x i t y r e s u l t s can be derived by methods invented by Godel t o give h i s incompleteness theorems, j u s t by proving them through a p p r o p r i a t e r e p r e s e n t a b i l i t y ( r e a d : expressabi l i t y ) statements.
om
By careful l o g i c a l d e s c r i p t i o n s of a p p r o p r i a t e l y chosen computation models many lower complexity bound r e s u l t s have been proved f o r l o g i c a l t h e o r i e s which in some cases meet e x a c t l y known upper bounds ( i . e . complexity of e x i s t i n g algorithms f o r
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t h e s o l u t i o n o f t h e problem under c o n s i d e r a t i o n ) . I n t h e r e a l m o f pure p r e d i c a t e l o g i c t h e f o l l o w i n g two theorems resume e s s e n t i a l l y t h e s t a t e o f t h e a r t :
d x n p l e x i t y Theomri d o h P t l e d i x  S U h i t g C h n n e n . The d e c i s i o n problem w i t h r e s p e c t t o s a t i s f i a b i l i t y o f t h e f o l l o w i n g s o l v a b l e subclasses o f p r e d i c a t e l o g i c has t h e i n d i c a t e d p r e c i s e ( l o w e r and upper bound) c o m p l e x i t y where c denotes some c o n s t a n t and n t h e i n p u t l e n g t h : Monadic p r e d i c a t e l o g i c : N T I M E ( c ~ " ~ ~1 VmAVm n Monadic : DTIME( " " ) vm*2vm
: NTIME(
V"A"
: NTIME(C")
"
"
)
Examples o f c l a s s e s w i t h NPcomplete s a t i s f i a b i l i t y problem a r e VpAq w i t h i y t q , V"Aq w i t h OZq,A"
and VAm.
Complexity Theotlem d o h K m m nubc&zannen. The s a t i s f i a b i l i t y problem f o r Krom f o r mulae r e s t r i c t e d t o t h e f o l l o w i n g c l a s s e s has t h e i n d i c a t e d c o m p l e x i t y : Monadic i s complete f o r P 2 V ~ (even A ~ v A") i s complete f o r p o l y n o m i a l space V ~ A (even ~ V ~ A ~ V )i s complete f o r DEXPTIME
b u t V"AkVm i s i n P f o r e v e r y f i x e d k AVA i s i n P The u n s a t i s f i a b l e Krom f o r m u l a e i n AVA a r e complete f o r n o n d e t e r m i n i s t i c l o g a r i t h m i c space. The u n s a t i s f i a b l e Herbrand formulae ( i . e . prenex normal forms whose m a t r i x i s a c o n j u n c t i o n o f a t o m i c o r negated atomic f o r m u l a e ) a r e a l s o comp l e t e f o r n o n d e t e r m i n i s t i c l o g a r i t h m i c space. T h e r e f o r e n o t o n l y f o r t h e u n d e c i d a b l e cases, b u t a l s o f o r t h e c o m p l e x i t y o f dec i d a b l e cases o f t h e Entscheidungsproblem Krom s t r u c t u r e p l a y s a d e c i s i v e r o l e . Complete ( r e f e r e n c e s and) p r o o f s f o r t h e above two theorems can be found i n Lewis (1980), F u r e r (1981), Denenberg & Lewis (1982), Lewis & Statman (1983); we l i m i t o u r s e l v e s h e r e t o a s h o r t comment on how t h e l o w e r c o m p l e x i t y bounds a r e o b t a i n e d t o g i v e t h e r e a d e r a f e e l i n g t h a t i n a s t r o n g sense u n d e c i d a b i l i t y and l o w e r comp l e x i t y bound r e s u l t s a r e s i m i l a r i n n a t u r e . The l o w e r NEXPTIME bounds f o r t h e monadic, t h e GodelKalmarSchutte and t h e SchonfinkelBernays case i n Lewis (1980) a r e o b t a i n e d by a d i r e c t d e s c r i p t i o n o f t h e acceptance problem f o r n o n d e t e r m i n i s t i c exponentialtimebounded T u r i n g machine computations as s a t i s f i a b i l i t y q u e s t i o n f o r f o r m u l a e o f t h e f o r m VAAVAM w i t h o n l y monadic p r e d i c a t e s i n t h e f i r s t two cases,of t h e f o r m V...VA...A i n t h e t h i r d case where p a r t i c u l a r c o m p l i c a t i o n s a r i s e f o r an a p p r o p r i a t e d e s c r i p t i o n o f t h e successor r e l a t i o n between t h e encod i n g s o f n a t u r a l numbers ( t h e s e r e p r e s e n t a t i o n resemble by t h e way those which had t o be i n t r o d u c e d by Jones & Selman (1974) f o r t h e i r automata t h e o r e t i c c h a r a c t e r i z a t i o n o f s p e c t r a . ) F u r e r ( p r i v a t e communication) o b a t i n e d t h e l o w e r c o m p l e x i t y bound N T I M E ( c ~ " ~') ~ even f o r t h e subclass A V n M f r o m a r e d u c t i o n t o i t o f t h e n o t o r i g i n c o n s t r a i n e d bounded domino problem which he has shown t o be o f e x a c t (upper and l o w e r ) c o m p l e x i t y NTIME(cn). The l o w e r DEXPTIME bound f o r t h e monadic Ackermann case has been o b t a i n e d i n d e p e n d e n t l y by F u r e r (1981) and Lewis (1980); Lewis achieves t h e r e s u l t by a d e s c r i p t i o n o f t h e nonacceptance problem f o r t h e a l t e r n a t i n g pushdown automata ( i n v e s t i g a t e d i n Chandra & Stockmeyer (1976) and Ladner & L i p t o n & Stockmeyer (1978) and a c c e p t i n g p r e c i s e l y t h e s e t s i n DEXPTIME), v e r y much i n t h e s p i r i t o f t h e r e d u c t i o n t e c h n i q u e e x p l a i n e d i n t h e n e x t s e c t i o n based on t h e f a c t t h a t t h e p r e f i x s t r u c t u r e i n t h i s case a l l o w s t o speak d i r e c t l y about t h e words t o be memorized i n t h e s t a c k ; a s i m i l a r d e s c r i p t i o n o f a l t e r 
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nating stack automata accepting p r e c i s e l y t h e s e t s in DZEXPTIME y i e l d s a lower n/log n f o r another i n t e r e s t i n g subclass of t h e p r e f i x c l a s s complexity bound cc \p'~\p'/\determined by imposing r e s t r i c t i o n s on the form of occuring atomic s u b formulae b u i l t u p from only binary p r e d i c a t e symbols, see again Lewis (1980); Fiirer describes lineartimebounded a l t e r n a t i n g Turing machines using t h e f a c t t h a t DTIME ( c f ( " ) ) ~ASPACE(f(n)) f o r every f and o b t a i n s thereby even formulae k in t h e subclass AV where k comes from t h e number of successors in universal branch s t a t e s . The complexity theorem f o r Krom c l a s s e s i s e n t i r e l y due t o Denenberg & Lewis (1982). They o b t a i n t h e i r lower complexity b o u n d s ( r e a d : completeness r e s u l t s ) in the monadic case from t h e observation t h a t Lewis'(1980) construction f o r the Ackermann c l a s s y i e l d s monadic formulae and t h a t t h e only conjuncts in the f o r mulae describing t h e nonacceptance problem of an a l t e r n a t i n g pushdown automaton which a r e not Krom a r e those used t o describe t h e a c t i o n of t h e machine a t universal branch s t a t e s ; b u t without universal branch s t a t e s one i s describing the nonacceptance problem of nondeterministic ( r a t h e r than a l t e r n a t i n g ) pushdown automata which i s hard f o r P . For the BernaysSchonfinkel Krom case a d e s c r i p t i o n of polynomialspacebounded Turing machine computations i s given again in the same s p i r i t a s the method explained in t h e next s e c t i o n using t h e f a c t t h a t the p r e f i x s t r u c t u r e in t h i s case allows t o r e p r e s e n t s t a t e s and symbols of the machine d i r e c t l y by constant terms; t h e space needed f o r t h e computation i s taken i n t o c a r e by t h e number of arguments of t h e (unique) p r e d i c a t e symbol which determines a l s o t h e number of universal q u a n t i f i e r s needed. The subcase with only 2 e x i s t e n t i a l q u a n t i f i e r s comes from t h e f a c t t h a t i t i s s u f f i c i e n t t o have machines working over an alphabet with only 2 symbols. S i q i l a r l y t h e lower DEXPTIME bound f o r t h e Maslov case i s achieved bv a d e s c r i o t i o n ofcomoutations of 1 inearspacebounded a ] t e r n a t i n q Turinq machjnesiwhich accept p r e c i s e i y t h e s e t s in DEXPTIME,the r e s u l t i n a formulae a r e even i n A V . A l p t comment has t o bemade on t h e G 6 d d c&eanean w L t h identity, i . e . t h e c l a s s V"A V" w i t h t h e e q u a l i t y symbol allowed. I t i s not known whether t h i s c l a s s has a recursive decision problem o r not whereas without i d e n t i t y i i s f i n i t e l y cont r o l l a b l e a s shown by Godel (1933) and Schutte (1933), (1934)tl;. Goldfarb (1981) has shown t h a t a t l e a s t t h e r e can be no primitive r e c u r s i v e decision procedure. His proof c o n s t r u c t s formulae F n ~ M Vw i t h e q u a l i t y describing i n i t i a l p a r t s of t h e graph of t h e Ackermann function; t h e Fn a r e (even f i n i t e l y ) s a t i s f i a b l e b u t not over domains with l e s s than a(n,O) elements f o r t h e Ackermann function (Y. Recently f o r a subcase of t h e Godel c l a s s with = , Goldfarb & Gurevich & Shelah (1983) gave a proof of f i n i t e c o n t r o l l a b i l i t y ; t h i s i s t h e s u b l a s s of formulae Q YF in N\V which r e q u i r e only f o r every unordered p a i r Ix,y) ( i n s t e a d of every ordered p a i r ( x . y ) ) a v such t h a t F(x,y,v) holds,formally speaking t h e Class of a l l formulae of form A A V((KXy + ~ K y x )& (KXY + G ) )
0
X Y
v
with a binary p r e d i c a t e symbol K and a q u a n t i f i e r f r e e formula G ; t h e a l l e g e d decision procedure i s not primite r e c u r s i v e , but i t i s not known whether t h e r e can be no p r i m i t i v e r e c u r s i v e one: u p t o now no formulae a r e known i n t h a t c l a s s which allow only "big" models l i k e Goldfarb's Fn mentioned above. We conclude t h i s panorama of main i d e a s , methods and r e s u l t s in c l a s s i c a l reduction and complexity theory f o r l o g i c a l decision problems by a h i n t t o an area of research which has not y e t found broader a t t e n t i o n d e s p i t e i t s n a t u r a l n e s s s many f a s c i n a t i n g open problems and t h e a v a i l a b i l i t y o f strong methods which could
(l)Gurevich & Shelah (1983) have p u t foreward a very e l e g a n t and s t r a i g h t forward p r o b a b i l i s t i c argument showing f i n i t e c o n t r o l l a b i l i t y of A2V" with = .
278
E. BORCER
e v e n t u a l l y be p u t t o use t o s e t t l e those problems. I am speaking a b o u t t h e comp l e x i t y o f t h e d u h i o n p4obLem 6011. decidxbee &COJL&A h.enZLicted t o 6om&e with 4impLe p X d d i e h O h r m W x oh atomic 4ub6om&e na3tuctu4e : a huge amount o f r e s u l t s i s known a b o u t d e c i s i o n problems f o r d e c i d a b l e f i r s t o r (weak) (monadic) second o r d e r t h e o r i e s and t h e c o m p l e x i t y o f d e c i s i o n procedures f o r d e c i d a b l e cases(1); we have seen how much work has been done i n p u r e p r e d i c a t e l o g i c t o determine t h e non o r s u b r e c u r s i v e c o m p l e x i t y o f t h e d e c i s i o n problem f o r f o r m u l a e c l a s s e s determined b y r e s t r a i n t s on t h e a v i l a b l e e x p r e s s i v e means l i k e q u a n t i f i e r , m a t r i x o r a t o m i c subformulae s t r u c t u r e . I n p a r t i c u l a r we have seen t h a t such s t r u c t u r a l c o n s t r a i n t s p l a y an i m p o r t a n t r o l e i n c u t t i n g down huge ( u p p e r and l o w e r ) c o m p l e x i t y bounds. The s i t u a t i o n t h a t a l m o s t a l l a c t u a l l y known l o w e r combounds f o r d e c i d a b l e t h e o r i e s a r e s o h u g e ( 2 ) may w e l l depend on t h e f a c t t h a t no n a t u r a l s t r u c t u r a l c o n s t r a i n t s a r e imposed on t h e formulae; f o r ex. what a b o u t a huge l o w e r c o m p l e x i t y bound f o r a t h e o r y i f a r b i t r a r y q u a n t i f i e r l e n g t h s and q u a n t i f i e r a l t e r n a t i o n s a r e a l l o w e d which h a r d l y appear i n mathematical p r a c t i c e ? To b r i n g t o a e t h e r t h e s e two l i n e s o f r e s e a r c h w i l l be a f r u i t  and s u c c e s s f u l enterprise.( 3) A LOGICAL DESCRIPTION OF MACHINE COMPUTATIONS
F o l l o w i n g Aanderaa (1971) and B o r g e r (1971) we b e g i n w i t h a d e s c r i p t i o n o f how machines M can be encoded smoothly i n f o r m u l a e aM. Through an a p p r o p r i a t e c h o i c e of M and aM we w i s h t o a c h i e v e two t h i n g s : t o g e t s y n t a c t i c a l l y s i m p l e f o r m u l a e aM whose l o g i c a l s t r u c t u r e r e f l e c t s t h e s y n t a c t i c a l s t r u c t u r e o f M, and  based on such 9 r e l a t i o n  t o make t h e p r o o f o f e q u i v a l e n c e o f t h e M  d e c i s i o n problem t o ( l ) See t h e e x c e l l e n t surveys Ershov e t a l . (1965), Rabin (1977), t h e book F e r r a n t e & Rackow (1979) and Kozen (1979) f o r r e s u l t s and r e f e r e n c e s . Joseph & Young (1981) c o n t a i n a d i s c u s s i o n o f r e l a t i o n s o f q u e s t i o n s o f p r o v a b i l i t y i n weak t h e o r i e s o f a r i t h m e t i c t o such computational q u e s t i o n s as whether P = NP o r NP = coNP. I f may be n o t e d a l s o t h a t ( u n  ) d e c i d a b i l i t y r e s u l t s f o r ( f i r s t o r d e r ) l o g i c a l t h e o r i e s can i n t u r n y i e l d r e s u l t s i n p u r e p r e d i c a t e l o g i c ; f o r ex. H e i d l e r (1973) o b t a i n s t h e s u r p r i s i n g r e s u l t t h a t p u r e e q u a t i o n a l l o g i c w i t h o u t any o t h e r p r e d i c a t e symbol a p a r t f r o m = and w i t h o u t any f u n c t i o n symbol b u t a l l o w i n g H i l b e r t ' s c h o i c e o p e r a t o r E t o b u i l d terms f r o m f o r m u l a e has an u n s o l v a b l e d e c i s i o n problem, by r e d u c i n g t o i t t h e ( u n d e c i d a b l e ) t h e o r y o f one symmetric r e l a t i o n .
( * ) Ex: n o n  K a l m a r  e l e m e n t a r i t y o f weak monadic second o r d e r t h e o r y o f one successor o r o f f i r s t o r d e r t h e o r y o f l i n e a r o r d e r (Meyer ( 1 9 7 5 ) ) , t h e t r i p l e exp o n e n t i a l l o w e r bound f o r t h e d e c i s i o n problem o f m u l t i p l i c a t i v e a r i t h m e t i c ( F i s c h e r & Rabin ( 1 9 7 5 ) ) , d o u b l e e x p o n e n t i a l f o r P r e s b u r g e r a r i t h m e t i c ( F i s c h e r & Rabin ( 1 9 7 4 ) J and f o r r e a l a d d i t i o n t h e completeness i n t h e c l a s s o f problems s o l v e d by a l t e r n a t i n g T u r i n g machines i n t i m e bound 2Cn u s i n g n a l t e r n a t i o n s f o r some c o n s t a n t c, see F i s c h e r & Rabin (1974), F e r r a n t e & Rackow (1975) and Berman (1977). ( 3 ) Some i n t e r e s t i n g r e s u l t s i n t h i s d i r e c t i o n : see G u r e v i c h (1965) and S c a r p e l l i n i (1982) where t h e r o l e o f q u a n t i f i e r r e s t r i c t i o n s on (un) d e c i d i a b i l i t y f o r some t h e o r i e s i s analysed, and B o r g e r & K l e i n e Buning (1980) where f o r e x t e n s i o n s o f m u l t i p l i c a t i v e a r i t h m e t i c i t i s shown t h a t r e s t r i c t i o n s on p r e f i x  s i m i l a r i t y type, q u a n t i f i e r , Krom, Horn and t e r m s t r u c t u r e c u t down a r b i t r a r i l y complex u n d e c i d a b i l i t y t o d e c i d a b i l i t y . One h a l f o f t h i s c l a i m i s p r o v e d b y a p p l i c a t i o n s o f t h e r e d u c t i o n method o u t l i n e d i n t h i s s e c t i o n t o v a r i o u s o t h e r c o m p u t a t i o n f o r m a l i s m s l i k e r e s t r i c t e d P o s t c a n o n i c a l forms, P e t r i n e t s ( d e s c r i b e d as f a c t o r replacement systems)etc. ( N o t e t h a t i n t h e meantime i t has been p r o v e d i n K o s a r a j u (1982) t h a t t h e r e a c h a b i l i t y problem f o r P e t r i n e t s i s i n d e e d r e c u r s i v e , a r e s u l t which i s needed f o r t h e above c l a i m . )
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t h e a M  d e d u c i b i l i t y q u e s t i o n i n f i r s t o r d e r p r e d i c a t e l o g i c a t r i v i a l one. As exp l a i n e d i n t h e course o f t h e p r e c e d i n g s e c t i o n v i a Skolem prenex normal form t h e data of M a r e r e p r e s e n t e d by t h e l o g i c a l terms c o n s t i t u t i n g t h e u n i v e r s e o f t h e i n t e n d e d model o f aM; t h e s t a t e s o f t h e f i n i t e c o n t r o l o f M a r e encoded by p r e d i c a t e symbols o c c u r i n g i n ab,. The q u a n t i f i e r f r e e m a t r i x o f aM w i l l be a conj u n c t i o n o f i m p l i c a t i o n s where t o each p o s s i b l e t r a n s i t i o n s t e p ( " i n s t r u c t i o n " , " r u l e " ) Ii d e f i n e d by t h e program M corresponds a c o n j u n c t pi o f aM a s s u r i n g t h a t i f an i n s t a n c e o f t h e p r e m i s e s i n
pi
r e p r e s e n t s a c o n f i g u r a t i o n C o f M,
then t h e c o r r e s p o n d i n g i n s t a n c e o f t h e c o n c l u s i o n s i n
pi
r e p r e s e n t s t h e immediately
succeding c o n f i g u r a t i o n a c c o r d i n g t o Ii. Consider a r e g i s t e r machine program M w o r k i n g o v e r 2 r e g i s t e r s . S i n c e t h e i r i n v e n t i o n by Minsky (1961) and Shepherdson & S t u r g i s (1963) t h e s e machines have become w i d e l y known, i n p a r t i c u l a r t h e f a c t t h a t t h e y a r e u n i v e r s a l f o r t h e computation o f a l l p a r t i a l r e c u r s i v e f u n c t i o n s . F o r convenience o f e x p o s i t i o n we assume w i t h o u t l o s s o f g e n e r a l i t y t h a t M c o n s i s t s o f i n s t r u c t i o n s Ii = ( i : do oi,go
t o (p,q))
oiEIal,a2,s1,s2,stop}
w i t h i n s t r u c t i o n numbers l ~ i ~o pre,r a t i o n symbols and numbers l s i , q i z r
o f t h e n e x t i n s t r u c t i o n t o be executed.
Execute Ii means: t e s t , i f t h e r e g i s t e r c o n s i d e r e d i n oi equals z e r o o r n o t ,
 i . e . +1 i n t h e j  t h r e g i s t e r i n case oi = aJ. ' 1 i n t h e j  t h r e g i s t e r i n case oi = s .  and t h e n go o v e r t o t h e n e x t i n s t r u c t i o n w i t h number J pi r e s p . qi i f t h e r e g i s t e r t e s t e d was equal z e r o r e s p . n o t . Assume a l s o t h a t
execute oi
Ir = ( r , s t o e go t o r ) , t h a t t h e s t a r t i n s t r u c t i o n i s I1 and t h a t t e s t s a r e executed o n l y i n s u b t r a c t i o n i n s t r u c t i o n s ( i . e .
oiEIal,a2)
i m p l i e s pi = qi
in
which case we w r i t e pi i n s t e a d o f (pi,pi)).
To
defines\,
we r e p r e s e n t n a t u r a l numbers n by means o f l o g i c a l terms b u i l t up
from a symbol 0 f o r an i n d i v i d u a l c o n s t a n t and a symbol ' o f a monadic f u n c t i o n ; f o r a l a t e r a p p l i c a t i o n we w r i t e these terms b o t h i n p r e f i x  and i n p o s t f i x n o t a t i o n , i . e . i n t h e forms 
n : = O& n times
:E
or

I...I v 0
n  t i mes
Any s t a t e i o f M i s r e p r e s e n t e d by a b i n a r y p r e d i c a t e symbol, c a l l i t a g a i n Ii. T h e r e f o r e an M  c o n f i g u r a t i o n C = (m,i,n) o f s t a t e i and c o n t e n t s m o f t h e f i r s t and n o f t h e second r e g i s t e r i s coded by t h e l o g i c a l a t o m i c f o r m u l a (m,i,n)
::
iii I. n 1 
t o be i n t e r p r e t e d as meaning t h a t t h e c o n f i g u r a t i o n C can be reached through an Mcomputation s t a r t i n g f r o m a p r e v i o u s l y g i v e n i n i t i a l c o n f i g u r a t i o n . D e f i n e uM : :A A ( P ~ A . . . A ~ ~w i t ~h ) t h e c o n j u n c t s XY i n s t r u c t i o n Ii on c o n f i g u r a t i o n s d e f i n e d by: x'1.y J (OIiy + O I j y ) xIiy
f o r I.= ( i , d o al,
f
A
(x'Iiy
f
pi
describing the e f f e c t o f
go t o j )
x I k y ) f o r Ii=(iSl,
go t o ( j , k ) )
S y m m e t r i c a l l y f o r i n s t r u c t i o n s w i t h o p e r a t i o n s a2 o r s2 by i n t e r c h a n g i n g t h e p o s i t i o n s o f x and y and u s i n g ' x i n s t e a d o f X I .
E. BORGER
280
I t should already be c l e a r from the above t h a t r e a l l y u i s nothing e l s e t h a n ( a definition of the e f f e c t o f ) the program of instructrons M formulated in f i r s t order logical terms; in f a c t one has t r i v i a l l y the following
RechcLLoian Cmm. For any Mconfigurations C , D holds: C +M D iff u M h i5 + D
bL
where C +M D means t h a t M , s t a r t e d in C , a f t e r a f i n i t e number of steps reaches D, and $L denotes deducibility in predicate logic. Phood. Based on Skolem's theorem(') on canonical interpretations of s a t i s f i a b l e formulae over the corresponding term domains discussed i n the preceding section one obtains the implication from r i g h t t o l e f t j u s t by observing t h a t the above indicated intended canonical interpretation over the natural numbers of m1.n as meaning C 3 (m,i,n) 1
obviously yields a model f o r the premisse uM A c, from which in t h a t model the follows meaning t h a t C +M 0. The other implication follows from the conclusion f a c t t h a t by the very definition of uM as e f f e c t of program M , any model for uM simulates every possible computation step of M ; t o say i t more precisely in any canonical model of uM over the naturals a n d f o r every single Mcomputation step producing D from C by l e t us say instruction I . , the corresponding conf r h C in the given model. junct pi assures the t r u t h of the inference of
n
Since 2register machine programs a r e known t o be universal f o r the computation of p a r t i a l recursive functions the above reduction lemma already contains a complete proof f o r the
C a t o U u h q . The Entscheidungsproblem f o r ( r e s t r i c t e d ) f i r s t order predicate logic i s complete f o r the recursively enumerable s e t s (Church(1936), Turing (1937)) even when r e s t r i c t e d t o the class VAVA(0,m) (Buchi (1962)) and Krom and Horn formulae (Aanderaa (1971), Borger ( 1 9 7 1 ) ) ( 2 ) ; thereby t h i s class i s a reduction class f o r sa t i s f i abi 1 ity . Indeed by contraposition a ( f o r ex. halting) configuration D cannot be reached by M from C i f f the formula uM A A ID i s s a t i s f i a b l e , and t h i s formula i s via prenexing equivalent t o the Skolem normal form of a Krom and Horn formula in VAVA(0,m).
As referred t o in the preceding section many other r e s u l t s on strong reduction classes have been obtained by variations and f u r t h e r refinements of the above method to describe over term domains suited t o the data s t r u c t u r e of the given machine model three things:  the e f f e c t uM of the program M  a s t a r t condition C  a (non) stoo condition ij (1) See f o r example Kreisel & Krivine (1967: pp. 1820) (L)
With respect t o prefix and Krom structure t h i s r e s u l t i s already optimal as i t i s with prefix AVVA; the l a t t e r i s obtained in l i t e r a l l y the same way by substituting 0 by a term *x f o r a new Skolem function symbol * t o be used as constant zero function.
28 1
Decision Problems in Predicate Logic
We want t o g i v e h e r e d e t a i l s f o r j u s t one o t h e r case t o i l l u s t r a t e t h a t such r e d u c t i o n s can y i e l d n o t o n l y u n d e c i d a b i l i t y phenomena f o r l o g i c a l d e c i s i o n problems based on t h e u n d e c i d a b i l i t y o f t h e c o m b i n a t o r i a l system described, b u t by s i m p l e c o n t r a p o s i t i o n a1 so Logical dechiniotz phoccddwren doh cornbimtotohiaL ciecOiun pmbLem reduced t o d e c i s i o n problems o f a s o l v a b l e c l a s s o f formulae. T h i s approach has been i n v e s t i g a t e d by Lewis (1975) f o r ( t h e emptyness problem o f ) c o n t e x t f r e e grammars, f i n i t e ( t r e e ) automata ( o f f i n i t e o r d e r ) and pushdown automata and f o r ( t o t a l i t y and t o t a l  e q u i v a l e n c e problem o f ) f u l l schemas; i n t h e l a s t case an i m provement o f Manna's (1968) r e d u c t i o n o f these problems f o r a b s t r a c t programs t o t h e d e c i s i o n problem f o r p r e d i c a t e l o g i c w i t h f u n c t i o n symbols ( b u t n o t = ) i s g i v e n which a v o i d s use o f f u n c t i o n s and r e s u l t s f o r f u l l schemata i n formulae o f a d e c i d a b l e c l a s s . F o r t h e o t h e r cases r e d u c t i o n s t o f o r m u l a e o f Monadic p r e d i c a t e resp. p r o p o s i t i o n a l l o g i c a r e g i v e n whose d e c i s i o n problem i s known t o be s o l v a b l e . We show h e r e t h e d e s c r i p t i o n o f c o n t e x t f r e e grammars. L e t a c o n t e x t  f r e e grammar M be g i v e n w i t h o u t l o s s o f g e n e r a l i t y w i t h r u l e s Sk + S . S and S . + ak i n Chomsky normal f o r m and axiom S1. We g i v e a d e s c r i p t i o n 1 J J o f M by a Skolemized f o r m u l a uM w i t h prenex normal f o r m i n t h e monadic subclass 2 o f V"A V; t h e t e r m i n a l symbols ak a r e r e p r e s e n t e d by i n d i v i d u a l c o n s t a n t s (denoted a g a i n b y ) ak t h e o p e r a t i o n o f c o n c a t e n a t i o n o f symbols t o words b y a b i n a r y f u n c t i o n symbol denoted by
'I()":
any t e r m
T
b u i l t up f r o m t h e a k by use o f " ( ) "
corresponds t o t h e u n i q u e l y determined word t o b t a i n e d f r o m T by j u s t c a n c e l l i n g a l l parentheses. Every v a r i a b l e 5 . o f M i s r e p r e s e n t e d by a monadic p r e d i c a t e 1
symbol (denoted a g a i n b y ) Si w i t h t h e f o l l o w i n g i n t e n d e d i n t e r p r e t a t i o n :
Sit
means
Si +Mt
t h e q u e s t i o n i f S1 fMt f o r some t e r m i n a l word t i s f o r m u l a t e d as q u e s t i o n
i.e.
whether some t can be parsed s u c c e s s f u l l y by M (read: whether i n v e r s e a p p l i c a t i o n s o f grammatical r u l e s o f M t o some t f i n a l l y y i e l d S1). D e f i n e uM :=
A A ( P ~ A . . . ~w~i t)h t h e c o n j u n c t s B~
describing the parsing e f f e c t
(i.e.
t h e e f f e $ t Y o f an i n v e r s e a p p l i c a t i o n ) o f t h e L  t h grammatical r u l e S.S o f M t o words d e f i n e d by: i j Six A S.y + S k ( x y ) f o r e v e r y r u l e Sk + S . S . i n M and l e t tMbe t h e J 1 J c o n j u n c t i o n o f a l l formulae:
Sk
+
f o r every r u l e S .a J k Then t h e f o l l o w i n g Ireduc.tion p ~ o p e h t yh o l d s :
S1 +Mtf o r some t e r m i n a l word t i f f +
PL
S. J
+
uM A tM+ VSlx
ak
in
1.1
as one can show paraphrasing
X
t h e a r g u m e n t a t i o n g i v e n f o r t h e p r e c e d i n g r e d u c t i o n o f r e g i s t e r machines. Theref o r e uM A tMA 1 V S1x i s a monadic f o r m u l a which i s s a t i s f i a b l e i f f M generates X
t h e empty language; and a prenex normal f o r m o f i t i s indeed i n t h e c l a s s V.. . V M V w i t h number o f b e g i n n i n g e x i s t e n t i a l q u a n t i f i e r s depending on t h e number o f t e r m i n a l symbols. P r e c i s e l y t h e same c o n s t r u c t i o n a p p l i e s t o f i n i t e t r e e automata w i t h f i x e d o r d e r k where S i f expresses t h a t M w o r k i n g on t reaches t h e r o o t i n s t a t e i; t h e i m p l i c a t i o n o f u , , c o r r e s p o n d i n g t o an i n s t r u c t i o n ( i ( l ) , Si!l)xl~...~Si(k)xk jth letter.
+
Sefjx l...xk
...,i ( k ) , j )
+
L
o f M i s then
w i t h a k  a r y Skolem f u n c t i o n f . r e p r e s e n t i n g t h e J
E. BORGER
282
COMPLEXITY RELATIONS BETWEEN PROGRAMS AND PROGRAM FORMULAE The i n t i m a t e c o n n e c t i o n between programs M and t h e i r l o g i c a l d e s c r i p t i o n ',,, ( b e t t e r : t h e l o g i c a l f o r m u l a t i o n o f t h e i r e f f e c t on g i v e n data, o f t h e i r semant i c s as i s s a i d i n computer s c i e n c e ) which has been e s t a b l i s h e d by t h e method exp l a i n e d i n t h e p r e c e d i n g s e c t i o n f o r t h e examples r e g i s t e r machines, c o n t e x t f r e e grammars and f i n i t e automata i s such as t o c a r r y o v e r n o t o n l y ( u n  ) d e c i d a b i l i t y p r o p e r t i e s b u t a l s o t o p r e s e r v e t h e c o m p l e x i t y o f c o r r e s p o n d i n g d e c i s i o n problems, a g a i n i n such a way t h a t t h e p r o o f o f t h i s c o m p l e x i t y p r e s e r v a t i o n i s o b v i o u s f r o m t h e c o n s t r u c t i o n . We i l l u s t r a t e t h i s f e a t u r e by two examples c o n s i d e r i n g a ) deghee compeexity o f d e c i s i o n problems, b ) i M e p m b i U t y phopehtien f o r machine problems and t h e i r t r a n s l a t i o n i n t o complexity bouna2 ( o h f i t h e mode,& o f t h e desc r i b i n g formulae. S i n c e t h e program
M
and t h e program f o r m u l a aM a r e " t h e same"

even more: i n t h e
r e d u c t i o n lemma e v e r y c o m p u t a t i o n s t e p o f M i s i n 11correspondence w i t h a l o g i c a l d e d u c t i o n s t e p u s i n g an i m p l i c a t i o n o f aM  i t i s n o t s u r p r i s i n g t h a t b y t h e r e d u c t i o n lemma any 0  h a l t i n g problem HD(M) : = IC
I
C
+M 01
i s 11 e q u i v a l e n t and t h e r e f o r e hec~vrcrivedyb o m o h p k i c t o t h e d e c i s i o n problem w i t h r e s p e c t t o deduci b i 1 it y o f t h e f o r m u l a e c l a s s FD(M) : = {aM
A
+
D
1
C arbitrary}. (1)
We say: any 0  h a l t i n g problem " i s " v i a o u r t r a n s l a t i o n a l o g i c a l d e c i s i o n problem ( w i t h respect t o d e d u c i b i l i t y ) . Since registermachines ( w i t h 2 r e g i s t e r s ) are universal f o r the computation o f a l l p a r t i a l recursive f u n c t i o n s a l s o the deducib i l i t y d e c i s i o n problem o f any c y l i n d r i c a l r e c u r s i v e c l a s s o f f o r m u l a e F can be shown " t o be" ( r e c u r s i v e l y i s o m o r p h i c t o ) a 0  h a l t i n g problem o f a 2  r e g i s t e r mac h i n e M w i t h a s i m p l e r e l a t i o n between F and M. We c o n c l u d e t h a t f i r s t o r d e r l o g i c a l d e c i s i o n problems a r e as " n a t u r a l " as h a l t i n g problems o f 2  r e g i s t e r machines. B u t th.ere i s more t o say a b o u t t h e degree t h e o r e t i c a l r e l a t i o n between l o g i c a l and c o m b i n a t o r i a l d e c i s i o n problems. F i r s t observe t h a t b y i g n o r i n g t h e v a r i a b l e s i n t h e d e f i n i t i o n o f oM,uMcan he viewed as a semiThue system  w i t h s u b s t i t u t i o n rules
Pi
and an a l p h a b e t c o n s i s t i n g o f l e t t e r s 1,O and I.( f o r e v e r y s t a t e o f M) J
 w o r k i n g on " c o n f i g u r a t i o n words" C j u s t l i k e M on c o n f i g u r a t i o n s C i n t h e p r e c i s e sense o f t h e r e d u c t i o n lemma above, which i s a g a i n o b v i o u s f r o m t h e semiThue " i m p l e m e n t a t i o n " uM o f M and now reads: C +M 0
i f f C +u

0. M I n t h i s sense t h e l o g i c a l d e c i s i o n problem f o r c l a s s e s o f f o r m u l a e a M ~ + CD
i s ( u p t o r e c u r s i v e isomorphy) a l s o t h e same as t h e word problem
{(SJD) IC
+o
Dl
M f o r t h e semiThue i n t e r p r e t a t i o n uM o f M.
(1) Note t h a t t h e s e c l a s s e s FD(M) as w e l l as t h e f o r m u l a e uM can be c h a r a c t e r i z e d by p u r e l y s y n t a c t i c a l l o g i c a l means w i t h o u t any r e f e r e n c e t o programs M o r c o n f i g u r a t i o n s D.
Decision Problems in Predicate Logic
283
In Borger (1979) (1983) f u r t h e r e q u a l l y natural and simple " i n t e r p r e t a t i o n " of M have been given !n terms of Thuesystems  j u s t add t o uM t h e inverses of a l l i t s PrOdUCtionS, Markov algorithm , Post normal c a l c u l i , Post correspondence problems, Turing machines, p a r t i a l implicational propositional c a l c u l i , Wang's non e r a s i n g Turing machines and o t h e r s and i t i s shown t h a t through a l l these i n t e r p r e t a t i o n s of M t h e manyone degrees i f not the r e c u r s i v e isomorphy types of corresponding decision problems l i k e h a l t i n g , word, confluence and so on a r e preserved; t h e proofs of these equivalences follow a general p a t t e r n developed t h e r e ; they a r e easy i f not t r i v i a l once one has grasped t h e "good implementation" of M i n t h e system considered. Therefore we have p u t foreward strong evidence f o r the following epistemological STATEMENT. LogicaL d e c h i a n p m b k m axe ab "mtuhae" M a n y aMeh k i n d a6 cambimto hiad d e c i b i o n p h o b L m . Let us mention two consequences of these c o n s i d e r a t i o n s . The f i r s t concerns Wang's (1962: pg. 54) problem "Whether t h e r e i s some natural undecidable s e t of formulas of the p r e d i c a t e c a l c u l u s w i t h a decision problem t h a t i s not of t h e maximum r e c u r s i v e l y enumerable degree". Our reply on t h e b a s i s of t h e above shown " i d e n t i t y " of (machine and o t h e r ) h a l t i n g problems  t r a d i t i o n a l l y taken a s representing ( r e c u r s i v e l y enumerable) degrees  with l o g i c a l decision problems i s t h a t Wang'h p m b L e m h n o t a problem about l o g i c a l decision problems but t h e pobLem whetheh thehe ate m t m k i & m e d i a t e degheen; t h e f a c t t h a t u p t o now a l l p r e f i x  s i m i l a r i t y and o t h e r " n a t u r a l l y " c l a s s i f i e d c l a s s e s turned o u t t o have e i t h e r a r e c u r s i v e o r a decision problem of maximum degree i s only just another among many examples in mathematics suggesting t h a t intermediate degrees do not ( y e t ? ) c o n t r i b u t e t o s a t i s f a c t o r y c l a s s i f i c a t i o n s of t h e complexity of construct i o n s occuring in mathematical p r a c t i c e . A second s i m i l a r conclusion can be drawn about many attempts in the l i t e r a t u r e
t o study decision problems of formal grammars with r e s p e c t t o t h e i r degree comp l e x i t y . In Borger (1983) we develop a method showing t h a t Post correspondence problems  t h e s e a r e most f r e q u e n t l y used f o r reductions t o show formal language decision problems t o be unsolvable  and any formal language decision problem t o which t h e former have been reduced e f f e c t i v e l y in a very strong s y n t a c t i c a l sense ' b e the same"; indeed t h e i r r e c u r s i v e isomorphy types coincide. Therefore here again i t turns o u t t h a t degree complexity does not c o n t r i b u t e t o t h e i n s i g h t i n t o formal language decision problems: j u s t d e c i d a b i l i t y o r u n d e c i d a b i l i t y of maximal degree i s t h e only r e l e v a n t question, a p a r t from i n v e s t i g a t i o n s on subrecursive complexity i n decidable cases. Let us conclude t h i s argument however with a p o s i t i v e example: the f a c t t h a t the h a l t i n g problems H D ( M ) and t h e decision problems of FD(M) a r e " t h e same" implies t h a t well known complexity r e s u l t s f o r metadecision problems of h a l t i n g problems i n t h e KleeneMostowski a r i t h m e t i c a l hierarchy c a r r y over automatically t o logical metadecision problems a s exemplified i n the following
C o h a U a h y . (Borger & Heidler (1976)) With r e s p e c t t o d e d u c i b i l i t y t h e following metadecision problems f o r l o g i c a l decision problems a r e of t h e i n d i c a t e d ( p r e c i s e ) a r i t h m e t i c a l complexity:  t h e emptyness problem i s nlcomplete  t o t a l i t y and i n f i n i t y problem a r e n2complete  c o f i n i t e n e s s , r e c u r s i v i t y and reduction c l a s s problem a r e n3complete A b h o h t and n&p&
pmoa doh T m c k t e n b m t ' n Meohem, promised i n the preceding s e c t i o n a s consequence of our reduction method, w i l l be given now. By t h e r e duction lemma we have already shown t h a t t h e c l a s s e s No := I F / C , F } F i n : = IFlF has a f i n i t e model}
284
E. BORCER
a r e r e c u r s i v e l y i n s e p a r a b l e : t h e r e c u r s i v e u n s e p a r a b i l i t y o f two h a l t i n g problems
HE(M) i s c a r r i e d o v e r t o t h e nomodel and t h e f i n i t e  s a t i s f i a b i l i t y prob
HD(M) and
lems through o u r r e d u c t i o n :
(1) C +MD
implies
( 2 ) C +ME
"
+
1(uM
A
PL
uM
C
A
A
c
l n
A
la)
has a f i n i t e model
I n f a c t f o r ( 2 ) n o t e t h a t C +ME i m p l i e s t h a t n o t C i n t h e r e d u c t i o n lemma i n t h a t case f o r uM
A
C
+MD;
A 1D
t a k e t h e model c o n s t r u c t e d
and c u t i t down t o t h e i n i t i a l
domain o f a l l numbers zt t 1, where t denotes t h e maximal r e g i s t e r c o n t e n t o c c u r i n g i n t h e t e r m i n a t i n g computation from C t o E, by d e f i n i n g ( t + l ) ' : = t t l . T h i s i s s t i l l a model f o r u,, A C A 1 D . The c l a s s e s No and I n f o f a l l i n f i n i t y a x i o m a r e r e c u r s i v e l y i n s e p a r a b l e because No and F i n a r e r e c u r s i v e l y enumerable b u t p r e d i c a t e l o g i c i s undecidable. To show a l s o t h e r e c u r s i v e u n s e p a r a b i l i t y o f t h e f i n i t e  s a t i s f i a b i l i t y and t h e i n f i n i t y  a x i o m p r o p e r t y we have t o m o d i f y s l i g h t l y o u r d e f i n i t i o n o f uM f o r M t o assure t h a t a l l n o n  p e r i o d i c computations o f M g e t i n f i n i t y axioms as t h e i r desc r i p t i o n ; because t h e n t h e d e s i r e d i n s e p a r a b i l i t y p r o p e r t y i s c a r r i e d o v e r f r o m t h e corresponding i n s e p a r a b i 1 it y p r o p e r t y o f a p p r o p r i a t e l y chosen machines M. The problem i s e a s i l y s o l v e d b y t h e o b s e r v a t i o n t h a t n o n  p e r i o d i c r e g i s t e r machine computations must have l a r g e r and l a r g e r numbers o c c u r i n g i n a t l e a s t one r e g i s t e r ; such " b i g " elements i n t h e models can be assured by t h e " g r e a t e r as" axioms discussed i n t h e p r e c e d i n g s e c t i o n , r e l a t i v i z e d however t o p o s s i b l e r e g i s t e r contents. F o r m a l l y d e f i n e 6 as b e f o r e uM w i t h t h e a d d i t i o n a l c o n j u n c t s ( f o r e v e r y M  i n s t r u c t i o n Ii): (xIiy
+
x'Gx) (yIix
+
x ' G x ) (XGY
+
x'GY)
ixGx
As b e f o r e t h e r e d u c t i o n lemma and f o r a p p r o p r i a t e D, E a l s o t h e above r e l a t i o n ( 2 ) h o l d ; i n a d d i t i o n we have t h e d e s i r e d r e l a t i o n t h a t i f C does n o t d e r i v e D i n M, then 6 A A l n has a model as b e f o r e b u t no f i n i t e one. F o r more s u b t l e quest i o n s !!bout c o n s e r v a t i v i t y o f s p e c i f i c r e d u c t i o n procedures see Aanderaa & Borger & Lewis (1982). F o r a n o t h e r i11 u s t r a t i o n ( l ) o f how o u r r e d u c t i o n technique c a r r i e s o v e r unseparab i l i t y p r o p e r t i e s f r o m machine problems t o formulae problems d e f i n e ( t h e "ThueV C ~ A ~ O M " T~ ) as uM w i t h ++ i n s t e a d o f +. I t i s easy t o adapt t h e p r o o f o f t h e r e d u c t i o n lemma f o r oM t o a p r o o f f o r t h e f o l l o w i n g
ReducGtion Lemm
6 0 ' ~Rhe
Thue uemian
T~
05
uM. F o r any c o n f i g u r a t i o n s D,E
of M
which a r e n o t r e a c h a b l e one f r o m t h e o t h e r i n M h o l d s :
(1) C
+MD
iff
I
T~ A
PL
( 2 ) C +ME
iff
+
PL
T ~ ,A
ln ,n
IT
A
+
A
+c
( l ) Taken f r o m Aanderaa (1971) and Borger (1975). Aanderaa's method i s d i f f e r e n t f r o m mine i n t h e r e s p e c t t h a t Aanderaa does n o t s e p a r a t e c o m p l e t e l y t h e i n p u t d e s c r i p t i o n f r o m t h e program f o r m u l a t i o n . Due t o t h i s f a c t h i s formulae a r e more complex than o u r s and h i s p r o o f o f what i n o u r f o r m u l a t i o n reduces a g a i n t o t h e r e d u c t i o n lemma becomes more i n v o l v e d and does n o t show t h a t t h e same argument works a l s o i n s u b r e c u r s i v e c o m p l e x i t y l i k e f o r Enunseparable En+lsets.
285
Decision Problems in Predicate Logic
Pmod. In ( 1 ) from l e f t t o r i g h t the claim follows from t h e case f o r r i g h t t o l e f t assume t h a t C 7LMD: then a canonical model f o r i s given by defining F a s meaning F fMD
T~
A
JD
uM; A
E
A
from
C
I n ( 2 ) from l e f t t o r i g h t conclude a s i n t h a t case f o r uM but using the implithe to infer c a t i o n s from r i g h t t o l e f t i n T~ and s t a r t i n g w i t h given computation C +ME. In the o t h e r d i r e c t i o n defining F a s meaning F +ME
r
y i e l d s a canonical model f o r which means: C +ME. The reduction lemma f o r
T~
T~
A
ID
A
c
F where t h e r e f o r e a l s o
c must
be t r u e
immediately implies my 1975version o f Aanderaa's
Theonem. For a r b i t r a r y r e c u r s i v e l y unseparable h a l t i n g problems H o ( M )
and H E ( M ) , in theory T w i t h T~ A TEA F a s nonlogical axiom theoremhood and l o g i c a l falsehood a r e r e c u r s i v e l y unseparable. Therefore t h i s theory i s e s s e n t i a l l y undec i d a b l e and consequently incomplete. Phood. Any r e c u r s i v e s e t R s e p a r a t i n g I F I h F ) and IF1 pT1F) would y i e l d the rec u r s i v e s e p a r a t i o n s e t {TITER) f o r H D ( M ) and H E ( M ) . For the same reason T can have no r e c u r s i v e supertheory. B u t then i t must be incomplete because otherwise i t would be r e c u r s i v e .
S u f f i c i e n t l y d i f f i c u l t h a l t i n g problems f o r a machine M(1ike t h e above r e c u r s i v e s t a t i n g t h e e f f e c t of t h e u n s e p a r a b i l i t y ) generate "program formulae" T~ A ID A program on i t s p o s s i b l e data t o g e t h e r with a commitment on one s t o p s t a t e which may and one which may not be reached and y i e l d i n g thereby e s s e n t i a l l y undecidable and incomplete t h e o r i e s . I t would be i n t e r e s t i n g t o analyze how simple such program formulae could become generating s t i l l incomplete t h e o r i e s . ( l ) .
r
Note t h a t by r e l a t i v i z i n g our program formula tM A A 'E t o a new "successor" r e l a t i o n S defined by A V Sxv a formula i s obtained which by t h e same arguments
x v as above can be seen t o have r e c u r s i v e l y enumerable b u t no r e c u r s i v e models i f H D ( M ) and H E ( M ) a r e r e c u r s i v e l y unseparable, and En+l  but no E n  models in the Grzegorczyk hierarchy i f H O ( M ) and H E ( M ) a r e Enunseparable En+lsets. (The comp l i c a t i o n by r e l a t i v i z i n g t o a successor r e l a t i o n i s needed because T~ A A i s a KrOm formula and a l l s a t i s f i a b l e Krom formulae a r e known t o posses r e c u r s i v e models ( s e e Aanderaa & Jensen (1973), Ershov ( 1 9 7 3 ) ) ) . T h u s we have a s h o r t proof e x h i b i t i n g a very simple s a t i s f i a b l e formula excluding recur
ln
(1) A very i n t e r e s t i n g r e s u l t . r e l a t e d t o t h i s question i s i n German0 (1976) where i t i s shown t h a t any theory i s incomplete i f i t i s r e c u r s i v e l y enumerable, c o n s i s t e n t and admits term r e p r e s e n t a t i o n s f o r a d d i t i o n and m u l t i p l i c a t i o n ( u s i n g =,O, Successor).
E. BdRGER
286
s i v e models, improving considerably much more involved e a r l i e r s o l u t i o n s by Kreisel (1953), Mostowski (1953), (1955), Rabin (1958) t o t h a t problem r a i s e d by H i l b e r t & Bernays (1939).
DESCRIPTION OF BOUNDED MACHINE COMPUTATIONS: COOK'S PROBLEM, SPEKTRALPROBLEM Take again t h e technique of f i r s t o r d e r d e s c r i p t i o n of e v e n t u a l l y unbounded machine computations a s explained i n s e c t i o n 2 and consider i t f o r f i n i t e computat i o n s : look a t the atomic formulae PI .q representing t h e t  t h configuration 1t C reached by M s t a r t i n g with Co = C a s propositional v a r i a b l e s := I t . with P31 ,q the same intended i n t e r p r e t a t i o n of
ct
I t . = 1 a s meaning CO c~. PY1.9 I f you d e f i n e now uM f o r given computation length a. and given i n p u t (memory) bound k a s before but using f i n i t e conjunctions over time 0 5 t 5 a. and r e g i s t e r contents 0 5 p. q 5 k i n s t e a d of universal q u a n t i f i c a t i o n , then t h e r e s u l t i n g formula u M,ge,k again f u l f i l l s the &duc.tion pmpehty t h a t f
c
~
f
iD i f f
u,,,,a,k
A
' C
+
na.i s
a tautology
However the length of u grows exponentially because t h e formula d e s c r i b e s M,k,k g l o b a l l y the e f f e c t of every machine i n s t r u c t i o n , i . e . r e f e r r i n g always t o the whole configuration say ( p , i , q ) and not only the content of the p a r t i c u l a r memory p o s i t i o n a f f e c t e d by I i . Indeed i n t h a t way we succeeded in giving compact formul a t i o n s of M  namely by Krom formulae  and i n t r i v i a l i z i n g completely t h e equivalence proofs showing t h a t t h e reductions work. I t i s easy however t o modidy 0u4 g l o b a l duchiption techniyue t o a Local one s i n c e f o r every reasonable notion of algorithm t h e execution of an elementary computation step ( i n s t r u c t i o n ) has a local c h a r a c t e r . I t i s s u f f i c i e n t t o consider the r e g i s t e r s not any more a s s t a c k s but a s containing s t r i n g s of symbols displayed i n l i n e a r order with a p o i n t e r showing the p o s i t i o n of the symbol on which t h e program a c t u a l l y i s working (Tur i n g t a p e ) . What we w i l l gain i s t o o b t a i n from our c o n s t r u c t i o n i n the same way a l s o a proof f o r Cook's theorem t h a t any computation in time bound a. , s t a r t i n g with input of length n , of any (even nondeterministic) Turing machine M can be described by a propositional formula u M,n,a. of polynomial length in M,n,a. Since the s a t i s f i a b i l i t y problem of Krom formulae i s in P(see Cook 1971), unless P=NP we have t o g i v e uv ,the K4om b t ~ u c t u h ef o r u ~ , ~ , ~ . W e can preserve however the Horn s t r u c t u r e f o r our program formulae, a p a r t i c u l a r l y i n t e r e s t i n g f e a t u r e s i n c e P contains a l s o t h e s a t i s f i a b i l i t y problem f o r Horn formulae. This comes o u t from the following ( c f . Aanderaa & Borger 1979): T h e o m o n ( a h o n t  ) How &chipZion 0 6 ~ i n ; R ecompuhtioiovld. There i s a Pcomput a b l e function a s s o c i a t i n g t o every d e t e r m i n i s t i c T u r i n g machine program M and any natural numbers n ( f o r i n p u t l e n g t h ) and a. ( f o r computation l e n g t h ) a Horn formula u ("pmgmm ~ o h m u h " ) , a Krom Formula aM ("n&~taht & ~ m u h "with so c a l l e d M, a. ,n "input" v a r i a b l e s x l , .. . , x n ) and a Herbrand formula wM,% ("&top ~ o t ~ n & ~ ' s) a t i s fying t h e following &duca%on pmpehty: l e t Co(q) be t h e s t a r t configuration of M with i n p u t q . Then
+i accepting config.
i s s a t i s f i a b l e f o r every M,a. 01sequence q of length n where a M,n ( 9 ) denotes t h e r e s u l t of simultaneous substit u t i o n of every x i by qi f o r 1 5 i 5 n . Co(q)
i f f uM,k
A
aM,n(q) A w
281
Decision Problems in Predicate Logic ( A Hehbmnd formulae. )
60m&
i s a conjunction of atomic formulae o r negations of atomic
Proof. F i x a r b i t r a r i l y Q , n and a d e t e r m i n i s t i c Turing machine M with s e t s of i n s m i o n s e t s I . = ( i , j , o . .,$. .) f o r 1 5 i 5 r , 0 5 j 5 m over the alphabet 1  J 1,J {ao,...,am}. To execute I i means a s usual: i f in s t a t e i the l e t t e r under the reading head (pointed t o by t h e p o i n t e r ) i s a . , t h e n do o i , j  which i s one o f : a k . J ( " w r i t e " a ) o r r resp. a.(move p o i n t e r 1 p o s i t i o n t o the r i g h t resp. l e f t )  and k go t o execute the i n s t r u c t i o n w i t h index $i .. Let a. = 0 , al = 1, a2 = b ("blank" ,J symbol) and without l o s s of g e n e r a l i t y l e t 0 be t h e accepting s t a t e of M . Let the i n i t i a l c o n f i g u r a t i o n s CO(q) f o r 01input sequences q of length n be defined by the i n i t i a l s t a t e 1 and t h e following f i n i t e tape w i t h p o s i t i o n s numbered by Q ,...,0 , l ) . .., I.: b. ..b b q b . . b
+
crc'+
Q times
1  n times
i . e . t h e p o i n t e r (reading head) i s in p o s i t i o n (with number) 0. To encode a r b i t r a r y Mconfigurations we introduce f o r a l l numbers t , k , j , i with 0 5 t 5 1 , Q < k 5 R , 0 5 j 5 m, 0 5 i 5 r pairwise d i f f e r e n t propositional tTt where Ct denotes with t h e following intended i~~Xeqme,?izfitn variable I i , Pk, k,j t h e configuration reached by M in t s t e p s f o r given Co = Co(q):
I t. = 1 t Pk = 1 Ti,j
i f f t h e i n s t r u c t i o n t o be executed i n Ct i s i i f f the p o i n t e r C t i s in p o s i t i o n k 1 i f f the tape c e l l ( w i t h number) k i n C t contains a . .
=
J
a s conjunction of t h e following formulae desDefine t h e pmgmm @un& c r i b i n g l o c a l l y  i . e . f o r any s i n g l e tape c e l l  f o r every Minstruction I i how execution of I i on C a f f e c t s the i n s t r u c t i o n address, the p o i n t e r p o s i t i o n and t h e content of the tape c e l l (we w r i t e f o r s h o r t n e s s 6 + y1 A y 2 i n s t e a d of (6
+
(i)
Y,)
A
(6
+
~2)):
f o r any w r i t e  i n s t r u c t i o n ( i , j , a h , $ i , j ) i n M:
IF
A
t
I.
P:
A
t
A
Pk
T:,~
+
T~+' +i,j
t
A
Tk1.j'
A
P:+'
A
klh
tt 1 +
T~+'
(successive s t a t e , pointer position, c o n t e n t of working c e l l ) ( " c o n t e n t of nonworking c e l l s remains unchanged")
T k i , j i
f o r any Q 5 k , k' 5 R and 0 5 j ' 5 m w i t h k' # k ( i i ) f o r any rightmove j n s t r u c t i o n ( i , j , r , $ . . ) : 1 ,J t t t+l t+l A T t + l A 'k A T k , j A ':',j' . A 'k+l k ,j ,J
A
T t+ k, 1 , j'
( i i i ) f o r any leftmove i n s t r u c t i o n analogously w i t h I 5 k  1.
E. BORGER
288
Define the n f ~ ~ t a h~tomunUaaa
M,n I; { i n i t i a l s t a t e i s 11 P:
encoding Co a s conjunction o f : { i n i t i a l p o i n t e r p o s i t i o n i s 01
Ti,o

G,2
f o r  2 5 k 5 0 o r n < k 5 a. Inoninput c e l l s a r e blank}
ft
';,l
lXk
xk f o r 1 5 k 5 n
{ i n p u t in c e l l s 1, ..., n}
of l I .9i f o r 1 5 i 5 r expressing t h a t Define the n t o p 6omunUaa w ~ a, s conjunction ~ a t time a. M could be in accepting s t a t e 0 b u t i n no o t h e r s t a t e . The reduction property f o r u A aM,,(q) A w ~ i s, e a~ s i l y proved following the M,9. approach explained in s e c t i o n 2: i f M reaches i n 9. s t e p s an accepting configurat i o n ( i . e . with s t a t e 0 ) from given Co = C o ( q ) , then t h e above i n d i c a t e d intended t r u e . Conversely any model i n t e r p r e t a t i o n obviously makes u M , % A ~ ~ , ~ A( q ) f o r t h i s formula simulates any Mcomputation of length (a. s t a r t i n g with Co(q) i n the sense t h a t f o r any t r u t h assignement A making our formula t r u e and f o r any t t C and t 5 .f with Co(q) +; C t where C has i n s t r u c t i o n i, p o i n t e r p o s i t i o n k and tape a . ... a . , A a s s i g n s truthvalue 1 t o the encoding v a r i a b l e s 1.; and L
J9.
J9.
2 9.. I t follows t h a t C9. must have s t a t e 0 because
T k , j h f o r 9. 5 h
A(1:)
Pi
L
=
0 for
a l l 1 <. i < r. This c o n s t h c t i o n e s t a b l i s h e s a l s o C o o k ' 6 Rheohm about NPcompleteness of the prop o s i t i o n a l l o g i c decision problem s i n c e i f M i s nondeterministic, proceed a s above b u t f o r every p a i r ( i , j ) take in u a s conclusion of the corresponding impliM,a. c a t i o n ( s ) t h e d i s j u n c t i o n over a l l p o s s i b l e 1  s t e p t r a n s i t i o n s ( i , j , o . .,$. .) 1.J 1 , J of M. (Note t h a t then uMP9. i s not any more a Horn formula.) We now show t h a t applying t h e same c o n s t r u c t i o n t o machine d e s c r i p t i o n s over f i n i t e domains y i e l d s t h e famous automata t h e o r e t i c c h a r a c t e r i z a t i o n of s p e c t r a . In f a c t the only thing t o show i s t h e following h g i c a l denchipLLan 0 6 ,jivzite cornpuhLLouzb Oweh divzite domcLim : For every n we a s s o c i a t e t o every r e g i s t e r (or Turing) machine program M and any s a formula ci of order n + 1 f u l f i l l i n g t h e following /reduction p m p e h t y : M accepts k in 'an(ks) steps i f f s f k a f o r every 2 5 k . Proof. We paraphrase our proof given f o r Cook's theorem. Let n,M,s be a r b i t r a r i l y fixed. To d e s c r i b e over a domain w i t h k elements  say k := { O , l , ..., k  1 )  t h e Mcomputation of length a n ( k s ) s t a r t e d with i n p u t k , we need an encoding of a,(ks) many successive "time moments" t o g e t h e r with t h e corresponding s i t u a t i o n of t h e computation. The i d e a i s t o c m t e by nuccenniwe poloeh n e t c o m t m t i o n  s t a r t i n g from t h e sary Cartesian product over k  Rhe needed a n ( k 6 ) o b j e c h of a type u n of order n , t o ohdeh thene o b j e c h in a & n U h m y and then t o d e s c r i b e t h e Mcomputation in the same way a s done before but using now a zero p r e d i c a t e Z and a successor r e l a t i o n S r e l a t i v e t o the previously defined l i n e a r ordering K, and an embedding F of k i n t o a segment of these untype o b j e c t s f o r d e s c r i p t i o n of t h e input. Formally the power s e t types over k S a r e defined by u1
:=w u
stimes
~ :=+ ( ~ ui)
IU n I
=
n
289
Decision Problems in Predicate Logic
...,
Over k = 10, k  1 ) t h e r e a r e e x a c t l y a n ( k s ) o b j e c t s of type u n where by o b j e c t s of type uo we understand s  t u p l e s of elements from k . We use x,y,z a s v a r i a b l e s of 1 resp. a s stuples and t . u , v , w , t ' , u ' , a s v a r i a b l e s of type o n f o r n type of v a r i a b l e s of type I f o r n = 0 ( u s i n g u = v a s abbreviation f o r u1 = V ~ ... A A u = vs, s i m i l a r l y A e t c . ) .
...
S
U
Dedine t h e formula Ord(Z,S) of o r d e r n + 1  expressing t h a t Z represents "zero" ( t h e f i r s t element) and S the "successor" r e l a t i o n w i t h r e s p e c t t o a l i n e a r ordering K of a l l type u  o b j e c t s over any ( f i n i t e ) domain  a s conjunction o f the n following formulae: ( ( K u v v Kvu v u
A A A
=
v)
A
A TKuu
(Kuv
A KVW +
K uw ) )
u v w VZU
A
u A
A ( Z U +
u
1VKvu)
A(Suv + + ( K u v
u v
{"zero" has no "predecessor")
V A
lV(Kuw
A
Kwv)))
{no element between successors}
W
Using Ord(Z,S) we can d e f i n e t h e program formula uM a s before using Z resp. S (predicate) f o r 0 resp. t + 1 and (almost) t h e same i n t e n d e d r n u n i n g doh v a r i a b l e s I i , P and T . encoding over any domain k = {O, k  1 ) resp. s t a t e i , J the p o i n t e r ( r e a d i n g head) p o s i t i o n and tape c e l l i n s c r i p t i o n a . f o r any time J moment t and any tape p o s i t i o n u of any Mcomputation s t a r t e d with C o ( k ) ( l e t  2 be t h e number of t h e leftmost c e l l v i s i t e d during t h e given computation and It( t h e o r d e r number of t i n t h e given ordering K ) :
...,
I i t i s t r u e i f f a t time It1 i n s t r u c t i o n i i s executed P t u i s t r u e i f f I u I = ( p o i n t e r p o s i t i o n a t time It1 )+ I T . t u i s t r u e i f f a t time It1 the tape c e l l with number 1.1 J letter a .
+ n.
contains t h e
J
Formally l e t M have i n s t r u c t i o n s I i ( O 2 i 5 r ) over the alphabet a.(O 5 j 5 m ) J with a. = b ( " b l a n k " ) , a l = 1, accepting s t a t e 0 of M and i n i t i a l configurations C o ( k ) defined by the i n i t i a l s t a t e 1 and unary encoding of k i n t o the tape ktimes ... b'l l'b
...
...
f
with the p o i n t e r (reading head) in p o s i t i o n (numbered) 0. For technical reasons which w i l l become c l e a r l a t e r choose s such t h a t f o r any k , i f M accepts k , then during t h e computation s t a r t e d w i t h input k i t w i l l never v i s i t the tape c e l l numbered a (ks) + k. Without l o s s o f g e n e r a l i t y we assume a l s o t h a t in the accept i n g s t a t e !ny computation becomes c o n s t a n t (formally I. = (O,j,aj,O) f o r a l l 0 < j < m; remember t h a t we t h i n k of Turing i n s t r u c t i o n s I i a s of s e t s of quxdruples. )
Dedine the program formula of t h e following formulae:
uM
a s universal q u a n t i f i c a t i o n of t h e conjuncticn
9 s
(i)f o r any w r i t e  i n s t r u c t i o n s e t ( i , j , a h , + i , j ) i n M: t ' A Pt'u A Tht'u I i t A P t u A T . t u A S t t ' + I J +i,j {successive s t a t e , p o i n t e r p o s i t i o n unchanged,new working c e l l content}
E. BORGER
290
I i t A P t u A T . , t v A v f. u A S t t ' + T t ' v f o r 0 5 j' J j' {content of non working c e l l s remains unchanged) ( i i ) f o r any rightmove i n s t r u c t i o n s e t ( i , j , r , @ i .) in M:
zm
,J
1.t 1
A
Ptu
A
T.,tv
+
I
T.tu
A
J
A
J
t'
$i,j
Stt' A
A
SUU'
Pt'u'
T.t'u
A
J
A
T.,t' J
for 0 5 j '
zm
( i i i ) f o r leftmove i n s t r u c t i o n s analogously To d e f i n e t h e i n p u t r e p r e s e n t a t i o n we make use of the following embedding dohmula s t a t i n g t h a t t h e given domain (namely k = {O, ...,k1) f o r some k ) i s embedded i n t o a segment of the ordering of t h e onobjects by some function ( w i t h graph)F:
V Fxu { e x i s t e n c e } x u A A A (Fxu A Fyu t x A
A
A
A ( F X U A Fxv
A A
+
u
=
v ) {uniqueness]
x u v =
y) {injectivityl
X Y IJ A
A(KUV u v w x y
A A A A
A
Kvw
A
Fxu
A
Fyw
+
VFzv)
{range i s a segment}
Z
Uedine t h e ntaht domula
aM,s
a s conjunction of the above embedding formula and
the following formulae: A ( Z t + I l t ) {read: a t time p o i n t 0 i n s t r u c t i o n 1 i s t o be executed} t A ( Z t t V V ( P t u A Fxu A A ( K v u t 1VFyv))) t u x V Y {read: a t time 0 t h e p o i n t e r p o s i t i o n i s encoded by the f i r s t Fvalue} A ( ( J V F x u ) + T o t u ) ) u x X {read: a t time 0 every tape c e l l (with number) i n the range of F has tape i n s c r i p t i o n a l = 1 and any o t h e r the blank symbol a. = b . 1
A ( Z t + A((VFxu + Tltu)
t
by saying t h a t a t t h e l a s t moment no i n s t r u c t i o n De6ine t h e n t o p 60hmLLPa w M,s d i f f e r e n t from the a c c e p t i n g  s t a t e  i n s t r u c t i o n I. can be executed:
...
+ 111th A lIrt) t u We now show t h a t M e ConjuncLLon a := Ord(Z,S) A oM,s A aM,s A wM,s
A((1VKtU)
pmgmm, n.taht and n t o p 6omlLeae w i t h hebpect t o M e ckr&Lned by Ord(2,S) ~ U R d mM e heduction pmpehty.
06
ze.hoAWCebnOhbth4kJW
Indeed i f M accepts k in a t most a n ( k s ) . then {remember t h a t t h e configuration sequence ( C t : 0 < t 5 a n ( k s ) ) defined by Co(k) and M becomes c o n s t a n t once the i s c a l l e d } t h e above i n d i c a t e d accepting i n s t r u c t i o n set I. = (O,j,aj,O)o < intended meaning of K,Z,S,Ii,P,T. over k = TO,.T.,kll y i e l d s a model f o r a over k J together w i t h Fxu meaning I u I = p. + x (where p. and I u / a r e defined a s i n d i c a t e d above f o r the given k ) .
29 1
Decision Problems in Predicate Logic I n v e r s e l y any model M o f c a r d i n a l i t y k s a t i s f y i n g a, say o v e r t h e domain t k11, encodes any Mcomputation ( C :O 5 t 5 a ( k ' ) ) s t a r t e d f r o m n t Co = Co(k) i n t h e sense t h a t i f C has i n s t r u c t i o n number i, p o i n t e r p o s i t i o n k k = {O,l,
...,
and t a p e a .
...
~  e
, then
a
Iiut,
Putue+k and Tjhutuh+e
for
e 5
h <
j a n ( ks)a.
an(ks) a r e t r u e i n t h e model where u t denotes t h e t  t h o b j e c t o f t y p e on o v e r k w i t h r e s p e c t t o t h e g i v e n i n t e r p r e t a t i o n o f K and i s t h e o r d e r number o f t h e consequently M a t t i m e p o i n t K  s m a l l e s t Fvalue. Due t o t h e s t o p f o r m u l a w M,s an(ks) c a n n o t be i n any o f i t s s t a t e s 1 5 i 5 r, t h e r e f o r e a t t h a t moment i t must be e x e c u t i n g t h e a c c e p t i n g i n s t r u c t i o n Io. From t h e above g i v e n c o n s t r u c t i o n and w e l l known f a c t s a b o u t t h e Grzegorczyk and t h e y n  h i e r a r c h y ( c f . Rodding 1967) f o l l o w s : yn 5 SPECTRAn+l, whereas SPECTRAn+l 5 yn+l f o l l o w s f r o m an easy g o d e l i z a t i o n o f formulae w i t h types bounded by n + 1 i n t o a r i t h m e t i c a l statements o v e r +,.,2' w i t h q u a n t i f i e r s bounded by an; see o p . c i t . S i n c e as e x p l a i n e d f o r Cook's theorem t h e same c o n s t r u c t i o n a p p l i e s t o n o n d e t e r m i n i s t i c machines f r o m t h e s p e c i a l case n = o f o l l o w s :
Fimtohdm Spectm a m ULith m p e c t t o 06
u n a q mpmentaLion pmc.hdy
the
NPbe.12
pobLtLve nwnbehs.
S i n c e a f u n c t i o n i s p o l y n o m i a l i n t h e l e n g t h k o f t h e unary r e p r e s e n t a t i o n ktimes o f k i f f i t i s e x p o n e n t i a l i n t h e l e n g t h o f t h e b i n a r y r e p r e s e n t a t i o n o f k, t h a t c h a r a c t e r i z a t i o n can a l s o be s t a t e d i n t h e more f r e q u e n t l y used form:
WLth hebpect t o b i m q m p w e n t a f i o n t h e 6ih6tohCleh 6peCtM am phec.hdy those netn 0 6 pOb&iVe numbea which am accepted by a nondetemini6Lic Tuhing mckine i n e x p o n e m e (namely 2c"(x) f o r some c o n s t a n t c ) time. Using t h e s t a n d a r d encoding e(S) o f a f i n i t e s t r u c t u r e S = (k;R l,...,Rd) t e n a t i o n o f t h e encodings e(R1) d e f i n e d as {al,a2}word
,...,
e(Rd) o f R1,...,Rd
as conca
where f o r r  a r y R e(R) i s
o f l e n g t h kr which has i  t h d i g i t a2 resp. al
if R is
t r u e resp. f a l s e f o r t h e i  t h element o f kr w i t h r e s p e c t t o l e x i c o g r a p h i c a l o r d e r i n g , t h e domain c a r d i n a l i t y k need n o t t o be encoded s i n c e i t can be computed n o n d e t e m i n i s t i c a l l y i n p o l y n o m i a l t i m e f r o m e ( S ) . T h i s i s t h e reason why t h e above g i v e n p r o o f f o r t h e c h a r a c t e r i z a t i o n o f f i r s t  o r d e r s p e c t r a proves a l s o F a g i n ' s e x t e n s i o n t o g e n e r a l i z e d s p e c t r a : F o r any LENP n o t c o n t a i n i n g t h e empty word h o l d s L = e ( [ I } spectrum(a)) where a i s t h e f o r m u l a d e f i n e d i n t h e main cons t r u c t i o n f o r a n o n d e t e r m i n i s t i c T u r i n g machine M a c c e p t i n g L i n t i m e ks, b u t w i t h a d d i t i o n a l l y Ord(O,xx,y.y = x + l )  i n o r d e r t o r e s t r i c t a t t e n t i o n t o models o v e r k = [O,l, k11  and t h e f o l l o w i n g replacements i n t h e s t a r t f o r m u l a aM . ,s' r e p l a c e t h e embedding f o r m u l a by t h e new embedding 60hmLLen e x p r e s s i n g t h a t t h e domain k = [O,l, k11 'is embedded v i a F i n an o r d e r p r e s e r v i n g way i n t o a segment o f t h e s  t u p l e s : A V Fxu A A A A A(Fxv A Fyw P ( S xy++ Svw)) x u x y v w
...,
...,
{Note t h a t t h e o r d e r  p r e s e r v a t i o n i m p l i e s uniqueness and i n j e c t i v i t y o f F and t h e f a c t t h a t t h e range o f F i s an Ksegment}; r e p l a c e t h e i n i t i a l  t a p e  d e s c r i p t i o n by t h e f o l l o w i n g new i n i t i a l  h p e 60mLLen e x p r e s s i n g t h a t M s t a r t s a t t i m e 0 w i t h t h e encoding o f t h e monadic " i n p u t " p r e d i c a t e I  which i s o f l e n g t h k and i s i n s c r i 
E. BORGER
292
bed i n t h e tape c e l l s numbered by Fvalues u: A ( Z t + A A(FXU + ( ( I x
A
+
Tptu)
(1Ix
A
+
Tltu))
x u
t
A(Zt t
+
A((~VFXU) Totu)) u x f
{ b l a n k a.
o u t s i d e range ( F ) }
F i n a l l y bound a l l p r e d i c a t e symbols e x c e p t I by an e x i s t e n t i a l q u a n t i f i e r . Again t h e same c o n s t r u c t i o n a p p l i e s t o r u d i m e n t a r y p r e d i c a t e s , i . e . those number t h e o r e t i c a l r e l a t i o n s which can be d e f i n e d e x p l i c i t e l y f r o m t h e graphs o f "+" and a number t h e o r e t i c a l ' I  ' ' u s i n g Boolean o p e r a t i o n s and bounded q u a n t i f i c a t i o n s : p r e d i c a t e R i s s a i d t o have a ,$&5.t ohdeh mpkueevLtaLLtion i n 4.inite d o d m i f f some f i r s t o r d e r f o r m u l a % ( c o n t a i n i n g i n p a r t i c u l a r a p r e d i c a t e symbol T? o f t h e same a r i t y as R and e v e n t u a l l y a b i n a r y p r e d i c a t e symbol K) i s s a t i s f i a b l e o v e r e v e r y domain k := { O , l , ..., k ~ l }w i t h K i n t e r p r e t e d as < and i n e v e r y such model o f "p; t h e i n t e r p r e t a t i o n o f R i s t h e r e s t r i c t i o n o f R t o k. O b v i o u s l y t h e zero p r e d i c a t e xx.x = 0 and t h e successor r e l a t i o n xx,y.y = x + l have r e p r e s e n t a t i o n Ord(Z,S)Z resp. Ord(Z,S)S w i t h Ord(Z,S) as d e f i n e d above, whereas one can d e f i n e aR : z Ord(Z,S)
A
add f o r R = G+ : = hx,y,z.x+y
%
A
add
:: Ord(Z,S)
A
= z and
m u l t f o r R = G.
where add and m u l t a r e t h e r e c u r s i v e d e f i n i t i o n s o f
"+"
resp.
'I"
add : z A A ~ ( z + y (G+xyz ++x = z ) ) { r e a d : x + o = XI X Y z A A A A A ( s y y ' + ( G + X y ' Z ' t t v(d+XyZ A S Z Z ' ) ) ) { X + y ' = (X x y ' 2' y Z m u l t :E A A A(Zy XY z A
+
(c.xyz++
A A
A
A(Syy'
x y'
2'
y
+
y)}
+
x}
{ r e a d : x0 = XI
y = 2))
+ (G.Xy'Z'tt
f r o m Z,S:
v(G.XyZ
A
G+ZXZ')))
{xy' = Xy
Z
Since K l e e n e ' s T  p r e d i c a t e can be c o n s t r u c t e d as r u d i m e n t a r y p r e d i c a t e (see Smullyan 1961) t h e r e i s i n p a r t i c u l a r a f i r s t o r d e r r e p r e s e n t a t i o n ar o f T i n f i n i t e domains where f u r t h e r m o r e T(i,x,y)
i m p l i e s i , x < y. Therefore t h e
cl
complete nonemptiness problem I i I V V T ( i , x , y ) l f o r t h e r . e . s e t s W . = XY I x 1 3 y T ( i , x , y ) ) resp. t h e i r n2complete i n f i n i t y problem { i I g x 3 y T ( i , x , y ) }
is
11reduced t o t h e nonemptiness resp. i n f i n i t y problem f o r spectrum (nonemptyi) resp. spectrum ( i n f i ) nonemptyi
where
:: V V V(a z x y Xz.z=i
A
Z=i) A
(ar
TZXY))
{read: machine i f o r some i n p u t x has an a c c e p t i n g computation y l infi
:= machine i f o r some x has an a c c e p t i n g computation y A
A 1KYYl Y1
A
A (fzxyl
YI
{ y i s t h e l a s t element i n t h e model]
+
lKyly)
{no a c c e p t i n g computation f o r i n p u t x i s s h o r t e r than y l
293
Decision Problems in Predicate Logic
Note t h a t i n t h e a l m o s t Horn d e s c r i p t i o n o f f i n i t e computations nonHorn i m p l i c a t i o n s appear o n l y f o r t h e i n p u t d e s c r i p t i o n o r f o r n o n d e t e r m i n i s t i c i n s t r u c t i o n s . T h i s suggests t h e f o l l o w i n g LogicaL complexity meanurn do& Boolean dune.tiom: say t h a t a f o r m u l a F d e f i n e s a Boolean f u n c t i o n s f ( w i t h r e s p e c t t o i t s " i n p u t " v a r i a b l e s x = xl, x ) i f f f o r e v e r y 01sequence q: n f ( q ) = 1 i f f F(x/q) i s s a t i s f i a b l e .
...,
F i s c a l l e d pseudoHorn o r Horn i n i t s w o r k i n g v a r i a b l e s i f F ( x / q ) " i s " a Horn f o r m u l a f o r e v e r y 01sequence q. D e f i n e Horn c o m p l e x i t y o f f as l e n g t h o f a s m a l l e s t pseudoHorn f o r m u l a F d e f i n i n g f. T h i s c o m p l e x i t y measure i s n t m n g l y connected t o Cook'n pmbLem: The Horn c o m p l e x i t y o f any Boolean f u n c t i o n can be p o l y n o m i a l l y bounded by i t s a r i t y n and program s i z e and maximal r u n t i m e on any i n p u t sequence o f l e n g t h n o f any d e t e r m i n i s t i c T u r i n g machine computing t h e funct i o n ; s i m i l a r l y i t can be proved t h a t P # NP i f f o r e v e r y p o l y n o m i a l p t h e r e i s a f u n c t i o n f such t h a t i t s s m a l l e s t pseudoHorn d e f i n i t i o n i s a t l e a s t p  b i g g e r than i t s s m a l l e s t p r o p o s i t i o n a l d e f i n i t i o n . S i m i l a r r e s u l t s a r e known f o r network o r T u r i n g machine c o m p l e x i t y o f Boolean f u n c t i o n s ; i n f a c t we have t h e f o l l o w i n g
T h e o m (Aanderaa & Borger 1979) F o r any Boolean f u n c t i o n f, i t s Hohn complexity and i t s ( l o g i c a l ) nehuonk compLexity  and t h e r e f o r e T u r i n g machine c o m p l e x i t y ahe poLynomiaLtq e q l L i v a L e k W i t h o u t g i v i n g t h e whole p r o o f we want t o i l l u s t r a t e t h a t one d i r e c t i o n o f t h i s e q u i v a l e n c e , namely C H ( f ) 5 O(C,4(f)), amounts t o a l o g i c a l d e s c r i p t i o n o f a r b i t r a r y l o g i c a l network computations which can and has been done a g a i n by an approp r i a t e m o d i f i c a t i o n o f t h e r e d u c t i o n technique e x p l a i n e d i n t h e second s e c t i o n o f t h i s paper: S i n c e C r r ( f ) can be bounded by some l i n e a r e x p r e s s i o n i n t h e network c o m p l e x i t y o f f w i t h r e s p e c t t o l o g i c a l networks b u i l t up w i t h any complete. s e t o f b i n a r y Boolean o p e r a t i o n s , we need t o c o n s i d e r o n l y networks computing f w i t h b i n a r y o p e r a t i o n s say v , A and I ( S h e f f e r ' s s t r o k e ) . We show how one can a s s o c i a t e t o an a r b i t r a r y such l o g i c a l network N (computing Boolean f u n c t i o n f ) a Horn netw h k domanLLea on, a Kmm i n p u i dom& an which i s Horn i n t h e i n p u t v a r i a b l e s and a Hehmnd ouipLLt darn&
w such t h a t uN
A
an
A
w defines f w i t h r e s p e c t t o i t s i n 
p u t v a r i a b l e s and i s o f l e n g t h l i n e a r l y bounded by t h e c o m p l e x i t y o f N. By such a construction CH(f) CN(O(f)) i s proved. L e t N be an a r b i t r a r y l o g i c a l network w i t h nodes No,...,Nm
where N1,
...,Nn
are
e n t r i e s , No i s t h e node w i t h r e s p e c t t o which N computes f and t h e n o n  e n t r i e s a r e l a b e l e d w i t h v,
A
1.
or
Every node Nk i s encoded by v a r i a b l e s yk.uk w i t h t h e in
tended iuztehpm&ztivn Yk = f,i,Nk(q)
and
k

1Y k
f o r p r e v i o u s l y g i v e n values q t o t h e i n p u t v a r i a b l e s xl,
..., xn.
Define therefore
aN as c o n j u n c t i o n o f t h e f o l l o w i n g formulae f o r e v e r y node Nk l a b e l l e d w i t h op(Nk) a p p l i e d t o t h e d i r e c t l y preceeding nodes Ni,N.

0 5 i,j,k
p u t e d a t Ni,N.: J Case 1. op(Nk) = A : (yi Case 2. op(Nk) = v : yi Case 3. op(Nk) = Define
an
w := yo
A
i n t h i s order
J
5 m , d e s c r i b i n g t h e computation a t node Nk f o r t h e arguments com
I:
u.
1
A +
A
yj) yk
+
yk
yj u . + yk
+
J
as c o n j u n c t i o n o f a l l yi
ui yk
Yi
+
uk u j + uk (ui A u . ) + uk J Uk Y j Uk
+
xi and ui
+
++
l x . f o r 1 5 i 5 n and
The n e c h t i o n pmpehty f o r a l l qe{O,ll
A
reads:
294
E. BORGER
fN,No(q) = 1 iff
uN
A
an(q)
A
yo
A
l u o is satisfiable
where a s before a n ( q ) denotes an a f t e r s u b s t i t u t i o n of q i f o r x i . The proof of the reduction property follows the now well e s t a b l i s h e d p a t t e r n : from l e f t t o r i g h t the above i n d i c a t e d intended i n t e r p r e t a t i o n s a t i s f i e s a N A a n ( q ) A w . Conversely any truth assignement f o r which t h a t formula i s t r u e simulates t h e network computation i n t h e sense t h a t f o r every node N k of N: (i)
f N , N k ( q )= 1 implies A(yk) = 1
from which f
N,No
( 4 ) = 1 follows because A ( u o ) = 0 by
W.
The simulation property
i s shown by induction along t h e computation process of N: the base of t h e induct i o n a t e n t r i e s N i ( l 5 i 5 n ) i s assured by a n ( q ) , whereas f o r every node N k with d i r e c t l y preceding nodes Ni,N. i n t h i s order t h e claim follows from t h e inducJ t i v e hypothesis, the formulae corresponding t o t h i s node and f N , N k ( q ) = o p ( N k ) ( f N , N , ( q ) ,f N , N ( 9 ) ) . j 2 3 For a proof of the o t h e r claim C N ( f ) 5 O(CH(f) ( l g C H ( f ) ) ) s e e Aanderaa & Borger 1981.
Added 1 9 8 3 , j u l y . W.D. G o l d l a r b has j u s t shown t h a t t h e case w i t h = i s u n s o l v a b l e , e v e n when r e s t r i c t e d t o Krom
GGdelKalmarSchOtte
o r t o formulae w i t h
o n l y d y a d i c p r e d i c a t e l e t t e r s . See t h e p a p e r “The U n s o l v a b i l i t y o f t h e GGdel c l a s s w i t h i d e n t i t y ” s u b m i t t e d t o t h e J . o f Symbolic Logic.
Decision Problems in Predicate Logic
295
References A complete l i s t o f papers published i n t h e area o f (complexity o f ) l o g i c a l d e c i s i o n problem i s o u t o f t h e scope o f t h i s survey. However t a k i n g t h e union o f a l l papers c i t e d i n any o f t h e papers c i t e d i n t h i s l i s t w i l l g i v e you a r a t h e r complete picture o f the situation. Aanderaa, S t i l 0. (1966): A New Undecidable Problem w i t h A p p l i c a t i o n s i n Logic. Ph.0. t h e s i s , Harvard U n i v e r s i t y . Aanderaa, S.O. (1971): On t h e d e c i s i o n problem f o r formulas i n which a l l d i s j u n c t i o n s are b i n a r y . Proc. o f t h e Second Scand. Log. Symp., pp. 118. Aanderaa, S.O., Borger, E. (1979): The Horn complexity o f Boolean f u n c t i o n s and Cook's problem. i n : F.V. Jensen, B.H. Mayoh, K.K. M o l l e r (Ed.): Proceedings from 5 t h Scandinavian Logic Symposium, Aalborg U n i v e r s i t y Press, pp. 231256, Aanderaa, S.O., Borger, E. (1981): The equivalence o f Horn and network complexity f o r Boolean f u n c t i o n s . Acta I n f o r m a t i c a 15, 303307. Aandera, S.O. & Borger, E. & Gurevich, Yu. (1982: P r e f i x classes o f Krom formulae w i t h i d e n t i t y . i n : Archiv f. math. Logik 22, pp, 4349. Aanderaa, S.O., Borger, E., Lewis, H.R. (1982): Conservative r e d u c t i o n classes o f Krom formulas. The Journ. o f Symb. Logic 47, 110129. Aanderaa, S.O. & Jensen, F.V. (1973):On t h e existence o f r e c u r s i v e models f o r Krom formulas. Manuscript, U n i v e r s i t y o f Oslo. Aanderaa, S d l 0. and Harry 4. Lewis (1973): P r e f i x classes o f Krom formulas, Journal o f Symbolic Logic pp. 628642.
38,
Aanderaa, S . O . G1 Lewis, H.R. (1974): Linear sampling and t h e v j v case o f t h e dec i s i o n problem. Journal o f Symbolic Logic 39, pp. 519548. Abramsky, M o t t i (1980): The c l a s s i c a l d e c i s i o n problem and p a r t i a l f u n c t i o n s . i n : Archiv math. Logik 20, pp. 312. Ackermann, W. (1928): Ober d i e E r f u l l b a r k e i t gewisser Zahlausdrdcke. Mathematische Annalen 100, pp. 638649. Ackermann, W. (1954): Solvable Cases o f t h e Decision Problem, NorthHolland, Amsterdam. Ash, C.J. (1975): Sentences w i t h f i n i t e models. i n : Z e i t s c h r . f . math. Logik und Grundlagen d. Math. 21, pp. 401404. Asser, G. (1955) : Das Reprasentantenproblem i m Pradikatenkal k u l der e r s t e n Stufe m i t I d e n t i t a t . Z e i t s c h r i f t fir math'ematische Logik und Grundlagen der Mathematik, 1, pp. 252263. Bennett, J. (1962): On spectra, Doctoral D i s s e r t a t i o n , Princeton U n i v e r s i t y , P1.J. Berman, L. (1977). Precise bounds on Presburger a r i t h m e t i c and t h e r e a l s w i t h add i t i o n : p r e l i m i n a r y r e p o r t . Proceedings o f 18th Annual Symposium on Foundations o f Computer Science, IEEE Computer Society, pp. 9599. Bernays, Paul & Schonfinkel, Moses (1928): Zum Entscheidungsproblem der mathemat i s c h e n Logik, Mathematische Annalen 99, pp. 342.372. Borger, E. (1971): Reduktionstypen i n Krom und Hornformeln. D i s s e r t a t i o n . Minster see B e i t r a g z u r Reduktion des Entscheidungsproblems auf Klassen von Hornformeln m i t kurzen Alternationen. i n : Archiv f u r math. Logik und Grundlagenforschung 16 (1974), 6784. Borger, E. (1973): Eine entscheidbare Klasse von Kromformeln. Z e i t s c h r . f u r math, Logik und Grundlagen d e r llathematik 19, 117120.
E. Borger (1974): La z3completude de l'ensemble des types de reduction, in: Logique e t Analyse 6566, 8994.
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~(0,4)

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LOGIC COLLOQUIUM '82 G . LON< G. Long0 and A. Marcia (editors) 0 Elsevier Science Publishers B. V. (NorthHolland), 1984
MODEL THEORETIC ISSUES IN THEORETICAL COWUTER SCIENCE, PART I: RELATIONAL DATA BASES AND ABSTRACT DATA TYPES.
J.A.Makowsky Department of Computer Science, Technion  Israel Institute of Technology, Haifa, Israel
Table of Contents: Introduction. 1. Abstract model theory and computer science. 1.1 From syntax to semantics and back. 1.2Finding axioms. 1.3Comparing logics.
2.Data base theory. 2.1. What it is all about. 2.2.Safety (definiteness,domain independence). 2.3.Typed dependencies. 2.4.Implicational dependencies. 2.5. Decision problems. 2.6. Query languages. 2.7.Conclusions and some open problems. 3. Specification of abstract data types. 3.1.Introduction. 3.2. The axiomatic framework. 3.3. A complete specification language for rich semantical systems. 3.4. Typical models and initial algebras. 3.5. A complete language for semantic systems which admit initial semantics. 3.6. Relevance for specification of abstract data types. 3.7.A word on other applications.
Supported by t h e Swiss National Science Foundation Grant No. 82.820.0.80
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Introduction The following paper is an account of experiences I had in several attempts to capture problems posed in computer science, or more precisely by computer scientists. In most of the cases the computer scientist already thought that logic might help in stating problems more precisely, and ultimately, also in solving them, though they were usually suspicious about the impact such solutions would have on their direct practical involvement with programming, program analysis, program design or program veriflcation. Maybe a word on impact of foundational studies on applied science and engineering is needed here: Most electricians are not aware how much 19th century physics has contributed in making the portability of electrical appliances possible. The fact that two or three numeric parameters (voltage, power and the number of cycles in alternating current) contain all the information needed to decide, whether a given appliance can be used by plugging it into a given outlet, has become too common in every day life to be reflected upon. To remember that many years of research were needed to clarify this situation, is by now safely forgotten. Ultimately the problems posed by computer scientist to logicians, or for that matter to any one willing to spend time on foundational questions, is similar: What are the parameters needed t o e m r e portability and reliability of software? Needless to say, we are still very far from satisfactory answers. The current progress in technology even prevents computer manufacturers from reaching agreements, a s they were reached rather quickly, say, by manufacturerers of phonographs and, to some extent, videotape systems, on speed and size of the records (tapes) to be produced. But a deeper reason behind the problem consists in the absence of a definite model of the real world, here the programming environment. Though models of "computability" have been sufficiently clarified for deterministic sequential algorithms, provided their task can be unambiguously specified in some form of "natural scientific language", it is much less clear what "specipcation", "implementation" and "correctness" should mean. The problems involved are not exclusively problems of computer science. Any large scale design and implementation of a big organisational complex, from industrial to social engineering, touches upon the same fundamental questions. The only difference with computers stems from the fact, that they execute programs very quickly, and programs, which are used only once or a few times, become very soon obsolete. But we are generally inclined to expect that the time needed t o develop a program stays within proportions to the time i t runs and remains useful. This leads some to think, that also the foundational questions can be solved quickly. However, a short glance a t the history of mathematics shows us, that something like fifty years were spent till the basic notions of, say, point set topology were safely
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established and had their impact not only on mathematicians but also physicists and practitioners of applied statistics. In the following paper I will try to explain how I have learned to view certain problems in the foundations of computer science and I will do this with three apparently different aspects of current computer science research: Data base theory, algebraic specification of abstract data types and algorithmic logic. As I will try to show, they have much more in common than widely believed, a t least when looked a t from the point of view of " a b s t r a c t m o d e l theory". But I a m fully aware that the challenge of applied science is not met by declaring that some ready made theory captures all its problems. This is never so. The difference between pure and applied, say, in differential equations, consists rather in the motivation of the results than in the results itself. The pure mathematician is content with knowledge which contributes to his understanding of the internal problems of differential equations as such, and solving a particular one is seen by him as a challenge of his general understanding. To the physicist, most of the work actually consists in justifying his particular differential equation, its parameters and its solutions in terms of his physical problem, and a large part of the work of good applied mathematics consists in reaching an understanding between the two perspectives. In computer science, we meet the same situation and both sides are often tempted to underestimate the work involved in Listening t o e a c h o t h e r . This paper is also an attempt to illustrate this work. For reasons of space (in this proceedings) and time (the Damocles sword of the deadlines), the paper had t o be cut into two parts. The first deals with data base theory and specification of abstract data types, and the second one with various approaches to semantics of programming languages. The first part also includes a general introduction on abstract model theory and its potential use in theoretical computer science. The second part [Makowsky 19831 will also include a n expository chapter of some more technical parts of abstract model theory. Here is an outline of both parts: In chapter 1 we t r y to give a description of what abstract model theory is all about, and how it is connected to the fundamental questions of computer science cited above. In chapter 2 we t r y to exemplify this in the case of data base theory. As it turns out there are various intimate connections between f r n i t e m o d e l t h e o r y and data base theory, which have led people to think that either data base theory is just undergraduate logic or that the logicians try to sell i t as such. But the real problems in any applied science are neither defined by their mathematical difficulty nor by the methodologies used to solve them, but rather by the questions they try to answer. Data base theory tries to answer the questions about design, design criteria, optimization and specification of data bases and their queries. This chapter could not have been written without the patience of C.Beeri, M.Vardi and A.Zvieli.
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In chapter 3 we turn our attention to the problem of specifying abstract data types. There are various approaches to this problem, but the most successful, a t least in terms of fashion, is the one called algebraic. Here w e flnd an i n t e r e s t i n g i n t e r p l a y b e t w e e n category theory and universal algebra, which has led people to think that this is just disguised "abstract nonsense", but again, as above, the problem we address here is the clarification of concepts such as data type, implementation, specification, modular programming and their ramifications and only a close, sympathetic analysis of these problems can lead to satisfactory answers. The results reported here are joint work with B.Mahr. As it turns out, there is a common theme in these last two chapters: In both u n i v e r s a l Horn formulas play a n eminent role. In the last section of chapter 3 we try to give some explanation of this phenomenon. It seems to support some of the arguments put foreward by the proponents of logic p r o g r a m m i n g as the programming style appropriate for the fifth generation computers. However, some recent complexity results, such a s [ItaiMakowsky 19821 still nourish some scepticism with respect to the unrestricted use programming languages like PROLOG. The remaining two chapters form part two ([Makowsky 19831): In chapter 4 we turn to the technical parts of abstract model theory a s we see them fit the needs of various branches of program semantics and program verification. We attempt to give a general definitjon of predicate transformers, as they appear in the context of program correctness. The definition is parallel t o the definition of generalized quantifiers, which will turn out to be a special case. On the basis of a set of predicate transformers one can build various algorithmic logics, of which again the classical examples of dynamic logic, process logic and others are special cases. In chapter 5 we turn our attention to program correctness and programming logics. We use the various logics from the previous chapter to introduce a new type of semantics, which in contrast to operational or denotational semantics, maps programming languages into subsets of logics. Here the meaning of a program is the set of all statements in a predicate transformer logic which are true about it. This is clearly not new as such, but has never been defined in a general context. One of the advantages of such a general approach is, that this allows us also to compare various approaches to program semantics which hitherto were considered incomparable. This last chapter is to report about work which is still in progress, mainly in collaboration with N.Francez and S.Katz. I would like to thank the Swiss National Science Foundation, who supported me generously during the two years in which the material presented came into being. I would like to thank also E.Engeler and E.Shamir who encouraged me to look into foundational problems in Computer Science and to C.Beeri, A.Meyer and V.Pratt. whose interest and criticism in early stages of the work was extremely stimulating. I
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would also like 'to thank R.Fagin and J.Thatcher, who read and commented the almost final version. A last note on the references: There are over one hundred titles listed as I saw them fit my presentation. I did not try to give historic remarks, nor did I attempt completeness. I tried t o give the reader pointers to the vast literature, a s I came accross it during my random walk in the world of theoretical computer science. It serves as a basis for further backtracking: The transitive closure of this reference list surely covers much.
1. Abstract model theory and computer science. 1.1 From syntax to semantics and back. In the early days of modern logic, logic was perceived mainly syntactically. Propositional logic, first order logic and second order logic were given as formal languages and a main topic of research was the study of deduction rules and proof systems. There are various philosophical, sociological and even political reasons for this, usually subsumed under the name "Hilbert's program". G.Kreise1 has written extensively about Hilbert's program and the way it failed. From his analysis in [Kreisel 1968,19701 he drew several conclusions relevant for computer science which inspired the theses of R.Statman [Statman 19743 and C.Goad [Goad 19801. The former added a new dimensions to our understanding of the complexity of proofs and the latter used his experience gained in proof theory to speed up the synthesis of special purpose programs for hidden surface elimination, [Goad 19821, Hilbert's program wanted to reduce mathematics, and therefore all exact sciences, to the formal (or, as we would say today: algorithmic) manipulations of symbols. The ultimate hope behind this was, to find general purpose algorithms, which would solve all formally stated problems. A s we know today, Godel showed that this is impossible. But a t the same time modern semantics was born. The fashion had changed, and instead of the "God given" Natural Numbers, Tarski and his contemporaries moved to accept naive set theory as the basis of mathematics and proposed to explain logic in terms of set theory. The meaning of logical formulas was explained in terms of structures, relations, functions and in the case of first order logic this was justified by the celebrated conipleteness theorem. One of the corollaries of the completeness theorem is the compactness theorem, which was extended to uncountable sets by Mal'cev in 1936 and independently by Henkin in 1949. Among the many consequences of the compactness theorem is the existence of various "nonstandard" models of arithmetic and analysis, which later led to a very fruitful branch of logic called nonstandard analysis, which was first pursued by
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A.Robinson and led to various impressive results in analysis, Banach space theory, the theory of Brownian motion and even mathematical economics. But one of the first nontrivial applications of the compactness theorem was the characterization of the universal first order formulas in terms of an algebraic preservation property: A first order formula cp is logically equivalent to a universal formula if and only if cp is preserved under substructures. This result was followed by a n intensive program exploring the relationship between semantic properties of formulas and syntactic characterizations of such formulas. Motivated by algebraic practice the notion of substructure was replaced successively by unions of chains, products, reduced products, factors and many others and sufficient experience was gained to delegate this direction of research to the level of master theses and difficult exercises. The mathematical tools used to solve such problem, if there is a clean solution, usually are interpolation theorems, ultraproducts and Back and Forth arguments. In [ChangKeisler 19731 the reader may find what ever is known in this direction. But there is another way in looking a t this program: We could reverse the problem and start with any syntactically defined class of formulas together with their meaning functions and ask for a characterization of this class in terms of the preservation properties it has. Looked a t it this way, what we really are asking for is giving meaning to syntactic categories. It is this aspect of preservation theorems which I think is relevant to foundational questions in computer science. Very often the computer scientists start with syntactic restrictions and later try very hard to remove them, without understanding their significance. But, a s will be shown in chapters 2 and 3, those restrictions, originally imposed for technical reasons, can be characterized by preservation properties, which show that they are intimately connected with the implicit assumptions the computer scientists have made. The use of powerful set theoretic methods led also to another development. Already in the fifties Mostowski in Warsaw, Engeler in Zurich and Tarski and Henkin in Berkeley started to look a t various generalizations of first order logic involving infinitary constructs and generalized quantifiers and a n abundancy of logics appeared. It was Engeler, however, who first noticed the possible relevance of infinitary logics to computer science([Engeler 1967,1970]).This has since led to the development of dynamic logic, and we shall return to this topic in chapter 4 and 5 of this paper. In the sixties much of model theory was generalized to those newly discovered logics, and, based on earlier work by Mostowski, Lindstrom defined an axiomatic framework, sometimes called abstract model theory or higher model theory, in which we can study logics in general, compare their expressive power and prove characterization theorems for logics in terms of their model theoretic properties. The latter has very striking parallels with the above mentioned preservation theorems both in content a s well as in methodology. In the rest of this chapter we shall briefly describe this framework and give some key
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results, which we will apply in chapter 4 and 5 to the study of various dynamic logics. An account of the state of art in abstract model theory can be found in the forthcoming book [BanviseFeferman 19831, and an introductory survey in [Flum 19751. The purpose of abstract model theory can be summarized as follows: We want to be able to quantify over all possible logics satisfying certain properties and prove theorems about them. The theorems we want to prove can be (A) characterization theorems. (B) presentation theorems. (C) theorems relating various properties of logics. Examples for (A) are the Lindstrorn theorems, ([Lindstrom 1969]), characterizing first order logic in terms of the LowenheimSkolem theorem together with various properties such a s compactness or axiomatizabilty. The latter has been closely analyzed for its usefulness in computer science in [MandersDaley 19831 and in [Makowsky 19801. Examples for (B)are Birkhoff's theorem characterizing the varieties a s the equationally definable classes of algebras (cf.[Graetzer 1979]), Mal'cev's characterization of the quasivarieties (cf. [Mal'cev 19711) and Cudnovskii's theorem that every class of structures closed under substructures can be axiomatized by a class of infinitary clauses (cf. theorem 1 in chapter 3 and [Cudnovskii 19681).In some sense the result in [MeyerParikh 19811, showing that most dynamic logics for finitely branching programs can be embedded in the recursive part of countably infinite logic, also fits this category. An example for (C), finally, is that axiomatizability implies recursive compactness or that for countable logics the amalgamation property is equivalent to compactness. The former is a corollary of the Lindstrom theorems and the latter is in [MakowskyShelah 19831. The use of abstract model theory lies in its limitative character. It tells us that certain requirements are incompatible o r entail other limitations. It can tell us to what extent seemingly different approaches are nevertheless the same. Or it can give is a framework in which we can precisely compare concepts which hitherto appeared incomparable. 1.2 Finding axioms. Our first problem is to find axioms for logics. Logics will consist of quadruples L = ( T , % r , F l n l , i= ), where T is a class of v o c a b u l a r i e s or s i g n a t u r e s and Str is a function mapping every T E T into a subclass * ( T ) of all structures of vocabulary T such that if T,T' E T,T c T' then S t r ( ~c) S t 7 ( ~ ' ) . Here we assume that in T we have a partial order denoted by c . In all the cases we consider the elements of T are sets of symbols and c just is the subset relation. The usage of the term v o c a b u l a r y for what is called similarity type ( o r signature or even language) seems to capture what we really have in mind. The vocabulary is the most elementary part of logic, and it determines about what we will talk. In the case of first order logic it consists just of sets of relations symbols, function symbols or constant symbols,
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together with their arities, and in the case of many sorted logic, with their sort specifications. In other cases it may also specify variables to be second order, or make certain distinctions between logic with or without equality, or other special symbols. Sometimes it is convenient to think of T as a c a t e g o r y of v o c a b u l a r i e s rather than just a set or a class. This is especially the case when want to consider vocabularies which are more complicated than usually and we have to define the partial order on T in a more complex way. But these cases are still not very well developed. Now given T,the class % ( T ) tells us which structures of vocabulary 7 we are interested in. This may, in logic, often comprise all the 7structures, but in applications we very often impose restrictions. In chapter 2 we shall see that for data base theory we only consider finite structures, or finite reducts of expansions of the standard model of arithmetic, and in chapter three only certain countable structures, the reachable structures are of interest. Finally in chapters 4 and 5 various more complicated structures will enter the picture, including certain models of tense logic, probability logic etc. Again it may be convenient to think of % ( T ) as subcategories of a big category S r u c t = u S ~ T ( Tand ) of a functor T E
T
mapping T into S t m c t . We refer the reader to [Barwise 19741 for a detailed presentation. Also i+ni is a function which maps every T E T into a set of objects called f o n n t d a s . Again we require that for T , T ' E T , T C T ' we have that M ( Tc)M ( 7 ' ) .When choosing the set of formulas, we have to bear in mind to contradicting aspects: We want to say much about our structures, certainly as much as we need in our particular context. But we do not want to say too much, because we want to keep our model theory out of the difficulties of full second order logic. Finally I= is a relation on % ( T ) x ~ ( T ) ,which satisfies certain axioms: IsomOrphismAxiom If A B E ST(T),(P E M(T and ) AEB then At= iff BI= rp. ReductAxiomlf ( p ~ m n l ( ~ ) , ~ c ~ ' a n d thenAl=rpiff A~St~(~A ' )r r l = ( p . Renaming Axiom Let U,T E T and p : +~u be a r e n a m i n g , i.e. an isomor) is phism in the category of vocabularies. Then for each p E M ( Tthere ++ E Rd (u) such that for all A E S ~ T ( Twe ) have that Al= rp iff MI= p". Those axioms do not require too much. All examples which we shall encounter in this paper satisfy them. From the theorems in abstract model theory cited in the previous section, however, only example (B) can be proved with these axioms alone. In chapter 3 we shall use similar axioms to axiomatize the behaviour of sets of formulas rather than formulas. What makes abstract model theory into a theory are various additional closure properties, which we impose on the formulas, or rather on their models Mod,(rp)=IA E S t r ( ~ ) : A l =pi. Many model theoretic properties of various logics can be stated by only referring to the model classes Mod,((p) definable by their formulas. The compactness theorem and the LowenheimSkolem theorem are among them, and also various
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definability theorems. But here are some of the closure axioms: Atomic Axiom For every T E T the usual ratomic formulas are contained ill M
( T ) .
Basic Axiom For every
T E T the usual 7basic formulas (i.e. atomic and negated atomic formulas) are contained in M ( T ) . Boolean Axiom Ftnl(7) is closed under the boolean operations A , V , .. with their usual meaning, i.e. heir model classes are defined by intersection, union and complement respectively. QuantificationAxiom h z i ( T ) is closed under existential quantification 4 z with its usual meaning. The next two axioms assume some knowledge of the structure of the formulas if we want to state them naturally. They are the relativizatian axii~rnand the substitution axiom. We will discuss them in more detail in chapter 4. For the first part of the paper their exact definition is irrelevant. Examples of logics are first order logic, infinitary logics, logics with generalized quantifiers etc. All the classes of dependencies in chapter 2 can be viewed a s logics (though without all the closure properties) and in some sense also the s e m a n t i c a l systems of chapter 3. Behind the choice of closure properties lies the problem of iteration under various formation rules for formulas, in other words the choice of primitives for our logics. Already in the early days of infinitary logics did fieisel point out in [Kreisel 19681 that we need definability criteria to evaluate such choices, rather than just adding various constructs ad libitum. He advocated a line of research which not only led to unifying results for infinitary logics and generalized recursion theory but also to a deeper insight in general. It led to the very rich theory of admissible sets, as presented in [Barwise 19751. The corresponding problem we face in computer science has not even been formulated generally. In data base theory only [Chandra 1901] questions the choice of programming primitives and [ChandraHare1 19801 define general criteria for query languages. We shall study the latter in section 2.7. and show that in this case "Kreisel's program" can be followed to a large extent. For specifications of abstract data types [BurstallGoguen 19831 and [MakowskyMahr 19831 attack this problem In the second part of the paper we shall outline what can be done for semantics of programming languages, but we are still f a r from a general understanding. When we want to apply the framework of abstract model theory t o foundational problems in computer science, we observe quickly that what we hope to be logics are usually not closed under all the closure operations we have mentioned above. The striking example here is Hoare logic, which consists of statements about programs of a particular form, the correctness statements, but is not closed under any iteration or boolean operation. This leads t o an abundancy of logics which are hard t o compare, cf. [MeyerTiuryn 19811 and [Meyer 19801. The reasons for
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the absence of closure properties are sometimes not clear, but in other cases motivated by practical experience. Logics are usually closed under substitutions of predicate symbols by formulas, but in programming some formulas occur as test in programs, and those should remain simple, and it is clear that we do not allow termination statements of other programs to occur as test. The choice of the correct closure properties for logics affects the applicability of results from abstract model theory very much. Sometimes, however, the absence of closure properties can be compensated by a weakening of the theorems. OIten a theorem states that every formula p in a given logic satisfying certain conditions is equivalent to another formula in a different logic. In the absence of the Boolean axiom, this may be conveniently rephrased by stating that (p is equivalent to a boolean combination of such formulas. In more complicated cases, however, we have to allow the use of additional predicates. To be more precise we need some definitions: Definitions: (i) Let cp E M ( T u IRj) R E T be a formula and u c T . We say that (p deflnes R implicitly over u i f : Every ustructure A can be expanded to a T u [Rjstructure A’ such that A’ I=p and given two structures A,B E ~ T ( uT {I?{) with Al= (p,BI=(p such that At u = E l u, then RA=RB,i.e. R is uniquely determined by (p and u. (ii) Let p E M ( T [Rj) Ube a formula which defines R implicitly over u, R nary. We say that $(vl,vz,...,v,) E ~ ( U )with n free variables defines R explicitly, if for every A E ~ T ( T with ) Al= (p we have that Al= v 111.v~ ....,vn(R<=>$(v i,vz,...,
v,)).
With these definitions we can state a n even stronger closure property: Aclosure Axiom: Every implicitly defined relation has an explicit definition. Examples: (i) (Manysorted) first order logic satisfies all the closure axioms. The Aclosure is a variant of Beth’s definability theorem first stated in [Feferman 19741. (ii) First order logic without function symbols and with all structures finite, as we shall use it for data base theory in chapter 2 , does not satisfy the Aclosure axiom, a s pointed out in [Hajek 19761. (iii) Note that Aclosure is a stronger property than closure under substitution. (iv) I t is this Aclosure property which made various ideas of [Kreisel 19681 more precise. Kreisel’s work led to the definition of admissible sets, and H.Friedman showed a deep connection between Aclosed logics and logics built on admissible sets, cf. [MakowskyShelahStavi 19761. (v) An interesting application of Beth’s theorem to data base decomposition problems may be found in [Vqrdi 19821. More examples will be studied in chapter 4.
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1.3 Comparing logics. When we want to compare logics, we want to compare their expressive power, i.e. what subsets or relations of its structures are definable. We say that a logic L1 is reducible to a logic Lz if every formula of L , can be translated into a formula of Lz. Again our notion of comparability will depend on the various closure properties the logics in question have. For positive results we are usually interested in the highest possible degree of precision on the nature of this translation, whereas for negative results, on the contrary, we prefer ample freedom. Let us propose some definitions: Definitions: Let &=(7i,&t,ml,I= *) be logics for i=1.2. (i) L , is explicitly reducible to L,, if T , c Tz,for each T E T, Str2(r)c S T , ( T ) , and for every p E M , ( T there ) is 11 E M z ( r )such that MoG((o)n S t ~ ( ~ ) = M o d ~ ( l ( ' )
We write for this L , < L ~ . (ii) L1 is implicitly reducible to Lz, if TI c Tz, for each T E TI . % T ~ ( T )c str,(~),and for every implicit definition over T via c E i%i,(r') with T c T' there is an implicit definition over T via @ E F&(T') such that ~%d,~(p)n * z ( T ' ) 1 mod,(@) 1 T We write for this L , h , L 2 . (iii) We say that L , and Lz are explicitly (implicitly) equivalent, if both L I c L z and Lz
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2. Data Base Theory for the Relational Model. 2.1 Introduction One of the most frequent application of computers nowadays is in data bases. There are various ways of modeling data bases, such as the network model or the hierarchical model, but the most widely studied for theoretical purposes is the relational model. An excellent reference is the textbook [Ullman 19821. The entityrelationship model, cf. [Cheri 1976,1981] has not yet been really studied from a theoretical point of view. As much as I understand it, most of the theoretical results for the relational model carry over to the entityrelationship model, such as computable queries and dependency theory, but, to the best of my knowledge, no serious attempt has been undertaken to carry out such a task. In what follows we concentrate on the relational model. It consists of families of data base states, which are divided in acceptable or consistent states and inconsistent states. Those are distinguished by constraints or dependencies. The consistent states are the models of the dependencies. With data base states we can do two things: We can ask queries or we can perform transactions. To complicate matters this is usually done by many users a t the same time; we speak therefore of concurrent users. Transactions map consistent data base states into consistent data base states. They are usually decomposed into smaller operations which map consistent data base states sometimes into inconsistent data base states. There are two kinds of simple transactions: read only and write only. More complex transactions can be built from simple transactions by composition. Needless to say, all these operations should be computable. To sort out this mess a theory of transactions and concurrency control is in the making. The state of the art is described in [Date 19821, [Casanova 19811 and in the forthcoming book [Maier 19831. An excellent survey is [BernsteinGoodman 19821. In this chapter we are only concerned with a special case of read only transactions, queries and dependencies. Neither general transactions nor concurrency control play a direct role. Indirectly, however, they serve as a motivation in our presentation of dependency theory. Queries map data base states into relations. In [ChandraHare1 19801 an abstract definition is given, the computable queries, which is the basis of our presentation in this chapter.
Data base states are structures like for first order logic, but for practical purposes some restriction are necessary. First of all, the structures are finite. Second, we distinguish between the relations representing the tuples in the data bases and the aggregate functions such as arithmetic operations or linear order on the entries. And third, we are not really interested in the underlying universes but only in the relations a s such. In this chapter we shall not talk about the aggregate functions a t all. To ensure that it makes no difference whether we talk about
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relations or structures we introduce an invariance condition, called safety, which is discussed in section 2.2.
Dependencies are classes of data base states, usually the models of some first order sentences. They are grouped and classified according to syntactic criteria: Functional dependencies (FD), full implicational dependencies (FID), embedded implicational dependencies (EID), template dependencies (TD), multi valued dependencies (MVD) and embedded multivalued dependencies (EMVD) can be conveniently described as classes of first order sentences with specific syntactic restrictions. In the sense of chapter 1, dependencies usually form a logic, with the vocabularies ranging over sets of relation symbols only and all the structures being finite. Queries are definable relations in this logic, and we shall see in section 2.6. that the knplicit definition play an important role here. Preservation theorems in logic are theorems which characterize classes of first order formulas having some semantic properties by showing that those sentences are exactly the ones which allow a special syntactic normal form. They are special cases of presentation theorems in the sense of chapter 1. But in contrast to universal normal form theorems (such as every first order formula is equivalent to a prenex formula) which usually are constructive, the normal form theorems coming from preservation theorems are often nonconstructive. What we get is the following: The set of sentences Shaving some semantic property P is not recursive, but there is a recursive set So such that every sentence u E S is equivalent ( over some first order theory ) to a sentence u o ~ S O . Though the theorem is nonconstructive this has two advantages: (i) We can, with no loss of generality, restrict ourselves  or for that matter the programmer of a data base system  to dependencies of the form So,and (ii) by doing so, we know that property P is a priori ensured. If the property p is one which is of intrinsic importance to our database system, then the restriction to sentences from So will free the programmer from the correctness proof  or rather  force him t d choose his dependencies carefully and prove then correctness before he is allowed to write them down.
Now in logic, the choice of the semantic properties P is usually given in a natural way, say from algebraic considerations , and the problem is to find So. In data base theory the situation is reversed: We are given various candidates s,, a s the FD, FID, EID, MVD, TD, EMVD etc, and the problem we pose, is to define the corresponding properties P which both characterize so’and a r e g e n u i n e l y m o t i v a t e d by data base considerat i o n s . Tt is our firm belief, that the syntactic restrictions given to various classes of dependencies are only meaningful iff they correspond to a semantic property which reflects d a t a base p r a c t i c e . And it is such a property which should be called the meaning of a syntactic restriction. What we show here is giving meaning to being safe, typed and being a
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typed implicational dependency. What we propose furthermore is a program which consists of searching for the meaning of various other syntactic definitions of dependencies such as template, embedded etc. Fagin went a good way to do this for typed embedded implicational dependencies [Fagin 19821 by showing that they are faithful (i.e true in a product of finite nonempty relations iff true in each factor) and my previous [Makowsky 19811 proposed several such characterization, but their relevance for data base practice was not yet satisfactorally shown. In a forthcoming paper [MakowskyVardi 19831 more such results are collected. Our main results here are: (i) The complete characterization of equality generating dependencies based on separable dependencies wich have the subrelation property and are preserved under products. (ii) The complete characterization of full typed tuple generating dependencies based on separable dependencies with the intersection property and the duplicate extension property. The intersection property had been previously characterized in [MaierMendelzonSagiv 19791 a s the property which guaranties the uniqueness of the completion operation in connection with the chase. Separability, however, is introduced here to give meaning to the restriction to typed formulas. It is discussed in detail in section 3 and captures the idea of separation of sorts, or attributes. This chapter is organized a s follows: In section 2.2 we discuss a well known example from the above point of view, the definite formulas from [Kuhns 19691 and their syntactic characterization a s permissible f o n n d a s , as described by many authors, e.g. [Cooper. 19801, or as safe formulas ,as described in Ullman’s book [Ullman 19821. We also note that the definite formulas are not recursively enumerable, as was shown by [Di Paola 19691. In section 2.3 we follow the same pattern to propose a semantic characterization of typed, formulas. We also show that this class is not recursively enumerable. The results in this section are drawn from [MakowskyVardi 19831. In section 2.4 we discuss FID’s and FD’s and connect the typed FID‘s (TFID) to the intersection property of relations. The results here are continuations of our previous work [Makowsky 19811. We end our presentation with some final remarks and open problems. In section 2.5 we discuss decidability and complexity results for the consequence problem for various classes of dependencies. In section 2.6 we give a brief presentation of the theory of computable queries and in section 2.7 we draw some conclusions and present some more open problems.
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2.2. Safety (def3niteness.domai.n independence). Already in 1967 [Kuhns 19671 it was realized that first order formulas, which are relevant for data base dependencies or queries, should satisfy an invariance condition. Intuitively this condition says that it does not matter if we speak of a relation or of a first order structure containing this relation. Definitions: Let R be a finite relation on and let A = c A , , A 2 , ' ' ,&,R>
n,A(
be the corresponding relational structure. Let A * be the relational structure obtained from A by addition of exactly one element 4 to every sort A( and not extending R . Kuhns calls the formulas u which are true in A iff they are true in A* definite. A class r of data base states is called definite if it is closed under the formation of A*. Fagin [Fagin 19821 independently looked a t this property and called it &main independence. Let S denote the class of definite first order formulas, and 9 be the class of definite formulas with a t most k a o free variables. Di Paola showed Theorem 1: 9 is not recursively enumerable for any kro. For a proof one may also consult [Vardi 19811. Note that if we allow infinite relations, we only get that s* is not recursive. Here we have a nonrecursive set of formulas S and we would like to find a recursive set So such that every formula u E S is equivalent to a formula uo E s,. Let So be the set of safe formulas a s in Ullman's book [Ullman 19821 ( or equivalently the set of permissible formulas from [Cooper 19801 ). If we allow infinite relations, it follows easily from results in model theory that every formula of S is equivalent to a formula in So. In fact we have even more: Theorem 2: Let C be a first order theory. Call a formula C definite if i t is definite on the class of finite models of 8. Then the following are equivalent: (i) u is Cdefinite and (ii) In all finite models of C is u is equivalent to a formula in So. For the proof we define a n algorithm based on r e l a t i u i z a t i o n , which maps arbitrary first order formlas into safe formulas and which preserves equivalence (for models of C) if and only if the original formula was safe. This does not contradict theorem 1, since it merely says that the set of first order formulas, on which this algorithm does preserve equivalence, is not recursive. Formulas with 'free variables define relations. For first order formulas this gives us a special case of first order (explicitly) definable queries. The definition of definite is naturally extended to this case. We will return to definite formulas in the section on query languages.
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2.3.Typed dependencies. In this section we look at the class of typed and s a f e formulas, which we denote by To. Clearly this a recursive set of first order formulas. We propose to define a n operation on finite relations, which intuitively corresponds to the introduction of different attributes (sorts) for the arguments of the relation. Let R c A n be a finite nary relation over some domain A . Let rri(R) be the ith projection of R onto A . and q=ni(R)x[i] We define a new relation R on in the following way: ((a1,1).(az,2) ,...,(%,TI)) E R if7 ( a l , a z,...,u,,) E R .
n& i
We say that a first order formula u admits separation of attributes (sorts) or, shortly, is separable, if u is true about R iff it is true about R. A class of data base states r is called separable if it is closed under the formation of R. Remarks: (i) If all the rr,(R) are disjoint then R is isomorphic to R . (ii) Using (i) we see that separable formulas are definite. This is due to our definition of A, which is a projection. Had we defined it just to be a new copy of A , the results below had to be slightly modified. (iii) Functional dependencies are separable. Let T denote the class of separable first order formulas, and P be the class of separable formulas with a t most k r o free variables. Using a similer argument as in [Vardi 19811 one gets: Theorem 3: P is not recursively enumerable for any kto.
Problem: Is the class of EID’s which are in T recursive ? That separable formulas really capture the separation of attributes (sorts) is shown in the following theorem: Theorem 4: Let Z be a first order theory. Call a formula Eseparable if it is separable on the class of finite models of X . Then the following are equivalent: (i) u is Xseparable and (ii) X proves that u is equivalent to a formula in To. The proof is similar to the proof of theorem 2. 2.4. Implicational Dependencies. We are now in a position to define more classes of dependencies: Definitions: (i) A first order formula over a set T of relation symbols is a full implicational dependency (FID),if it is of the form Yf Aib,(z)
4
b (z)
where each bi is an atomic formula not containing the equality symbol, b is atomic possibly containing equality and each variable which occurs in b also occurs in some b,. Note that we do not allow the empty conjunction.
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If
b is an equality we also speak of equality generating dependencies (EGD), and if b is an instance of a relation symbol we speak of tuple generating dependencies (TGD). The functional dependencies (FD)are the EGD’swith only two hi's. (ii) The classes TFID of typed full implicational dependencies, typed tuple generating dependencies TTGD and typed equality generating dependencies TEGD are defined analoguously. (iii) The class of embedded implicational dependencies EID, consists of first order formulas of the form VZ A+@)
+
gV
A;j(T.V)
where the hi's are a s for the FID and the ci’s are atomic with all the variables from d occurring already in the b i g s . (iv) The class of embedded template dependencies ETD8 consists of the EID’s with only one formula cr, which is not a n equality. In contrast to. some papers in the literature we allow EID’s to be untyped. A special case of template dependencies are the inclusion dependencies IND, where there is also only one formula b i . (v) The classes TFID,TEID,TETD,TINDof typed embedded dependencies are defined similerily. An important subclass of TID are the Functional Dependencies FD. Let X be a set of first order formulas and E(X) denote the set of first order formulas which are equivalent to some formula in X. The followihg is a useful observation: Proposition 5: (Beeri and Vardi) Every typed full implicational dependency is equivalent to a conjunction of a TGD and a EGD. As was observed by Vardi and the author we have Theorem 6:Both E(FID)and E(TGD) are not recursive. Neither is E(FD).
A proof, due to Vardi, may be found in [Makowsky 19811. I t could also be proved using methods similar to [McNulty 19791. U n d e r what conditions can we axiomatize classes r o f data base states by dependencies of prescribed syntactic form ? Let us look first at
TFID’S.Clearly they are again definite and separable. They also are preserved under Cartesian products (the p r o d m t property). Given u E FD and a relation R and a subrelation Ro c R then u is true about R iff it is true about Ro. (This is not generally true for FID.) Let us call this last property the subrelation property, both a s a preservation property for formulas u as well as a closure property for classes of data base states r. The subrelation property is very strong and dependencies which satisfy it are invariant under losing any portion of your data bases. Its integrity can not be destroyed by deleting data. Note that the subrelation property is stronger than the substmcture property in model theory, because here we really take subsets of the relation, whereas in model theory we take subsets of the domains and consider the relation naturally induced on them. The substructure property is true for FfD..h
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fact we have ([MakowskyVardi19833): Theorem 7: A class r of data base states, closed under isomorphisms. is axiomatizable by a set of typed equality generating dependencies TEGD i f fr i s (i) separable, (ii) has the subrelation property and (iii) contains the trivial structure and (iv) is closed under products. The triwial s b t u r e , is the structure which has exactly one element of each sort and all the tuples satisfy all the relations. Since the compactness theorem is not true, if we only consider finite structures, theorem 7 can not be stated, like theorem 2 and 4, for single formulas. Similarily we can define the intersectionproperty, which requires that if u is true in two relations Rl,R2 then it is also true in RI n Ra. Again we have two versions of it, one as a preservation property and the other as a closure property for classes of structures. Clearly the subrelation property implies the intersection property, but the intersection property s e e m more natural: Not every subset of a library catalogue is necessarily a catalogue, but we definitely expect the intersection of two catalogues t o be a catalogue. The intersection property is true for TGD and for FD but not for FID. Proposition 8: If a separable formula has the intersection property, then it has the substructure property (but not necessarily the subrelation property). The proof is purely semantical and uses the fact that we can represent every substructure as the intersection of two relations by renaming. A last such property we want to consider is preservation (closure) under duplicate eztensions. This is like logic without equality, i.e. we allow multiple occurrence of elenients. More precisely, let a E A , b L A and h be a mapping such that it is the identity on A+] and h ( a ) = b . We have a natural extension of h to R. Now C A u f b 1,R u h(R)> is a duplicate extension of cA.R>. With this we have ([MakowskyVardi 19831): Theorem 9: A class r of data base states, closed under isomorphisms, is axiomatizable by a set of tuple generating dependencies TGD iff r is (i) definite (ii) closed under duplicate extensions, (iii) the intersection property and (iv) contains the trivial structure. For typed dependencies we have the following analogue to theorem 9, also from [MakowskyVardi 19831. Similar theorems can also be stated for the other cases. Theorem 1 0 Let E be a set of first order formulas such that (i) E is true in the trivial structure, (ii) is separable,
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(iii) has the intersection property and (iv) preserves duplicating extensions. Then C is equivalent to a set of typed tuple generating dependencies TTGD'S.
All the properties above but the closure under products have natural justifications in terms of data base practice. We had previously characterized TFfD in terms of the Armstrong property (cf. [Fagin 1982, Makowsky 19811 and a strong form of the finite model property [Makowsky 19811,but theorem 10 seems more natural. The Armstrong property is version of the weak generic structures, a s dealt with in chapter 3, adapted t o database theory, The finite model property in question is related to the class of securable formulas as defined in section 2.5. 2.5. The Consequence Problem
For various classes of dependencies the consequence problem has been studied. In general it is stated as follows: Given a finite set C of dependencies in D , and a single dependency o F D , can we decide whether o is true in all (finite) relations satisfying 2? This is closely related to the consequence problems in logic, with the difference that here we are mainly interested in finite models over relation symbols only and that the class of formulas D , is of very low quantifier rank. Additionally, if we look a t typed dependencies, we can not use variables repeatedly in different positions. Though a logician would expect undecidability results, if the formulas involve both existential and universal quantifiers, there is still place for many decidable subcases. Clearly if the dependencies are boolean combinations of purely universal and purely existential formulas, the consequence problem is decidable ([BernaysSchonfinkel 19281,see also [Lewis 1979,1980]).This class of formulas has also been extensively studied in model theory (cf.[Tharp 19741 and [Makowsky 1975]), and they were called securable or c o n t i n u o u s formulas. They have many nice properties: Proposition 11: Let s be the class of securable formulas. Then (i) S is closed under boolean operations and (ii) The class of valid and of finitely valid formulas in S coincide. Clearly we get from this that the consequence problem for securable formulas is decidable. In fact, the exact complexity of this consequence problem is known. The reader not familiar with complexity classes should consult [GareyJohnson 19791. Theorem 1 2 ([Lewis 19801)) There are constants c > d > l such that the consequence problem for securable formulas without function symbols or equality can be solved in N71AfE(cn)but not in NTIME(dn). Securable formulas have the finite model property and are closed under boolean operations. This leads us t o the following problem: Problem: Do the properties (i) and (ii) proposition 1 1 characterize S up t o logical equivalence, i.e. given s satisfying (i) and (ii), is i t true t h a t
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every formula in s' is equivalent to a formula in S ? If this is not the case, i s there such a maximal class for (i) and (ii), or what properties have to be added to get maximality? Note that by a folklore result in model theory [Shoenfield 1967, problem ~ O C p.971 , S is characterize