G. Lolli ( E d.)
Recursion Theory and Computational Complexity Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 14-23, 1979
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected] ISBN 978-3-642-11071-9 e-ISBN: 978-3-642-11072-6 DOI:10.1007/978-3-642-11072-6 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2010 st Reprint of the 1 ed. C.I.M.E., Ed. Liguori, Napoli & Birkhäuser 1981 With kind permission of C.I.M.E.
Printed on acid-free paper
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C O N T E N T S
.
S HOMER
: Admissible Recursion 'Iheory
Pag.
7
B. E.
: Computational Complexity and recursion theory : A Survey of S e t Recursion
Pag.
Pag.
31 101
Pag.
109
: C o n s t r u c t i o n s i n t h e Recursively Enumerable Degrees Pag.
172
: Recursively I n v a r i a n t Recursion meory
229
D. G.E. R.I.
JACOBS
NORMANN
SACKS SOARE
W. MAASS
: P r i o r i t y Arguments in Higgler Recursion
Pag.
CEN TRO I N TERNAZIONALE MATEMATICO ESTIVO (c.I.M.E.1
ADMISSIBLE RECURSION THEORY
STEVE HOMER
ADMISSIBLE RECURSION THEORY Steve Homer DePaul University INTRODUCTION The purpose of these lectures is to develop some deeper results in a-recursion theory which will hold in somewhat more general setting than L(a) and in particular in many other admissible sets and structures.
In addition, I will briefly mention
some applications to structures which arise from other areas of recursion theory and which are inadmissible. An important underlying idea behind many of these theorems is the notion of a dynamic argument. by this is the following:
In general what is meant
In w-recursion theory, the starting
point for all of these generalizations, the exceedingly strong closure properties of w are used in almost every construction, often without a second glance. Almost any operation on finite sets yields a finite set and in particular the image of a finite set under any function is finite. Uhen we consider a-recursion theory, a C1-admissible, we of course loose a great deal of the closure present in w-recursion theory. have a certain weak closure property
However we still
- the image of an a-finite
set under a CI (La 1 function is a-finite. But as soon as we do a construction in which a C (La) (or C 3 (La) .) function arises,
...
and such does occur in almost every priority argument, then we immediately run into trouble. (We will see a concrete example of this shortly when we start talking about Post's problem in this setting. While we have lost the strong closure properties in
L(a)
we have gained the use of much work in set theory, mainly by ~ o d e land Jensen, in which many deep properties of L have been developed.
It is just these properties of L which often save us
when we run into trouble because of a lack of closure under certain functions. Now some properties of L work, under suitable conditions, in other settings, while others are really peculiar to L. icular many of the deeper results about L depend
In part-
on taking
Skolem hulls of certain sets in L(a), taking the tranitive collapse of that Skolem hull, and being able to determine exactly what that transitive collapse looks like, namely, an initial segment of L(a)
(an L (y) for some ~ ( a ) .
These collapsing arguments tend to be very specific to L and almost never work in other settings.
In particular, they fail
when the universe is changed by constructing L relative to a given predicate.
So, to get constructions to work in more gen-
eral settings, we need to eliminate, if at all possible, these collapsing arguments peculiar to L.
We want to give a "dynamicn
argument (basically more similar to the original one for w), which views r.e.
sets is being listed and increasing in an
effective manner, not as being defined by a El-formula which is really the crucial property making collapsing arguments work. In order to make this more explicit
I want now to turn to a
couple of concrete examples.
Both examples depend upon the same
method and so the second will be given in much less detail. After presenting these examples I will indicate some extensions and applications of these results to other areas and in particular to admissible sets and structures. We will also, and of course this is the reason these methods were first developed, gain some knowledge of the structure of a-r.e. degrees. In what follows I am assuming a familiarity with the basic facts and definitions of a-recursion theory.
These are
given in the first of Sacks' lectures in this volume.
For a
more detailed account see the papers by simpsonpg or ~hore(141. 1.
Post's Problem for a-Recursion Theory Let o be a C1-admissible ordinal.
We want to prove the
following theorem. Theorem: There exists two a-r.e. sets which are incomparable
.
with respect to < This theorem was originally proved by Sacks -a and Simpson 1111. Their argument depended heavily on using properties of Skolem hulls in L. I will present another way of proving this theorem based
largely on ideas of R. ~horeb41,(161
.
applied to this problem by Simpson t191
Shore's ideas were The proof is more
dynamic, more "constructive" if you will, than the original proof and, as we will see, is more adaptable to other settings. It is not so dependent on the special properities of initial segments of L. The argument will be presented by starting out with the basic ideas for solving Post's problem from w-recursion theory.
I assume some familiarity with that argument.
As we try to
carry out the argument in the setting of L(a) we will meet with various difficulties for which we will propose solutions. Finally, we will put all of this together to get the actual construction and proof. Now, we are going to construct two a-r.e. We require that A g a B and BfaA.
sets A and B.
In fact we will construct them
to satisfy the stronger incomparability, A dWaB and B & , A . That is, for each e e L(a), we want to ensure that
SAe : {elA
+B
and
A bit of notation is necessary here. form
Requirements of the B are called A-requirements, requirements of the form Se
~ft
are B-requirements. In these requirements we are identifying a set C with its characteristic function,
For any set CSa, we let
F
= a-C.
In the construction of A and
B we let 'A (Ba) = set of elements enumerated in A (B) by stage a.
=
Finally let A"
A ~ ,B = x i s some standard p a i r i n g function. I t follows from t h e d e f i n i t i o n of complexity measure and t h e
where
recurs$veness of f t h a t $ is recursive. To s e e
4 s a t i s f i e s t h e theorem, assume t h e r e e x i s t s some
index e f o r 4 and a
y
suchthat f o r a l l
x,
x > Y
oe
5f
(x)
(x)
.
L e t z be
.
y such t h a t nl
(2)
= e. Hence, f o r
this z
since
(be ( 2 ) ( f
( z ) by t h e clefinition of $.
But nl
(2)
= e, hence
and s i n c e 4e is total, t h i s i u a contradiction. I n Section 2 we w i l l prove t h e following: 1.4
Theorem (Rabin E231).
a r e c u r s i v e function.
Let O be a complexity measure and f
Then t h e r e e x i s t s a recursive 0-1 valued
function such t h a t f o r any index e f o r $
f o r a l l but a f i n i t e set of x. W e w i l l prove Theorem 1 . 4 i n a more general form (i.e.,
f
p a r t i a l recursive i n s t e a d of f r e c u r s i v e ) . One may i n t u i t i v e l y f e e l t h a t t h e s i z e of an output i n d i c a t e s the complexity of t h e computation required t o o b t a i n t h a t output. Theorem 1.5 t e l l s us t h a t we c a n n o t recursively bound t h e complexity of a computation by its output. 1.5
Theorem.
Let
be a complexity measure.
Then t h e r e does
n o t e x i s t a recursive function k such t h a t f o r each e
k ( ~ , $ ,(XI 1 ) Qe(x) f o r a l l but a f i n i t e set of x. Proof.
Suppose such a recursive k e x i s t s . Then t h e
complexity of any 0-1 valued recursive Ge must s a t i s f y
f o r a l l b u t a f i n i t e set of x. recursive. $
Clearly, k(x,O) + k ( x , l ) is
Hence, by Theorem 1.4 t h e r e is a 0-1 valued r e c u r s i v e
such t h a t f o r any index e f o r
f o r a l l but a f i n i t e set o f x.
$I
Formulas (1) and (2) y i e l d an
obvious contradiction. Although Theorem 1.5 tells us that we cannot recursively bound t h e complexity of a computation by i t s output, our next r e s u l t shows t h a t the opposite does hold.
Namely, t h a t w e may
recursively bound t h e value of a computation by its complexity. 1.6
Theorem.
L e t 4 be a complexity measure.
Then there e x i s t s
a recursive function h such t h a t f o r each e
f o r a l l b u t a f i n i t e s e t of x. Proof. -
Define t h e a u x i l i a r y function H(erxry)
H is e v i d e n t l y recursive.
t
$,(XI i f Oe(x) = Y 1 otherwise
The desired function h is defined as
h(x,y) = max H(e',x.y) e 'LX Clearly, h i s recursive.
.
To see t h e conditions o f t h e theorem a r e s a t i s f i e d suppose x
2
e and t h a t 9, (x) is defined. Thus,
Since x
2 e,
it follows t h a t
max ~ l ( e ' . x , @ ~ ( x 2 ) ) @,(XI e 'zx
I f qe (XI i s undefined, then t h e r e l a t i o n h (x,4,(x) 1 holds.
Oe ( X I s t i l l
I t then follows t h a t t h e set f o r which t h e r e l a t i o n i n
t h e theorem f a i l s i s bounded by e.
0
W e conclude t h i s s e c t i o n v i t h s e v e r a l t e c h n i c a l lemmas t h a t
w i l l be used in a l a t e r chapter.
Intuitively, in a single
complexity measure, when w e 'combine1 computations t h e complexi t y of t h e new computation i s r e c u r s i v e l y bounded by t h e complexity of t h e component computatione. 1.7
-
Lemma
( F i r s t Combining L8ma).
Let O be a complexity
measure and l e t C(e,y) be a recursive function such t h a t i f Ge(x) and $el ( x ) a r e defined ( f o r a given x ) , then s o i s @ ~ ( e , e n(x) ) a l s o defined. such t h a t
Then t h e r e i s a recursive function h
@c(e,el) ( X I 2 h (x, ae (XI ,@e ( X I
)
f o r a l l but a f i n i t e set of x. Proof.
Define t h e intermediate function
(XI i f p ( e , e l ,x,u,v)
ee (XI=
u and
eel (XI
= v
= otherwise
p is evidently recursive. h(x,u,v) = h i s a l s o recursive. tions, let x
Define t h e function h by max p(;,;* {Z,5'~1
,x,u,v)
To s e e t h a t h s a t i s f i e s t h e desired condi-
2 e,e8. From t h e d e f i n i t i o n of h
I f both Qe(XI and Oe ,(XI a r e defined, by hypothesis, s o is
@c(eve' (XI defined.
Fmm the d e f i n i t i o n o f p and s i n c e x
2 e,e'
Should e i t h e r Oe (x) o r Oe, (XIbe undefined, then the i n e q u a l i t y
holds independently of whether o r not
(x) i s
,eS)
defined.
It follows t h a t t h e set of a l l x f o r which the i n e q u a l i t y fails
i s bounded by max Ce,e'l.
D
Another form involves a C w h i c h is a function o f a s i n g l e argument. 1.8
.
Lemma (Secand Combining Lemma) Let @ be a complexity
m a s u r e and l e t C(e) be a r e c u r s i v e function such t h a t i f $,(XI i s defined
then
(XI i s defined.
Then t h e r e i s a recur-
s i v e h such t h a t
f o r a l l but a f i n i t e set of x. Proof. -
Define t h e function p(e,x,y) =
p is evidently
@C(e) (XI i f 4,
(XI = y
otherwise
recursive, Define the function h a s
h(x,y) = max p(e',x,y) Eev(xl h i s r e c u r s i v e and f o r any x 2 e,
(whethel o r n o t < @C(e) (XI 1
0(,
(XI i s defined).
i s bounded by e.
Hence, the set {xi h (x,Be (x) )
D
I t should be clear t h a t Lemmas 1.7 and 1.8 can be general-
i z e d t o any f i n i t e mnber o f arguments.
W e prove a last v e r s i o n
which allows us t o e l i m i n a t e t h e f i r s t argument of t h e bounding function. 1.9
Lemma
(Third Combining Lemma).
Suppose t h e same hypothesis
a s Lemma 1.8 holds and i n a d d i t i o n (Pe(x) 2 x f o r a l l x. Then t h e r e i s a r e c u r s i v e f u n c t i o n h such t h a t
(XI 5 h (Oe (XI
BC
f o r a l l b u t a f i n i t e set of x. Proof.
Define th@ function p(e,x,y) =
p is e v i d e n t l y r e c u r s i v e . h(y) = h i s a l s o recursive.
2
x
(XI i f Qe(x) = y otherwise
Define t h e f u n c t i o n h by max
p(el,x*,y)
{e*' , x g ~ . )
For x 2 e
h(Qe(x)) = Since Oe(x)
@C(e)
max
( e ' ,x'(ee(x) 1
~(e',x*,@~(x)
e,
max p ( e ' ,x* ,Be (XI 1 { e *,xm5Qe(x) 1 It follows t h a t t h e set Ex lh (ae (XI )
< OC
(XI
(XI 1 i s bounded
by e. 2.
Blum-Rabin Compression and Speed-Up Theorems In
t h i s s e c t i o n we prove r e s u l t s on t h e complexity of compu-
t a t i o n s t h a t a r e b a s i c a l l y due t o M. Blum [ l l
.
The first r e s u l t ,
c a l l e d t h e Blum Theorem, t e l l s us t h a t i n any complexity measure t h e r e a r e a r b i t r a r i l y complex p a r t i a l r e c u r s i v e functions, i - e . ,
than a
functions such t h a t any way of computing them takes more predetermined p a r t i a l recursive g
almost everywhere- A s a
special case we prove Theorem 1.4 of Section 1. The second major r e s u l t is t h e Blum
Compression Theorem
This e s s e n t i a l l y tells how lneasured'sets
functions and
a r b i t r a r y step-counting functions a r e interweaved; i n p a r t i c u l a r , when t h e measured funotions.
collection is a sequence of step-counting
We conclude Section 2 with a proof of B l u m ' s well
known Speed-up Theorem. W e proved i n Theorem 1.3 t h a t f o r a r b i t r a r y complexity
measure 4 and recursive function
there always e x i s t s some
f
0-1 valued + s u c h t h a t any way of computing
f ( x ) s t e p s o r an inEinite set of x.
+
takes more
than
W e next prove t h a t such an
inequality holds on not j u s t an i n f i n i t e s e t but for a l l but a finite
set.
This r e s u l t is further generalized from recursive
t o p a r t i a l recursive functions. 2.1
Theorem ( B l u m i l l ) .
L e t 4 be a complexity measure and g a
p a r t i a l recursive function.
Then there corresponds t o g a 0 - 1
valued p a r t i a l recursive function f , with t h e same domain a s g, such t h a t i f e i s any index f o r f then cOe (XI 2 g ( x )
f o r a l l but a f i n i t e set of x. Assuming t h a t we've proven Theorem 2.1 we have 2.2 g
Corollary (Rabin 12311. be a recursive function.
Let 4 be a complexity measure and Then there corresponds t o g a 0-1
valued recursive function f such t h a t for any index e f o r f
f o r a l l but a f i n i t e set o f x. Proof.
L e t g be recursive.
By Theorem 2.1 t h e r e e x i s t s a
p a r t i a l recursive f such t h a t f o r any index e f o r f t h e set { x ~ @ ~ E F. W e next s show t h a t f s a t i s f i e s a l l requirements of t h e theorem. Let
F
3
U
w e see f i s w e l l defined,
By examining t h e construction
dom(f) = dom(g) and has range { O , I ) .
By t h e e f f e c t i v e nature of
t h e construction, f i s p a r t i a l recursive. Remark.
A value f ( k ) f o r t h e function defined above i s
dependent on input k and on t h e index R of g. Hence, it i s a p a r t i a l recursive function of two arguments, m(a,k). follows from t h e s 3 t h e o r e m
that
OS (El (k) = m(E,k)
It
f o r some
W e w i l l use t h i s result l a t e r on i n t h i s s e c t i o n .
r e c u r s i v e s.
W e complete t h e proof of t h e theorem by proving t h e follow-
ing* Claim. ix
I
L e t e be any index for f .
Then t h e set
Oe(x) < g ( x ) I is bounded. Proof. -
Assume the contrary.
That i e , f o r some index e f o r Then t h e r e w i l l be a
f t h e set {xlOe(x) :g ( x ) 3 is i n f i n i t e .
s t a g e so p a s t which no index s m a l l e r t h a n e w i l l e v e r be cancelled.
Since {xi Oe (XI< g (x) 1 is i n f i n i t e , it must have a
member x ' > so
. Since x'
is i n t h e s e t , @,(x1), hence Oa(x') i s
defined. Furthermore, by a b a s i c property of p a i r i n g f u n c t i o n s =
S1
<XI
,aQ (x') > 2
X'
>
So
F r m t h e d e t a i l s of t h e c o n s t r u c t i o n e i t h e r (a)
A t s t a g e sl
(b)
A t s t a g e 2,
,e
i s added t o A
or
s2 < sl, e is added t o A.
In e i t h e r case, t h e r e e x i s t s a stage added t o A. However,
By t h e c o n s t r u c t i o n , , $,(XI . Hence, t h e s e t i s bounded by e.
0
Returning t o t h e p r o o f , we d e f i n e t h e f u n c t i o n R(y) = max { h ( y ) , h ' ( y ) l .
A3
i n t h e proof o f
Theorem 3.3,
Our n e x t r e s u l t i s an a s s e r t i o n t h a t f o r any complexity measure Q
t h e step-counting f u n c t i o n s (9,)
t o t h e recursive functions.
a r e sparse r e l a t i v e
I n terms of t h e previous r e s u l t s ,
Borodin's Gap Theorem t e l l s u s t h a t t h e r e i s no r e c u r s i v e r, where r ( x ) > x , t h a t w i l l expand every p o s s i b l e complexity c l a s s .
The proof here is that of Young I301. Theorem (Borodin [31).
3.6
In any complexity measure Q for any
recursive function r with r(x)
2 x, for all x, there exists a
recursive monotonically increasing function t such that
Proof.
Let {Qel be the enumeration of the step counting
functions of measure Q.
We define function t by
t(0) = 0 t(x) = max[t(s)+m(s,x,t(s))
Is
r (y+m), otherwise .) Hence,
m(s,x,y) is recursive.
It follows from
its definition that t
is also recursive (and monotonically increasing). We neft show Ct = Crot 4,
E Crot
. Then we
when x > xo.
. Since r(x) 2 x, Ct 5 Grot. Suppose
know that for some xo > e, Qe (x) 5 rot (x)
But
= max[t(s)
+
min V {(Qe(x) Lr(t(s)+m)) m ezs + (oe(x) 5 t(s) + m) 1 I s < XI.
Thus, f o r x > x, > e , t ( x )
2 Qe(x) making
+e
E
Ct.
Hence,
We next show t h a t f o r any sequence of increasing recursive functions, t h e r e is a complexity c l a s s comprised of t h e union of t h e c l a s s e s bounded by functions from t h e sequence. 3.7
Theorem (McCreight-Meyer 1191
measure.
I
Suppose {fe
of recursive functions
e < w)
.
L e t Q be a complexity
i s a recursively enumerable s e t
such t h a t f o r each e , e l , x
f o r e > el. Then t h e r e is a
recursive function k ( x ) such t h a t
Observe t h a t t h e function k (x) = f x ( x ) i s g r e a t e r than f e ( x ) f o r each e klx)
on a l l but a f i n i t e set of x.
.
For when x
2 e,
2 f e (x) Hence, f o r each e t h e s e t {XIk (x) < f e (x)
f i n i t e and we concLude t h a t
U
is
C
Ck. e fe However, t h e r e may e x i s t some +I f o r which (1) Q I ( x ) 5 k ( x )
f o r a l l b u t a f i n i t e set of x, and yet ( 2 ) GQ(x) > f e ( x ) f o r each e on an unbounded s e t of x.
This would cause + I t o be i n Ck but
, thus, implying only U C C Ck. fe e 'e The way we avoid t h i s d i f f i c u l t y i s t o "guess" f o r each
not i n t h e union of t h e C
and some e t h a t f e ( x )
1 Q E ( x )f o r a l l but
a f i n i t e s e t of x. I f
w e discover t h a t f o r some x, f e ( x ) < Q I ( x ) , w e t r y t o assign a
value t o k ( x ) which i s l e s s than Q Q ( x ) . We a l s o "guessn again t h a t f o r a l a r g e r e ' ( e ' > e l t h a t f e , (x) ) Q Q ( x )f o r a l l but a f i n i t e set.
If
, then t h e r e w i l l be an e fe (x) f o r a l l but a f i n i t e s e t . However,
QI i s a c t u a l l y i n
eo such t h a t @ & ( X I
f
0
it w i l l a l s o be t h e case t h a t f
eo
u C
(x) 5 k ( x ) f o r a l l but a f i n i t e
set.
Hence
union o f t h e C set.
@a B
Ck.
E
i s not i n t h e
On t h e o t h e r hand, i f
t h e n f o r some e , f e (x) < Q R (x) f o r an i n f i n i t e
e
( x ) f o r an i n f i n i t e set and hence,
This w i l l f o r c e k (x) < Ck-
A r e c u r s i v e f u n c t i o n k w i l l be d e f i n e d i n terms o f
Proof.
The c o n s t r u c t i o n w i l l compute k(m)
a c o n s t r u c t i o n g i v e n below.
Throughout t h e c o n s t r u c t i o n we w i l l be accumulating
f o r any m.
t h r e e s e t s K , I, and TO The s e t KS constructed.
A
,at
d e f i n e d a s follows.
s t a g e s,
represents t h e function k(x) being
p a i r <x,y> i s placed i n KX a t some s t a g e x i f and
o n l y i f k ( x ) = y.
The set IS, a t s t a g e s , r e p r e s e n t s encodings
o f p a i r s such t h a t we have made t h e guess f k ( x ) f o r a l l but a f i n i t e s e t .
A t stage s
no more t h a n 21s guesses.
The s e t
QR(x)
t h e r e w i l l be an e x i s t e n c e
TO^
a t s t a g e s, c o n s i s t s o f
encodings of p a i r s f o r i n c o r r e c t -"guesses". S e t s K < ~ , I < ~ and
TO 4t ( X I .
The index e i s l a t e r popped when an assignment
t o 4 g ( t ) i s made making Q e ( x ) > $ point
( x ) f o r some x. A t t h i s g (t) index e once a g a i n becomes u n p o p p a b t e . The s e t TO', at
s t a g e s , a c t s os a c o l l e c t i o n of t r i p l e s coming from Q" which have been d i s c a r d e d . < v , e , r > is e j e c t e d from QS The s e t
TP',
,
s'c s ,
Whenever a t r i p l e o f t h e form it i s done s o by p l a c i n g it i n t o TO".
a t s t a g e s , r e p r e s e n t s t h e f u n c t i o n $g ( t ) b e i n g
constructed.
A p a i r <x,y> i s placed i n t o TpS
only i f
4
( x ) = y.
and TP',
r e s p e c t i v e l y , j u s t p r i o r t o s t a g e s.
g (t)
Sets
QCS
, TO"
a t s t a g e s i f and
and TP E TO' + pel ( b )
-
-
ml
&
Qe(b) > m
I.
That is, e i s t h e poppable index of h i g h e s t p r i o r i t y , l e s s t h a n o r e q u a l t o s, such t h a t at2 unpoppable i n d i c e s o f h i g h e r p r i o r i t y t a k e no more t h a n m s t e p s on i n p u t b.
F u r t h e r , index e ,
i t s e l f , t a k e s more t h a n m s t e p s on i n p u t b. If v = 0
t h e n set TpS = T P < ~and go t o s t a g e s + l . Here, no
such index e x i s t s . Otherwise, s e t
-
TO' & a e ( b ) > m e = rnin { ~ v , e , l >E e & v v ' < v[ ( < v 1 , e v , ~E> as TO') * Qel(bl 5 m11
-
-
e i s t h a t index w i t h p r i o r i t y v
causing a change t o be made.
I f t h e r e i s more t h a n one such e l i g i b l e index, e i s t h e s m a l l e s t
one. Set TpS = TP E Q'
-
such t h a t f o r +g ( t ) TO" f o r a l l s > so.
Of
Index e i s s t a b l e a t O ( 1 ) i f t h e r above i s t h a t e becomes s t a b i l i z e d a f t e r so
o(1).
We a l s o s a y
.
An index e i s s a i d t o be u n s t a b l e i f it is n o t s t a b l e . That
is, e i s unstable i f f o r a l l stages s there e x i s t s a stage
s' > s
such t h a t f o r g , g l and r , r l E {0,11,
and where
g # g l o r r # r t . For a f i x e d e an e - t r i p l e i s a t r i p l e o f t h e form < g , e , r >
f o r some g and r E I 0 , l ) .
I f index e i s
Claim.
3.9.4
s t a b l e ( a t 0 o r 1) t h e n t h e r e
e x i s t s a s t a g e so such t h a t a f t e r s t a g e so
,
(1) e has a l r e a d y become s t a b i l i z e d ( a t 0 o r 1 1 , and
a l l i n d i c e s w i t h h i g h e r p r i o r i t y than e , and which
(2)
s t a b i l i z e , have done s o a l r e a d y . Proof.
Immediate.
Lemma.
3.9.5
0
Then f o r any index e i f e is
Let $t be t o t a l .
u n s t a b l e t h e n t h e s e t s A = {XIa e ( x ) > $t (x) ) and are infinite.
A' = { ~ l @ ~ (> x$)g l ( t ) (x) 1
Suppose e were u n s t a b l e . Then by t h e d e t a i l s of
Proof.
t h e construction t h e s e t s B = {S
I
an e - t u p l e goes from an unpoppable t o a poppable s t a t e
a t s t a g e sl, and B1=
is
I
an e - t u p l e g e t s popped a t s t a g e s ) ,
a r e both i n f i n i t e . tion.
Observe
then a t stage
Suppose
A
were f i n i t e t o o b t a i n a c o n t r a d i c -
5 A.
s e t C = {XIS = < x , Q t ( x ) > E B}
For i f x E C ,
s = < x , a t ( x ) > an e - t u p l e goes from an unpoppable
t o a poppable s t a t e . By t h e d e t a i l s o f t h e c o n s t r u c t i o n , t h i s o c c u r s o n l y i f a e ( x ) > $ t ( ~ ) . Hence, x E A. Thus, s i n c e C
5
A,
and A i s f i n i t e , C is f i n i t e . Define t h e map f ( x ) = < x , a t ( x ) > t o s e e B
s E B, t h e n a t s t a g e
C
f [C].
For i f
s = < x , a t ( x ) > , an e - t u p l e goes from an
unp ppable t o a poppable s t a t e . C l e a r l y , x
E
C and s = f ( x ) E f [ C ] .
However, s i n c e C is f i n i t e , f [ C ] i s f i n i t e , hence B must a l s o be f i n i t e
c o n t r a d i c t i n g our i n i t i a l observation.
For t h e second h a l f suppose A ' were bounded t o d e r i v e a contradiction.
Define t h e mapping k ( x ) = < x , $ J ~ (( X ~ I)> t o s e e
B 1 C k[A1]. For suppose
s
E
B1.
Then a t s t a g e s an e - t u p l e
g e t s popped.
By t h e c o n s t r u c t i o n s = < ~ , c p ~ (( x~) > ) .
t h a t is, Oe ( x ) > (g, ( t ) (x)
i f we show x E A ' .
.
We a r e done
By t h e construc-
tion Oe
(*
( X I > 9g(t) (xt
.
By d e f i n i t i o n +g, ( t ) (x) = min { @ g ( t )(x) , Q t ( x )+Ot (x) 1, and t h u s
(**I
+g(t)
(XI
@g'( t )( X I
'
Combining ( * ) and (**I w e have v e r i f i e d t h a t B' C k [A']. A s i n t h e f i r s t h a l f t h e assumption t h a t A '
i s bounded
t o g e t h e r w i t h t h e f a c t B1 C k [ A 8 ] l e a d s t o t h e conclusion B' i s finite. This ends t h e proof of Lemma 3.9.5. Lemma.
3.9.6
Let
0
Then f o r a l l i n d i c e s e , i f e is
be t o t a l .
s t a b l e a t 0 t h e n t h e s e t s A = { x l o e ( x ) > I $ ~ ( x ) and ) A'
= {xloe(x) >
Proof. C = {s
2
(g8
(t)
(x) 1 a r e f i n i t e .
Consider t h e set
e l d u r i n g s t a g e s , e i s poppable}
U
{sls ( e l .
Since e i s s t a b l e t h e s e t C i s f i n i t e , hence, s o i s nl[Cl. To s e e t h a t A f nl[Cl must be a s t a g e
x E A. Then $ t ( ~ )i s d e f i n e d and t h e r e
so = < x , O t ( x ) > . I f so
x = a l ( s O ) E al[Cl. construction
, let
2e
I f so < e , so E C, and hence, then by t h e d e t a i l s of t h e
s i n c e it is d i s c o v e r e d a t s t a g e so t h a t Q , ( x ) > @ ~ ( x ) ,
t h e index e must be poppable a t s t a g e so.
.
5
Since A x = a 1 ( S 0) E n1 [C] proved t h e f i r s t h a l f . Consider, n e x t
Hence,
nl [C] and al [Cl i s f i n i t e , we
t h e set
F = Ex
I
<x,O> > s o l
where so i s t h e s t a g e o f C l a i m 3.9.4.
Since
is increasing
i n t h e second argument a l l t h e x ' s i n F appear i n t h e c o n s t r u c t i o n of $
f o r t h e f i r s t time a f t e r s t a g e so.
g (tl
Claim.
3.9.7
Proof.
Tf x
€
F
then Q e ( x )
L $g '
(t)
'
is g' (t) t h e r e a r e two subcases.
S i n c e t h e value of $J
min {$Jg(t)( x ) '41.t ( x ) , Q t ( x ) I
.
Subcase (1)
$ g l ( t ) (XI i s $Jg( t ) (XI
.
From t h e d e t a i l s of
(x) i s made a f t e r g(t) it is done by popping an index of tower p r i o r i t y than
t h e c o n s t r u c t i o n , i f an assignment t o $ s t a g e so
,
e . F u r t h e r , from t h e d e t a i l s
of t h e c o n s t r u c t i o n , t h e assignment
i s made t o $g ( t ) ( x ) such t h a t $g ( t ) ( x ) ) Oe (x) result
.
Hence, t h e
follows.
Subcase ( 2 ) . t h a t $t ( x ) < Oe ( x )
.
+
+ g , ( t ) ( x ) i s +t ( x )
Qe(x).
Ot ( x )
+ OT (x)
Suppose
.
We have t o show
i n s t e a d t h a t $ t ( x ) + Q t( x )
Then +t ( x ) < Oe ( x ) and from t h e d e t a i l s of t h e constru-
construction
e would be poppabZe a t s t a g e
< x , Q t ( x ) > . This,
however, c o n t r a d i c t s t h e assumption t h a t e i s s t a b l e a t 0. From Claim 3.9.7
- (xl<x,O> 2 s o ) . c t h e former, A'
it follows t h a t
XI$^,(^) ( X I
< Oe(x) 1
But t h e l a t e r s e t i s f i n i t e , hence, s o i s
.
This completes t h e proof of Lemma 3.9.6. Lemma.
3.9.8
L e t +t be t o t a l .
0
Then i f e i s s t a b l e a t 1 t h e n
t h e sets A = {XIee ( X I > $t ( x ) 1 and A ' =
{xi Ot ( x ) > $g , ( t ) (XI 1 a r e
finite. Proof.
Consider any x having t h e p r o p e r t y t h a t <x,O> > so,
where so i s t h e s t a g e of Claim 3.9.4.
Since
is i n c r e a s i n g
i n t h e second argurrlent, it f o l l o w s t h a t p r i o r t o s t a g e "<x,O>" no mention of i n p u t x h a s e v e r been made i n t h e c o n s t r u c t i o n .
3.9.9
Claim.
I f x i s such t h a t <x,O> > so t h e n Q e ( x )
5 $t(x)
and Q e ( x )
5
$g 1 ( t )
'
Once we've proven Claim 3.9.9 A , A 1 C_ {xl<x,O> f s o ) ,
hence A,A1 a r e f i n i t e
Before showing Claim 3.9.9 3.9.10
it f o l l o w s t h a t
we i n t r o d u c e
For any x s u c h t h a t <x,O> > s
Claim.
,,
Qe
1
(XI
I$t ( x ) ,4g1 ( t ) ( X I ) f o r a l l e ' having higher p r i o r i t y t h a n
-< min
e and such t h a t e ' i s unpoppable. Proof.
S i n c e e n h a s h i g h e r p r i o r i t y t h a n e , e ' must be
s t a b l e a t 0.
From t h e d e t a i l s o f t h e proof o f Claim 3.9.7,
<x,O> > so then Oe, (x) ( Q g 1( t ) ( X I
.
since
It must a l s o be t h a t
Q e , ( x ) f $It(x) f o r o t h e r w i s e , e ' would be poppable c o n t r a r y t o o u r assumption.
Hence, 6,'
( x ) f min {$t ( x ) ,$g,
(x)) for a l l x
such t h a t <x,O> > so. We r e t u r n t o t h e proof o f Claim 3.9.9. where <x,O> > so.
Let m = max { Q e , ( x )l e '
Consider x fixed has h i g h e r p r i o r i t y
than e and e ' i s unpoppable a f t e r s t a g e sol.
m < min {$It(x) ,$Ig,
(*I
From Claim 3.9.10
(XI)
I f we can show
(**I
(Pe
we a r e done.
For by ( * ) and Qe(x)
5
(XI ( m
(**I,
min C$t ( x ) ,$g, ( t ) (XI 1
which v e r i f i e s Claim 3.9.9. 3.9.11
Claim.
A t s t a g e <x,m> is t h e earliest s t a g e a t which
$g (t) (x) c o u l d be d e f i n e d . Proof.
It i s c l e a r t h a t $ g ( t ) ( x ) cannot b e d e f i n e d b e f o r e
s t a g e <x,O>, s i n c e x never a p p e a r s f o r c o n s i d e r a t i o n before. Suppose + g ( t ) ( x ) was d e f i n e d a t s t a g e < x , y > , y < m. Then it must have been t h a t s i n c e m = Q e , ( x ) (e' having h i g h e r
p r i o r i t y than
e)
that m =
Qe,(k) > y = $Si(t) ( X I .
But t h i s c a n n o t o c c u r by
t h e d e t a i l s o f t h e c o n s t r u c t i o n . Hence, Claim 3.9.11 A l l t h a t remains is t o v e r i f y ( * * ) .
i s proved.D
C e r t a i n l y , Qe,( x ) 5 m
f o r a l l e ' having h i g h e r p r i o r i t y t h a n e and unpoppable. By Claim 3.9.11
a t s t a g e <x,m> i n d e x e would b e popped.
But t h i s
c o n t r a d i c t s t h e s t a b i l i t y o f e a f t e r s t a g e so. T h i s completes t h e proof o f Lemma 3.9.8. The proof o f Claim 3.9.3
D
f o l l o w s from Lemmas 3.9.5,
3.9.6
F o r a l l t h r e e imply t h a t f o r $t t o t a l and e any
and 3.9.8. index,
{x
I
@ e ( ~> ) $t
(XI 1
i s f i n i t e i f and o n l y i f (X
1
Qe( x ) > $g, ( t ) (XI 1
is f i n i t e .
4.
Speedable and
Nonspeedable S e t s
I n t h i s s e c t i o n w e examine some c l o s e r e l a t i o n s h i p s between a b s t r a c t complexity a n d p u r e
recursion theory.
In particular,
we c o n s i d e r c o n n e c t i o n s between s p e c i a l t y p e s o f sets c a l l e d speedable and nonspeedable and classes o f r e c u r s i v e l y enumerable
(r.e.) sets.
The f i r s t theorem i s S o a r e ' s 1281 p u r e r e c u r s i o n
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 nonspeedable sets. Namely, s e t A is nonspeedable p r e c i s e l y when t h e c o l 1 e c t i o n o f r . e .
sets h a v i n g nonempty i n t e r s e c t i o n w i t h t o 0'.
IIJe c a l l any
i s Turing r e d u c i b l e
s a t i s f y i n g t h e l a t t e r c o n d i t i o n semilow.
A second theorem t e l l s
i n e v e r y r.e. T u r i n g d e g r e e . :larques,
indices of
u s t h a t t h e r e are nonspeedable sets Althouqh t h i s was f i r s t shown by
t h e proof g i v e n h e r e
is t h a t o f Soare.
h i s s e m i - l o w n e s s c r i t e r i o n makes t h e proof s i m p l e r . )
( H i s u s e of
We nex*
c h a r a c t e r i z e t h e s e r.e. Turing degrees t h a t c o n t a i n speedable sets.
Namely, an r.e. d e g r e e h a s a speedable s e t i f and o n l y i f
its Turing jump is above
9'.
We conclude t h e s e c t i o n w i t h a
d i s c u s s i o n o f t h e s t r u c t u r e o f t h e speedable s e t s and t h e i r r e l a t i o n t o w e l l - s t u d i e d c l a s s e s of r . e .
sets.
W e assume t h e r e a d e r is f a m i l i a r with t h e b a s i c n o t i o n s o f
o r d i n a r y and r e l a t i v i z e d r e c u r s i o n theory.
In particular, the
concepts o f r e c u r s i v e l y enumerable i n a set, Turing degree, jump of a s e t and a degree, O w , Turing r e d u c i b i l i t y
(zT),m-reducibil-
i t y (2nd and m-completeness a r e assumed. For d e t a i l s we r e f e r t h e r e a d e r t o Roqers [ 2 4 1
.
From now on {Re) and (
B ~ denote ~ 3 some c a n o n i c a l enumeration
of t h e r e c u r s i v e l y enumerable
sets and t h e s e t s r e c u r s i v e l y
enumerable i n B, r e s p e c t i v e l y . 4.1
Definition.(Blum and Marques 1 2 1 ) .
speedable i f f o r a l l e such t h a t Re =
A
An r.e.
s e t A is
and f o r a l l r e c u r s i v e
f u n c t i o n s h, t h e r e e x i s t s e' such t h a t Re,= A and
{XI
Qe(x) > h (x,Qe, ( x ) ) 1
is infinite.
I n t u i t i v e l y , i f A is s p e e d a b l e then f o r e v e r y a l g o r i t h m " e n f o r A and every r e c u r s i v e h t h e r e i s a n o t h e r a l g o r i t h m " e n " f o r A which i s a n h speed-up o f t h e f i r s t ( i . e . , i n f i n i t e l y often) 4.2
Definition.
n o t speedable.
Qe(x) > h (x, G e l (x)
. An r . e .
set A i s c a l l e d nonspeedabte i f it i s
More p r e c i s e l y , r . e .
s e t A i s nonspeedabte i f and
only i f t h e r e e x i s t s e such t h a t Re = A and a r e c u r s i v e f u n c t i o n h such t h a t f o r a l l e 1 where Re,
= A
,
x E A * (De(x) 5 h ( x , Q e , ( x ) ) f o r a l l b u t a f i n i t e set.
E s s e n t i a l l y a n r.e.
set A
i s nonspeedable i f t h e r e e x i s t s a
r e c u r s i v e h and an i n d e x e f o r A s u c h t h a t e v e r y a l g o r i t h m e' f o r A
c a n n o t be a n h speedup o f e. Given a set A t h e weak j u m p of A i s t h e set
Definition. HA =
el^^ " A
A set
# g).
A i s c a l l e d semi-Zou i f i t s weak
jump i s Turing r e d u c i b l e t o 0' (HA
L~ 0 '
.
S e v e r a l o f o u r r e s u l t s r e a u i r e t h e w e l l known l i m i t theorem of Schoenfield [261. 4.1
L i m i t Lemma.
Given S r w , S
L~ 0' i f and o n l y i f t h e r e i s
a r e c u r s i v e p r e d i c a t e S (e,s) such t h a t S ( e ) = l i m S ( e , s ) e x i s t s S
f o r a l l e. Proof (*)
. Suppose
and e q u a l s S ( e ) . L e t
S ( e , s ) i s r e c u r s i v e and l i m S ( e n s )e x i s t s
A,Z
S
b e d e f i n e d by
-
is r . e .
Since
t h e r e is a r e c u r s i v e f s u c h
and 0' i s m-complete
t h a t < e , t >E A * f ( < e , t > ) 9 0'. Hence,
g(e) *
and
Thus, S
. Then
3 t < e , t >E A
&
S(e,t)
2T 0 ' . Assume S f T 0' and t h a t S ( e ) = Cell
S ( e ) * 3 n 3m < e , l , n , m >
E
Re
&
1
Kn
5
0'
( e l f o r some el.
- 5'
0' & Km C
& K~ c - O' & K~ 5 5' el (By t h e d e f i n i t i o n o f r e c u r s i v e i n 0 ' where {Ki) i s a n enumera-
5(e) * 3 n 3m <e,O,n,m>
E R
t i o n of the f i n i t e subsets of w ) . 1
r e c u r s i v e p r e d i c a t e Os ( e ) where
S i n c e 0' i s r.e. 0' =
l i m 0;.
there is a
Define
i
1 if 3h.m < s <e,l,n,n> RS & KnC_ 0; S(e,s) I el 0 otherwise Clearly S(e,s) is recursive and from S = tell0',
&
,K
-
5 0;
lim S(e,s) S
0
exists and equals S(e).
We next prove Soare's recursion theoreticcharacterization of nonspeedability. Namely, nonspeedable sets are those r.e. sets whose complements are semi-low. Theorem (Soare [ 2 8 1 ) .
4.2
A recursively enumerable set A is
nonspeedable if and only if Proof
el^, "
# 81 $ 0'.
Let A be nonspeedable and let el and h satisfy
(*):
= Re
Define recursive function f by Rf
Definition 4.2. and Hz(e,s) by 1 HA(e,s) = 0
i
if (3X) [X E R:-R~
A
(XI > h (x,@f (el(x)
and 0 1
U
1
otherwise
We show that Hx(e) = lim HX(e,s) s is the characteristic function of Hz. By the Limit Lemma 4.1 HX(e,s) is clearly recursive.
HJi LT 0'. Suppose x
€
Re n
z (i.e.
e E Hz). Then x E :R
- 'A
= R:-RS
el Thus, since Re = A, 0 (x) diverges and since 1 el Rf(e) , h(@f(e) (x),x) is defined. Hence, HJi(e,s) = 1, s >so. Suppose Re n = (i.e., e 9 HA) then Re C_ A and hence,
for all s > so. Re
5
Rf (el
= Re U A = A. By the nonspeedability of A, since A = Rf(e),
is bounded by some xo. Hence, there will be some stage so such s S (= A ) for all s > so. Hence, (A Q x0)n(R:-$ ) 1 that A x0 Rel = $3 and thus HE(e,s) = 0 for all s > so.
(-1
Assume
is semilow,
By the Limit Lemma 4.1
that is, Hz= (elRen
# 91
there are Hji(e,s) and HE(e) where
zT 0'.
.
Hx (e) = lim Hx (e,s) s G(x,y,e) as follows: (1) If Qe(x) # y
Let el be such that Re = A and define 1
set
G(x,y,e) = 0,
t = min [Qe (x) = s or Hji(e,s) = 11 s2x 1 (a) If Qe (XI = t set G(x,y,e) = t let
(2) If Qe(x) = y 1
A
(b) Otherwise,
set
G(x,y,e)=O.
Observe that t exists in (2) when Qe (x) = y. Re = A or else x 1 for all but a finite set of s.
hence, either x
E
E
5 =
Let x
jB, HJi(e,s) = 0 for s > so.
2
,
x) to see that el and h
E
A where
h(Qe(x) ,XI.
x > maxis0,e,Qe (x)1, to see that Q (x) 1 el h (x,Qe(x) = max {G(x,Q~(x),e I e XI. Since x > e,
Re
For suppose Re = A. Then
witness the nonspeedability of A. since Re n
5
E
and then Hx(e,s) = 1
Re n
Define h(x,y) = max {G(x,y,el) le'
For then x
For then
G (x,Qe(x),el.
But since x > so, by the definition of G,
(XI. 0 el An easy method of demonstrating nonspeedability is given = @
in the following: 4.3
Corollary (Soare [281).
able it suffices for all e
to provide enumeration
set A to be nonspeed-
{ ~ ~ lof ~A such < ~ that
there are infinite sets of s such that
implies Re n Proof.
For an r.e.
RE-A~#
jB
x # 9. Define
RE-A~# 9
1
if
0
otherwise
Hx(e,s) = We will see that Hx(e) = lim Hx(e,s) exists for all e. Hence, s by Theorem 4.2,and Limit Lemma 4.1, A is nonspeedable. Suppose e
E
Hz
then there is an x E Re n
x.
Hence, for all
s > so. x E R:
n
6'
and t h u s Hi(e,s)
HJi(e,s)=L Then by h y p o t h e s i s
= 1.
Suppose f o r a l l s > so,
# $3 and t h u s e E Hz.
Re n
0
The n e x t r e s u l t was f i r s t shown by Marques i n [17]. I n 1281
Soare e x p l o i t s t h i s semi-lowness c r i t e r i o n t o o b t a i n a s h o r t e r proof. 4.4
Theorem (Soare I281, Marques 1171).
s e t s i n e v e r y r.e. Proof.
There a r e nonspeedable
Turing degree.
r.e. set i n degree a and l e t BS be some
L e t B be an
r e c u r s i v e enumeration o f B. Let IFn) be some c a n o n i c a l sequence of d i s j o i n t f i n i t e sets such t h a t (1)
IF,^
= n + 1 n # m, and
(2)
Fn n Fm = $3,
(3)
t h e f i n i t e set Fn i s r e c u r s i v e l y o b t a i n a b l e from
n.
We d e f i n e A by a c o n s t r u c t i o n . Stage 0.
S e t A'
Stage s + l :
=
0.
Define
min[*R:-As] e
I f n E B
set
A'+'
l e t xn
B',
-
u {xn).
= A'
i f such x e x i s t s otherwise
min Ix
E
Pn-
U
{dzle < n l l .
X
End o f c o n s t r u c t i o n . 4.41
Claim.
Proof.
s > so
,if
de = l i m d:
s For any e
n E B'+'-B~
construction
,s
e x i s t s f o r a l l e.
t h e r e i s a s t a g e so such t h a t f o r a l l t h e n n > e.
By t h e d e t a i l s of t h e
> so cannot be added i n t o AS.
Hence t h e
o n l y way i t can be changed is by being r e p l a c e d by
a smaller
d:
member of Re.
I t f o l l o w s t h a t a f t e r some s t a g e sl, dg is s t a b l e .
4.4.2
A i s nonspeedable.
Claim. Proof.
Suppose t h a t f o r an i n f i n i t e s e t o f s R:-As
# 0.
By the details of the construction it must be that Ren Hence, A's 4.4.3
Claim.
&
A
$3,
nonspeedability follows from Corollary 4.3. n E B * Fn n
A
# $3.
By this claim and the fact that B
g
and
A
IT B.
IF
fl
n
AI
( 1 we have
Hence, A and B are in the same Turing degree.
Proof (of Claim 4.4.3).
(*)
If n
E
B, let n E B'+~-B
for
some s~+l.Since IFn[ = n+l it follows that some xnE Fn is added to A at stage s+l. Hence, Fn n A # 8. ('1
then n
Since the {I?,) E
are mutually disjoint, if x
E
Fn n A
B.
This ends the proof of Theorem 4.3. Definition.
An r.e. set A is h i g h
if A' ET 0' and Zow if
-
A' ET 0'. An r.e. degree a is h i g h (Zoo) if it contains a high (low) set.
An r.e. set which is
4.5 Lemma. Proof.
low is semi-low. Let B be any set and let HB= el^,B # $3). We will
show that (1)
H~
H~
where
&,
and
H~
(2)
&"
B1
denotes m-reducible. From these it follows that HB
and that if B' ( T 0' then HB For (1) e
E
HB * Re n B # $3
For (2) e
E
nB * :R
&"
B'
L~ 0'. * :R (,) #
for some recursive f, *£(el H ~ . # $3 * 3m,n,x <x8nIm>ERe& Kn C - B & Km 5 !ij
& K~ c B & K~ 5 i where * 3m,n E R r (el R~ { 13x <x,m,n> E Re * B' 9
f(e) E B'
Marques proved
where
f (el = -
U
that if an r.e. degree contained a speedable
set then it must be high. Soare [ 2 8 1 completed the characterization by showing that
each such degree contains a speedable set
4.6
, Soare
Theorem (Marques 117I
[28])
.
An r. e. degree a
-
-
contains a speedable set if and only if a' > 0'. Proof.
-
Suppose a is low. Then for some A
(*)
E
-
a, A is low
and by Lemma 4.5 A is semilow. By Theorem 4.2, A is nonspeedable.
*
2' > 0-'. Then choose 9). (Here ( ~ is~ some 1
Suppose
A = 1 x 1 n ~ B~ f
-
B E a and let enumeration of the finite
sets where "xn recursively encodes set Kx.) Clearly, A ET B and 0'
.
' ,eE ( B '
> is
By p r o p e r t i e s o f p a i r i n g
f u n c t i o n s ( c f . G6del [ 6 ] ) , 0 ' < a ' < a , and hence a ' > max {a0,a1,a2).
From t h e d e t a i l s o f t h e c o n s t r u c t i o n
must be c a n c e l l e d no l a t e r t h a n s t a g e a ' . f #
@E,
E'
However, t h i s i m p l i e s
c o n t r a d i c t i n g t h e hypothesis.
N e remark t h a t i n b o t h B l u m ' s and R a b i n ' s r e s u l t s (a = w )
t h e complexity o f t h e c o n s t r u c t e d f u n c t i o n exceeded t h e g i v e n
0
f u n c t i o n on a l l b u t a ' f i n i t e '
set of w.
In t h e generalization,
however, one analogue t o f i n i t e was a - f i n i t e
The reason f o r t h i s is t h a t t h e
t h e o t h e r was bounded (a-Blum). excluded set { $ I QC ( 8 ) < g ( $ )
(a-Rabin), w h i l e
a l t h o u g h bounded does n o t appear
a-recursive f o r g nontotal.
I n 1291, Stoltenberg-Hansen
proves
an analogue t o t h e B l u m Theorem where f i n i t e i s , i n f a c t , Furthermore, it i s a l s o s e e n i n [291 t h a t
r e p l a c e d by a - f i n i t e .
even when a i s n o t admissable.
an analogue h o l d s
For any a-complexity measure 0 and a - r e c u r s i v e
Definition.
f u n c t i o n s t h e a-comptexity c l a s s bounded by s i s : C
= {(,I(,
Thus C 0
is t o t a l
, or
&
@,($)
simply Cs
5 s ( $ ) f o r a l l b u t a n a - f i n i t e set
,
o f $1. when 0 i s understood, i s t h e s e t
of a l l a - r e c u r s i v e f u n c t i o n s whose complexity i s bounded by s on a l l b u t an a - f i n i t e s u b s e t o f a. We n e x t prove a g e n e r a l i z a t i o n of t h e Borodin Gap Theorem Our r e s u l t , t h e a-Gap Theorem, t e l l s us t h a t t h e r e e x i s t s no uniform way t o i n c r e a s e bounds on a-complexity c l a s s e s t h a t w i l l always y i e l d l a r g e r
classes.
r e s u l t is t h e a - s t e p
Another way of i n t e r p r e t i n g t h e
counting functions a r e sparse r e l a t i v e t o
the a-recursive functions. 5.4 a-Gap Theorem (Jacobs [ l o ] ) .
a ( $ ) and r ( B , y )
(with r ( $ , y )
L
For a l l a - r e c u r s i v e f u n c t i o n s
y)
we can f i n d an a - r e c u r s i v e
f u n c t i o n s such t h a t
(i) s ( B ) 2 a ( $ ) f o r a l l (ii)For each
E
$ c a
< a , OE($)
, and
2 ~ ( $ 1i m p l i e s
>- r(B,s(B)
f o r a l l b u t an a - f i n i t e set o f 0 . The a-Gap Theorem may also b e viewed i n terms of a-complexit classes.
5.5
Corollary.
Assume t h e h y p o t h e s i s o f t h e a-Gap Theorem.
Then w e can f i n d an a - r e c u r s i v e f u n c t i o n s s u c h t h a t (i)
~ ( $ 1> a(B) f o r a l l 8 < a , and
(ii)
'XBS(B)
Proof.
0
C ~ ~ r ( ~1 , s ( ~ )
-1 L e t t and to b e a s i n t h e p r o o f o f Theorem 5.2.
An a - r e c u r s i v e
f u n c t i o n s ( B ) w i l l be d e f i n e d i n terms o f a
c o n s t r u c t i o n below.
For each s t a g e o < a , t h e construction a s p l i t s i n t o s u b s t a g e s 6 < a , where t h e v a l u e s sB a r e computed. A f t e r enough s u b s t a g e s a r e performed, some sa i s chosen as t h e 80 value of s (a)
.
Construction. S t a g e 0.
Set s ( 0 ) = a ( 0 ) .
Stage a. S u b s t a g e 0.
(3 S e t so = a(o)
S u b s t a g e B. a = s u p t ( u , s:) + 1 set sg 6 a&:
2
rP,(a)
i s a-r.e.
@&
(01 > r ( a , s ( o ) 1 .
Cv z ~ ' l v € r n g ( t ) l .
and bounded below a*; hence, applying Fact 5.2
E& is a - f i n i t e .
is
implies
Let t ( &=) E ' and l e t E,=
Proof. E,
s(a)
Define F& = t - l [ + l
~ ( E ) I .
t
-1
-1
is a-finite.
bounded above by al. Since -1( E ' ) o > a2 i m p l i e s ta
5
i s a - f i n i t e and EE 5 dom(t
a - p a r t i a l r e c u r s i v e , E,
hence by F a c t 5.1 F,
= {a < a l t ( u )
&'
1;
s o t h a t F, i s
Choose a1 < a
r n g ( t ) , l e t a 2 be such t h a t
E
+.
5.4.4
Claim. Suppose
By d e f i n i t i o n of u1 , a 2
of a. I f OE ( a )
E'.
2 s (a)
C = {al@,(a) 2 S ( U )
hold).
6
@,(a)
(i.e.
.
2
E
< a, t h e s e t
r(a, s(a))I
where t h e i m p l i c a t i o n f a i l s t o
Since s and r a r e a-recursive,
a-recursive;
and
2 @,(a) ( r ( o , s ( a ) ) ;
we conclude t h a t f o r
i s bounded above by some a,
, ti1( E ' ) (.
by t h e d e t a i l s of t h e c o n s t r u c t i o n
( s i n c e sa = s ( u ) ) it c a n l t be t h a t s ( u ) a hence it must be t h a t r ( o , s ( a ) ) < @ , ( a ) From Lemma 5.4.3,
s ( a ) i s defined
By Lemma 5.4.1,
o > max {al,a2).
a t substage t(a) >
i s max {al,a2}.
The required a,
it follows t h a t C i s
hence, C i s a - f i n i t e .
Before v e r i f y i n g t h e l a s t p a r t we introduce preliminary 5.4.5
For a < a , t h e set H = {@,(a)It(,) ( t ( a )
Lemma.
til(t + (E))
Proof.
& @& ( a ) + )
is bounded by some
6' < a
&
.
-
F i r a < a and l e t Du = { c l < t ( u ) ~ t ( y l ( ~ ' &) +
a)+}. @ t Zll ((&
D,
i s a-r.e.
and bounded below a*; hence,
by Fact 5.2, Do is a-finite.
Define g,:
a*
+
a
by
( ~ ) i f t ~ ~ ( c ~ )( 0t) )~ ~ - ~ t, (E') otherwise g, is a-partial recursive and Do g, [D,]
5
dom(ga); thus by Fact 5.1,
bounded by some 13 '
is a-finite, and
from the observation that H = g, [Do]
.
The lemma follows
.
Finally, Proof.(of Lemma 5.4.1): defined for all o' 1s; 18 < a)
< a.
Fix a < a and assume s(oj is
By a simple induction, the sequence
is strictly increasing.
This regarded as a one-to-
one order preserving map of a iinto a, implies that rng(si) is cofinal with a.
Let 8' < a be as in Lemma 5.4.5,
8' > (OE(o) It(€) 5 t(o) of 1s;)
there is a
BO
&
til(t(€))+
< a, where s
0
6
OE(ol+). > 8'
that is, By cofinality
. Thus,
at substage
-< r(u, s" ) . Hence, by 0 0 u at most substage B0 of a, s(a) is defined; concluding s(o) = s , @a for some 8, < a B0 , for all investigated E, @€(a)
< si
.
Most of the major results of abstract computational complexity theory (as found in Hartmanis and Hopcroft [7]) have been shown in [ 9 1 to generalize (in one form or another) to a-recursion theory. Besides those presented here, these include Blum's Speed-up, Lewis-Landweber-Robertson's Nonrecursively Enumerable Complexity Class and McCreight-Meyer's Honesty and Union Theorems.
The generalization of the last result, the Union
Theorem, required a rather extensive overhaul of the w-Proof. Namely, instead of a typical cancellation argument, the generalization necessitated a cancellation atop a finite injury priority
construction.
Further, the listing provided by a* did not
suffice here.
Consequently, a shorter listing together with a
blocking strategy (cf. [27]) had to be implemented.
References [I] Blum, M.,
"A machine-independent theory of the complexity of
recursive functions," J. ACM [21
Blum, M.,
and Marques, I.,
14 (1967) pp.
"On complexity
322-336. properties of
recursively enumerable sets," J. Symbolic Logic 38 (1973) 579-593. [3] Borodin, A. ,"Computational complexity and the existence of complexity gaps," J. ACM [41
Constable, R.,
[5] Davis, M.,
2
(1972) 185-194.
"The operator gap," J. ACM
19
(1972) 175-183.
Computability and UnsoZvabiZity, ~ ~ G r a w - H i l l
(1958), New York, N. Y. [61 Godel, K.,
The Consistency of the Continuum HypothesCs,
Princeton Univ. Press, Princeton, New Jersey, 1940. [7] Hartmanis, J.,
and Hopcroft, J., "An overview of the theory
-
of computational complexity," J. ACM 18 (19711, 444-475. [81
Hartmanis, J.,
and Stearns, R. E., "Computational complexity
of recursive sequences," IEEE Proc. Fifth Annual Symp. on Switching Circuit Theory and Logical Design, 1964, 82-90. 191
Jacobs, B. E., a-ComputationaZ CompZexity, Ph.D. Thesis, New York Univ., Tech. Rep. IMM-408, Courant Inst., NYU, 1975.
[lo] Jacobs, B. E.,"On J. Symb. Logic, [ll] Jacobs, B. E.,
generalized computational complexity,"
42
(1) (1977) 47-58.
"The a-union theorem and generalized primi-
tive recursion," Trans. AMS [12] Jacobs, B. E.,
237
(1978) 63-81.
"a-Naming and aSpeedup Theorems," Notre Dame
J. Symb. Logic, XX (2) (April 1979) 241-261. [13] Jacobs, B. E., " a-Speedable and non a-speedable sets," Can. J. Math. ,. to appear. [14] Jockusch, C. G., "The degrees
of hyperhyperimmune sets,"
J. Symb. Logic 34 (1969) 489-493. [15] Kripke, S., I, 11,
Transfinite recursion on admissible ordinals,
(abstracts),
J. Symb. Logic
29
(1964) 161-162.
[16] Kripke, S., The Theory of Transfinite Recursion, unpublished unpublished lecture notes by A. Thomas Tymoczko. [17.] Marques, I., "On degrees of unsolvability and complexity properties," J. Symb. Logic (1975) 529-540. [18] McCreight, E. M.,
CZasses of Computabte
Functions Defined
by Bounds on Computation, Ph.D. Thesis, Computer Sci. Dept., Carnegie-Mellon Univ., Pittsburgh, Pa., 1969. [19] McCreight, E., and Meyer, A,, "Classes of computable functions
Defined by bounds on computation," Preliminary
report, Proc. ACM Symp. on Theory of Computing (1969) 79-88. [20] Meyer, A., and Fischer, P., ~8Computationalspeedup by effective
operators:
J. Symb. Logic
[21] Moll, R., and Meyer,
A.
37
(1972) 55-68.
R., "Honest bounds for complexity
classes of recursive functions: J. Syrnb. Logic
39
(1974)
127-138. [22] Platek, R., Foundations of Recursion Theory, Ph.D. Thesis, Stanford, 1966. [23] Rabin, M. O., "Degrees of difficulty of computing a function and a partial ordering of recursive sets," Tech. Rep. 2, Hebrew Univ., Jerusalem, Israel (1960). [24] Rogers, H.,
Theory of Recursive Functions and Effective
Computability, McGraw-Hill, 1967, New York, N. Y. [25] Sacks, G. E., Righer Recursion Theory, Springer, Berlin, to appear. 1261 Shoenfield, J. R., Amsterdam, 19 [271 Shore, R. A.,
Degrees of UneoZvabitity, North-Holland,
. "Splitting an a-recursively enumerable set:
Trans. Am. Math. Soc.
204
(1975) 65-78.
1281 Soare, R., "Computational complexity, levelable sets,* J. Symb. Logic [291 Stoltenberg-Hansen, V.,
42
speedable and
(1977) 545-563.
"On computational complexity in
weakly admissible structures," Tech. Rept. ISBN 82-553-0342-1 Inst. of Mathem., University of Oslo. [30] Young, P.,
gap and
"Easy constructions in complexity theory: speedup theorems: Proc. AMS
37
(1973) 555-563.
CEN TRO I N TERN AZIONALE MATEMATICO ESTIVO (c.I.M.E.
A SURVEY OF S E T RECURSION DAG N ORMANN
A SURYEX
OF SET HXCURSION.
Dag Normann, University of Oslo, Norway.
a-recursion-theory, p -recursion-theory and recursion in normal type-3 objects has one thing in common, one investigates certain notions of computability on certsin sets, L(*), L(p) ~ n dthe type-1 objects I respectively. But while in d - and p-recursion theory o m mqy carry out the resp. L(p) , we nust e;o outside I in order to capture the finer points concerning . are several ways to do this. recursion in, say 3 ~ There One way is to follow Sacks ( this volume ) and construct a hierarchy for the subsets of I recursive in 'E, or in analysis within the structure L(4)
, This involves bringing in a certain' general, in 3 ~ F. superstructure of sets, e.g. ordinals and subsets of I, and history has shown that any finer analysis of 3~ involves the construction on one superstructure or another. In this paper we will take the full and most general consequence of this. We will start with an arbitrary set x and an arbitrary relation R and we will ask : About which other sets do x , R contain perfect inforaation? An alternative way of putting the question is : iVhich sets are *recursivet in q , H ?
It turns out that when we have answered this question we have not only constructed a sensible notion of set-recursion but we have also fully captured and generalized recursion in norvnal functionals. So we have given an alternative to the defirdtion of recursion in normal functiollals given by Sacks ( this volutne )
.
We will now do a bit of History-forging and develop Set-recursion as it ought to have beendeveloped. * We have given a relation lf and sets x = (xl, and we will just find out which,operations on 2 relative
... 3 2
to R
we may within reason call set-recursive. We must
forget absolute notions of finite computations and agree upon the following crucial point : Once we accept a set x as an argument of ? set-recursive function then this function may think that x In
a-
is finite. and p-recursion theory we investigate r.e.
sets
nnd recursion sets but the notion of a computation is rather
innlicit. In Sackst definition of recursion in 3~ we feel that from a notation for an Hbset Ifa and an index for a lst order formula over Hi
, there i a
p we
may compute the set defined by y
some sort of transfintte procedure
leading from (8,e) to the set defined. Here we will take the full computatio~vrlpoint of view. We will define a class of pernittable computations by induction over the ordinals. This scheaatic approach is the same aa Kleene used when he first defined recursion in Mgher types. At the botto~nwe need a few combinatorial operations : f( 2 ) = xi ( 1 4 ) is set-recursive with indcr (I,j4k) is sat-recursive with
index ( 3 ,i, j,k> X'e will describe two methods of constructing new set-recursive functions from old ones.
In the first we really think that
x is finite : If f is set-recursive with index e,, g(;)
=
ygl (f(y,x2, ... ,x 1 ) n
we let
be aet-recursive
with indt~x (4,e,,k>
.
Less controversial is to close the set-recursive functions under composition : are set-recursive with indices If fl, vfn el, 'en respectively and they all accept the same nuqber
...
.*.
of arguments, and h
is set-recursive for n arguments and
with index d then
.!?(?I
= h(fl(?),
. .f n( 1 )
is set-recursive with
index ( 5 ,d ,el, We have not brought in the R
... ,en,k>
yet but the sirnplest
way to do so is just by : f ( 2 ) = xin R is set-recursive relative to R
with
index ( 6 ,i,n> The class of functions we have defined so far is well-
.
known and called the functions rudimentary in R But we have taken the liberty to enumerate the llalgorithms"for the rudimentary functions by the indices and as every recursion 3 theorist knows, given a n .argument x and an index e for an algorithm we may uniformly in ( {elR(;))
2
and
e compute f e) (?)
. So we add the last part of the definition :
f(e,?)
=
R 4 Eel (x) is set-recursive relative to R with index
Theoretically we have n o w given seven clauses in an R inductive definition of the relation {el (34) = y , and such notions as denght of computation and subcomputations are easily defined. We may also prove the recursion-theorem and other standard helpful1 results in recursion theory. O u r first lemma is trivial, but it is the most important result about set-recursion Lemma
t
-
There is a set-recursive function f such that given -b
e,x,x
is defined if and only if [el (y,?)
then f (e,x,;)
are defined for all y E x
, and
then
Proof Let g(e,x,E'I = yk/xx[e3(~,z) Then g(e,x,'jZ3 = P C? Vy x Ee3 (y,3 = 16 f is constructed from g using a rudimentary function. Tho following results are simple but tedious and we leave them for the reader : 1.
The function f(x) = ordinal rank of x recursive.
2.
If x
3.
We 3ay set-recursively decide if a set x nuaber or not.
4.
g(x) = TC(x) is set-recursive. If x is infinite ( < = 7 rank x . ) w )
5.
=
is set-
is infinite then w is set-recursive in x
the closure of x
.
is a natural
then Cl(x)
under finite subsets, is set-
recursive uniforaly in x 6.
If
x = I end
a
.
is sn F-notsti on for H~ then R
.
8a
is
uniformly set-recursive in I, a relative to Y 6. shows that if A 6 3E , F, a then A is set-recursive in I, a relative to F , The opposite is also true, but the argument involves a cod in^ machinery which we won't give here. Theorem Let A S I, a s 1
. Let
Y
bc
R
normal type-3 functional.
is recursive in F, a if and only if A
Then A
set-recursive in I, a relative to F A
set or class A
relative to H
.
is
+
is called semi-set-recursive in x
if there is nn index e such that
re3R (x,;) 2 0 We then also have the following result : A G Tp(2)
xEA
r.e, in 3 ~ F, ,
l\I Q ]
.
Let M, = Mc(y) Then each I , is recursive in x YGX some y c x , and they will fully play the role of H-sets.
and
We will not go deeply into degree-problems here, that is dealt with by Sacks. For each infinite set x with suitable closure-properties there will be the following
natural degree-sfructure to investieate say on X
gl
= ( a: 38, J G
x
I e x > =.c
3
using
R
-b
set-recursive reducibility set-recursive reducibility modulo elements of a given z + x
-C
set-recursive reducibility modulo elements of X: ( corresponding to Sacks* absolute version of Post's
problem). Thls ought .t;obe a rich area for future research.
CENTRO I N TERSAZIOh ALE MATEMATIC0 ESTIVO
(c.I.M.E.
PRIORITY ARGUMENTS I N HIGHER RECURSION THEORY
GERALD
E. SACKS
Centro Internazionale Matematico Estivo
Priority Arguments in Higher Recursion Theory
Gerald E. Sacks
1.
Regularity of a-R.E.
Degrees
This is the first of six lectures on priority arguments in higher recursion theory.
The term "priority argument" refers to
a mode of combinatorial reasoning introduced independently by Friedberg [l] and Muchnik (21 to obtain a positive solution to Post's problem, the existence of two recursively enumerable sets such that neither is recursive in the other.
I will try to show
how their idea lifts to three generalizations of ordinary recursion theory.
The three are:
admissible ordinal;
a-recursion, where
8-recursion, where
0
a limit ordinal; and Kleene recursion in type 3.
a
is a
C1
is little more than
F,
a normal object of
Some of the results to be described are new and, in the
case of Kleene recursion, not hitherto announced.
Others are old
but viewed, it is hoped, in the light of a new day. One obstacle encountered in all three generalizations is the failure of typical recursively enumerable objects to be regular. Today's lecture focuses on that problem in the case of
a-recur-
sion theory. But first a whirlwind review of the fundamentals of recursion.
For any set
x,
order definable subsets of
let X.
L,
Fod(X)
a-
be the set of all first
G6del's universe of construc-
tible sets, is definable by iterating
Fod
through the ordinals.
L(i) = V{L(~) 1
1
L=V{L(6)
6 < A}
6
(A
is an ordinal}.
The Levy hierarchy of formulas of is .A bounded.
G
no
(also
is
G
a L(a)
Co)
A formula is
Bn-l.
where
or
Cn
An
is said to be satisfies
function from
Bn
is
into
by means of some
Then
f[a]
{f (b)
I
A formula
(ExIG, where
if it is of the form
(x)G,
".
Cn n
admissible (or simply admissible) if
C1
L(a)
is as follows.
if it is of the form
replacement.
C1
L(a)
(=
ZF
if all of its quantifiers are
A formula is
is
is a limit).
To elaborate, let
f
be a
L(a) whose graph is definable over formula with parameters in
C1
b E a))
is a member of
L(a)
L(a)
.
for all
a E L(a). A function g : a --> graph of be:
g
is
C1
a
over
is said to be
L(a).
a-recursive if the
Suppose A C a.
a-recursive if the characteristic function of
recursive, and
a-recursively enumerable if
a-finite if
a-finite, then
A E L(a). g[Hl
(=
If
{g(x)
g
I
is
is
A
Z1
A n H
L(a). is
x E HI)
is
If A
is
a-recursive and
H
a-finite.
If
A # 0,
is
enumerable iff
A
then A
is the range of an
A
a-finite.
C1
is
is
over
a-recursive and
last is nothing more than a restatement of the ity of
is said to
A
aL(a),
H
is
This
admissibila-finite, then
a-recursively
a-recursive function.
There is a natural, one-to-one correspondence between a and
L(a)
that is
A1
over
this correspondence yield:
L(a).
Standard arguments based on
(a) an effective coding of the
finite sets by ordinals less than
a-
ar and (b) an enumeration
theorem for the are
a-recursively enumerable sets.
a-recursive functions t
ties:
for each
I
It(%)
6, let Kg
x < ~(6)); then
out repetitions of all index of the versal,
v
be the
{K6
1
withthe following propera-finite set
6 < a}
is an enumeration with-
a-finite sets.
a-finite set
Kg.
6
is the canonical
(b) means there exists a uni-
a-recursively enumerable set W.
sively enumerable set A,
I
A = {x such a
and
6
(a) means there
For each
there exists a
6
a-recur-
such that
<x,6> E W};
is termed an index for A,
and the situation is
summed up by saying A = W6. From now on let f : a ->
function set B c a such that (1)
H, J, K,
a
=
y
f Gwa B.
6, y < a.
&
H c B
&
J c a-B]
The matter is rendered in symbols by
The enumeration theorem for
a-recursively enumerable
a-recursive rednction procedures.
{ ~ } ~ ( 6 )is said to be defined and equal to replaced by such that f = { E } B
R
.
in
B
if
CA
a-recursive in C1
,
a-recursive in a
iff
sets yields one for weak
with
WE.
A
Thus
f gwa B
y
admissible a
if (1) holds
iff there is an
is (by definition) weakly
the characteristic function of CB.
A
a-recursively enumerable R
(EH)(EJ)[ E R
for all
denote a-finite sets.
is said to be weakly
if there exists an f(6)
...
A,
E
a-recursive is weakly
Driscoll [ 3 ] showed that there exists a such that Gwa , restricted to the
recursively enumerable sets, is not transitive.
a-
Shore [ 4 1
characterizes all such
a's.
The following notion of reducibility is transitive. .a-recursive in
B;
in symbols A
(3)
Clearly
(EH) (EJ) [ E
B ->
A
A <wa are said to have the same
A < B a
and
B,
and
y]. w
makes it possible to lift some combinatoric tricks from
to
a.
Note that there exists a nonregplar,
enumerable set iff Proposition 1.3.
a
a
*
*
< a. is the least 8
sively enumerable subset of Proof.
Suppose 6 < a
some
*
.
enumeration of map of
a
Let range of
A If A
such that some
is not
a-finite.
a-recursively enumerable and
were not
a-recur-
a-finite, then any
A C 6
for
a-recursive
6.
f : a -> is
f3
without repetitions would define a one-one
into
f
is A
a-recursively
a
*
be a one-one,
a-recursive map.
a-recursively enumerable but not
Then the
a-finite.
R. B. Jensen has shown by means of a collapsing argument that 1.3 remains true when
a .is replaced by an arbitrary limit
ordinal. Theorem 1.4
(Sacks [S]).
Let
Then there exists a regular, the same Proof. A*
a-degree as
Let
be
a-recursively enumerable.
a-recursively enumerable
B
of
A.
Following Maass [6].
be as in 1.2.
A
It is safe to assume
f : a ->
A*
0 E A.
be a one-one, onto,
Let
a - r e c u r s i v e map, a n d . l e t r e c u r s i v e map.
The d e s i r e d
(Ey) Ix To s e e t h a t
p : a ->
E B~
by
Y
Note t h a t
Suppose
is not
B n T~
i n f i n i t e sequence
y
a-finite.
a*
be one-one, into, and
a-recursive.
Define
t fl for each
E
< a*.
otherwise Thus
{R€
recursive enumeration of all
I
E
*
< a. ) is a simultaneous a-
a-recursively enumerable sets.
[clA is defined similarlyf thus
is defined lclA(x)
is
I
undefined otherwise.
for all
E
< a
*
x < a.
and
It is helpful to think of an union of a noncontracting, sets. each
:R
XEU
Let
I
RE
a-recursively set
R
a-recursive sequence of
be an
as the
a-finite
a-recursive function such for
E,
is simply that part of
RE
enumerated prior to stage
the simultaneous enumeration of recursively enumerable set, then approximated as follows. enumerated in
B
Let
[E] ( Y )
B
a
be the least
a
is defined by
greater than
exists since (13) equals
= a*.
1 6
let
R " ( X , ~ , ~ ) be a predicate .A
such that
Then
According to 3.1 (relativized to that
g
is a one-one map of
There exists x,y
,
xo
a
kA
A) onto
there is a 8.
such
Then
@-recursive function of
E A'
g GwB A
{x0) A (y) is defined.
x,
such that
A A {xl} ( 2 ) = {x0}. (g(z)).
Let
{xl}
Then
6
BA
x
C A'.
0
There is a paradoxical element in the above proof, namely the observation that weakly with
it
I,
is
a,
in 2.4.
I
The limit of
The assumption of weak The set of
a's
such that
6-recursively enumer-
(The proof of 1.3 works perfectly well
is less than B*.
Since
In addition the B*
B2
->
B
B such
5.
Kleene Recursion in
3~
Today's lecture and the next are devoted to Kleene's theory of recursion in normal objects of type 3.
All that
follows applies equally well to type 4 and higher, but it seemed discrete to avoid higher type complexities in order to achieve a more valiant exposition at the type 3 level.
Today
is largely review, and tomorrow is regular sets and Post's problem. 2w
is the set of all reals.
cate for sets of reals. false otherwise, where domain is
w 2w
Thus
is the equality predi-
3~
s
E(x,y)
x, y C 2W.
Let
is true if F
F
and
be a function whose
and whose range is contained in
[13] defines recursion in
x = y
w.
Kleene
by means of schemes not unlike
those in common use at the first type level. schemes amounts to generating computations.
Unraveling those If
F
is normal,
then the computations lend themselves to a natural enumeration in the form of a hierarchy that resembles L. means
3~
is recursive in
F.
F
is normal
Since "recursive inw has not
yet been defined, it may be best to think of a normal object as consisting of a pair
< 3E,F>,
where
F
is arbitrary.
The use of hierarchies to characterize Kleene recursion in the normal case originates with Shoenfield [14] and Grilliot [151. The idea is to construe
F
as a jump operator acting on sets
of real's, and then to generate the sets recursive in
F
and a
arbitrary real in a fashion analogous to the generation of the
hyperarlthmetic sets by iteration of the Turing jump through the recursive ordinals.
Nothing could be more reasonable once
Kleene had shown that the hyperarithmetic sets (of integers) are just those sets recursive in Let
X C 2W.
The
F
2 ~ .
jump of
X,
denoted by
jX,
is the
effective disjoint union of certain definable subsets of
1
Let
be the
e-th
function
(e-th
according to some
standard GGdel numbering of formulas) from graph is definable over formula whose
X. let jx
all
where
<X,a>
encodes the
whose
z
2W
y
E
z
and
z
E
X,
x
is a real variable and
&
<e,a,O>
X {ell(a) = 01. such that
a E
where de-
exists a
a
constructive in F, a
all real b. sive in in
F, a
follows.
is recursive in
3 ~ F, , a, then there such that
If(b) lF < a
for
It is a consequence that a set R c 2@ is recur-
3 ~ F, , a
iff there exists an ordinal a
such that
R
is one-one reducible to H
constructive as
For some e,
for all real b. A
an
e
set R
is recursively enumerable in F, a
such that for all b,
if there is
The downward arrow
means "convergesn or "is defined".
4
upward arrow t
means "diverges" or "is undefined". {el (3~,~,b)+,then lie) ( 3E.F.~) 1 !dlF; and
(2)
R(c,d)
c $3 oF,
if
R(c,d)
holds and
R(c,d)
holds and c
d 9
oF,
then there exists a
R (c,d) such that: then d E
d
oF
and
such that
oF.
is an artful variation of the predicate:
d
indexes an immediate subcomputation of the computation indexed by
d.
If
a
P
A,
E
B
then
lows from (1).
0
is defined by
(Eb)[
E
Bl
by ( 2 ) .
The converse fol-
Corollary 6.3. (i) (in F)
(Moschovakis)
The class of recursively enumerable
predicates of reals is not closed under existential
auantification.
6.3 (ii) follows from 6.2 and the existence of a nonrecursive, recursively enumerable set of reals. K~a was defined above for an arbitrary real type 3 F. the form degree
a
relative to a normal
The definition extends easily from reals to sets of
8. If 1 a 1 (mod F),
and
Lemma 6.4 ( 1 2 5 1 ) .
If
= u, K
Hu
qF
then :H
and
a
have the same
is (by definition) .:K
> y, then rlF > sup {rrH,
I
IJ
< y}.
(In addition the supremum can be computed effectively from The proof of 6.4 is technical, but not difficult.
y.)
It
begins with the observation that
and makes use of the fact that
qF
is an
F-cardinal.
The
parenthetical portion of 6.4 follows from 6.2. Theorem 6.5
.
([251)
recfp(n) = recfF(K1).
The proof of 6.5 is based on 6.4
(including the parentheti-
cal part) and 6.2. Let
H
be a set of the form
characterization of
:K
8. The
Harrington
begins with a slight variant of 6.2:
there exists a W e 1 number
n
such that
for all
e
If
A.
and
In) (F,e,~,blZ, then the ordinal
1 in1 (~,e,~,b) 1
is
said to be a Moschovakis witness to the divergence of {el (F,H). (Harrington 1221 1.
Theorem 6.6 for all
er
if
{el(F,H)
is the least
:K
such that
diverges, then there is a Moschovakis
witness to its divergence at or below
a.
The proof of 6.6 combines the details of the proof of 6.2 with the following basis theorem of Harrington Suppose A reals; if that
&
Kechris 1221:
is a recursively enumerable (in F, H)
set of
2 W - ~ is nonempty, then it has a member c such s be undone for the sake of Rn*
The original priority method of Friedberg
and Muchnik has the property that each requirement i s injured at most finitely often. We illustrate the finite injury method by proving the FriedbergMuchnik theorem using a variation of Sacks [I81 which i s more powerful than the standard method and which will be used in $3.
In our constructions, the
requirements {R ) will be divided into the negative requirements e ec w
N
e
=R
2e
which attempt to keep elements
structed, and positive requirements
e'
out of
the r. e. set A being conwhich attempt to put
= R~e+i
elements
KO A.
The negative requirements will be of the form C
# Q!e(A),
where C is a fixed nonrecursive r. e. set, so that the negative requirements together a s s e r t that C d
T ~ .Sacks observed that the
requirement Ne can
be met by attempting to preserve agreement between Cs(x) and Oe,.(AS;x)
rather than disagreement as one might suppose.
(The point i s
that i f we preserve this agreement sufficiently often and i f C = Q (A) then e C will be recursive contrary to hypothesis. )
The positive requirements
will ensure a s in Theorem l.l that A i s siriiple and hence nonrecursive.
F o r every nonrecursive r. e. set
Theorem 2.1 (Friedberg-Muchnik):
C there i s a simple set A such that C $ A (and hence j#
s1 such that l(e, s) > p.
To compute
It follows by induc-
tion on t 2 s that
and hbnce that
Q,
e,s
( A ~p); = Q e ( ~ sp) ; = ( P ~ ( AP) ; = C(P). Since s
clearly holds unless C (x) t
# Cs(x)
> s f , (2. 2)
for some t 2 s and x 5 p ; but if x and t
a r e minimal then our use of "5I (e, t)" r a t h e r than nition of r(e, t ) i n s u r e s that the disagreement C (x) t
< l ( e , t)" i n the defi-
# @e ,t ( A ~ ; x ) is
Note that
preserved forever, contrary to the hypothesis that C = ee(A).
even though the Sacks strategy is always described a s one which preserves agreements, it is crucial that we preserve at least one disagreement a s well whenever possible. Lemma 2.3. Proof. -
(\de)[lim r(e.s) exists and i s finite]. s
By Lemma 1.1 choose p = Fx[C(x)
sufficiently l a r g e such that, f o r all s (Vx
# Se(~;x)].
Choose s *
2 sB,
< ~ ) r @ ~ , ~ ( A= ~%(A: ; x ) 41,
( V x 5 p)[Cs(x) = C(x)l, and
Ne is not injured a t stage s. Case 1. s
,
( V s 2 s*)[\,,(A,;
p) undefined].
Then r(e,s) = r ( e , s l ) for all
sB.
C a s e 2.
4e,t(At; p) i s defined f o r some t 2 s'.
Then 4
e,s
(As; P) = q , t ( A t i ~ )
f o r all s 2 t because l(e,s) 2 p, and so, by the definition of r(e,s), the computation @ (At; p) is preserved and N i s not injured after stage sf. e ,t e Thus ae(A; p) = S
Hence
(As; p).
, r(e, t) = limsr(e, Lemma 2.4,
Proof. max(r(i):
we
e,s
But C(p)
# ae(A; p).
Thus
s).
( V e)[We
infinite d We
nA
# $1.
By Lemma 1.2, let r(e) = limsr(e,s) and R(e)
i (e l .
Now if
(3x)[xt
We & x
> RCe)
& x
=
> Ze] then
* A # $. Note that
i s simple.
i s infinite by the clause IBx> 2e
in (2.1). and hence A
Sacks invented-the above preservation method (which plays a crucial role in the l a t e r infinite injury argument) to prove the following theorem. Theorem 2.5 (Sacks Splitting Theorem [18]): such that C i s nonrecursive.
Let B and C be r. e. s e t s
Then t h e r e exist r.e.
sets A0 and A1 such
that (a)
AOU A 1 = B and A O n A l =
(b)
G
$,
and
I T ~fori ,i = 0 , i .
Proof. -
st
Let
and C such that Bo =
$
and
( c ~ )be ~recursive ~
enumerations of B
-
and ) B ~ +Bs~ I = 1 f o r all s. It suffices to give
recursive enumerations {A
}
i,s s c
0'
i
= O,l,
satisfying the single positive
requirement P : x c B stl -Bs
@ [ X ~ A ~ o, r~x + c A ~i , s t l l *
and the negative requirements for i = 0 , 1 and a l l e,
: N
c # \(Ai).
Define A = $. Given A.1,s define the recursive functions ll(e.s) i,O i Let x c Bsti Bs and r (e,s) a s above but with A in place of A i,s so i Choose <el, i*) t o be t h e least <e, i ) such that x ( r (e, s ) and enumerate
-
' Ai-i*,s+i.
If ( e l , i * )
.
fails to exist, enumerate x c A0 , s t l ' This
defines Ai, i = 0,1. i To s e e that the construction succeeds, define the injury s e t Ie a s above but with Ai i n place of A. i
= 0, i and a l l e ,
It follows by induction on (e, i ) that, for
(1) C
# @e(Ai)8
i (2) lim r (e, s ) exists and i s finite, and s (3) 1:
i s finite.
D
It can be shown [26, p. 5251 that the r. e. s e t s Ai a r e automatically low, namely -
A: s T$I, where the jump of A was defined e a r l i e r to be
A At = {e: e r We ). By setting C = B in Theorem 2.5, it follows that any nonrecursive r. e. set B can be split a s the disjoint union of low r. e. s e t s A.
and A which a r e Turing incomparable and such that i
d g ( ~ =) dg(AO) U dg(Ai).
Thus, there i s no minimal r.e. degree.
Finite injury arguments a r e characterized by the fact that the injury s e t Ie i s finite for each e.
In $3 we will consider c a s e s where I
infinite although usually recursive.
e
is
Note that Lemma 2.2 holds by virtually
the s a m e proof as above if we a s s u m e "Ie recursiven in place of "Ie finitet1. This i s what allows the infinite injury method to succeed.
3. THE INFINITE INJURY PRIORITY METHOD
Shoenfield [22] and
, independently,
Sacks [19], [ZO], [21] discovered a
technique for handling a r e q u i r e m e n t which may be injured infinitely often (namely, the injury s e t Ie m a y be infinite).
Sacks considerably developed
this technique into what he called the "infinite injury priority method" and he used i t to prove many important results on r. e. degrees, the most striking of which i s the density theorem which a s s e r t s that for any r.e. d
CI
< 5 t h e r e i s an r. e. degree 2 such that
sketch of the method.
cJ,
$(i. s) for a l l i we enumerate x in A(') just if x r Bs+l s+l Let A =
us
Te C A
because N
5
e
i s injured by P. only if i < e. 1
Thus
A
s T ~ ( c e )because if x e A ( < ~ ' , say x t A(<e), s then x e Ie just
Fix e and assume by induction that C i
.
As '
Note that we have
5a
=*
# ei(A)
and A ( ~ ) B(i) for all h
< e. Then A ( < ~=* ) B(<e) i s recursive and hence I i s recursive. e Lemma 3.2 ( ~ n j u r yLemma):
C
# %(A).
Proof. Assume for a contradiction that A
lims ~ ( es) , = m.
Fixing
contrary to hypothesis. i ( e , s)
ie
C = .Qle(A). Then
a s an oracle we compute C, so C i s recursive
To compute C(p) f o r p
t u find
some s such that
> p and (Vx p.
,
Let
AStt Bsn q = {e)stt (p) = {els (p). We will prove by induction on t that for all t either
A
and hence that f(p) = q. (i) o r (ii).
2 sn
Suppose that x destroys the last of the computations
Now i f x enters A V B a t any stage s t 1 such that s
s must have been e-maximal, s o both (i) and (ii) hold for t = s.
t
S then
But x can
destroy a t most one of the computations, s o the other holds at t = s t l .
F u r t h e r m o r e , x cannot e n t e r A u B a t stage s + l f o r s r(e, s) 2 x by clause (ii) in the definition of ~ ( es). ,
/ S,
s > s " , since
Q
This construction can be modified in a number of ways.
F i r s t , one
can construct an r. e. sequence of r. e. degrees {2i: i e w} such that a a i s a minimal pair for each i wiP-j requirements a s in
5 3,
Next, by allowing infinitary positive
one c a n construct a minimal pair
degrees which a r e high (i. e.
5.
# j.
.
5 ,i of r.
e.
= 2")(see Lachlan [4]).
=
EMBEDDING DISTRIBUTIVE LATTICES IN THE R. E. DEGREES
Using a fairly e a s y modification of the preceding method we will now replace the Diamond lattice of Theorem 4.2 by lattice.
any countable
distributive
Since any countable distributive lattice can be embedded in the
countable atomless Boolean algebra i t suffices to prove the following. Theorem 5.1 (Lachlan-Thomason [29]): countable atomless Boolean algebra
There i s an embedding of a
8 into the r. e. degrees
f? which p r e -
s e r v e s sups, infe, and least element. Proof. -
Let {a.: i c w} 1
be any uniformly recursive sequence of
recursive s e t s (i. e., ( ( x, i ) :x c ai} i s a recursive relation) which f o r m s an atomless Boolean algebra 73 under U tains o and has jd
a s i t s only finite member.
A i , i c y and define A
= { < i,x (I
we immediately have
, n , and complementation, con-
> :x r A .
& i c a)
We will construct r.e.
sets
f o r a c 73. Notice that
"g)
(5.1)
dr(Aa
(5.2)
a Z B => de(A,)
= ds(AJ
We will further meet f o r a l l i.a,
N
Aa
8
= {jfp
a ! , , j : {j
I
v dg(Ap) dg(Ag)
.
, and
the requirements
= f total
< A -T a n e '
5 f
These requirements i n s u r e
Note that (5.1)-(5.5) guarantee that the m a p a
-,
dg(A ) i s the desired Q
embedding, and (5.5) guarantees that the map i s 1: 1. negative requirements insure (5.5) suppose: (2) a
$
f3, s a y i r a
- B.
(To s e e that the
(1) dg(Aa) 5 dg(AB): but
Then dg(Ai) (dg(Aa) I dg(A ) by (5.2) and (l),
but d g ( ~ , )5 dg(A- ) by (2).
B
B
Hence dg(Ai)
dg(A
(5.4)' contradicting A nonrecursive. ) i
-)
B"B
= dg(A4 = 2 by
The strategy f o r meeting the negative requirements N a s before.
Denoting N
e = ( i l . i2.j a s in
5 4.
)
(aoB.j)
by Ne
, where
a!
=a
ii'
(a9B.j
= a.
I2
,
begins
i
and
we define the restraint function by induction on e exactly
However, new difficulties in proving Lemma 3 (that N
e
i s satis-
fied) require greater c a r e i n enumerating elements for the positive requirements.
To meet requirement P we will appoint followers
x r w so-called because the eventual enumeration of x in A. will satisfy
' (although x may be cancelled before this happens).
If x i s a follower
of P. and y a follower of P. then we say x has higher priority than y (x< y) 1 1 if i < j o r i = j and x was appointed before y. x4 y
followers x and y existing a t stage s that Construction of A
i'
Stage s t 1 Jv 8
< y.
Do nothing.
.
Requirement
P i s satisfied i f
# 8.
A W.
1,s
iff x
i r o:
Stage s =, 0.
A.
We will arrange for all
Requirement P requires attention if P
i s not
satisfied and either x
(5.6)
and x
W.
t
> r ( < i , j ) , s ) f o r some uncancelled follower
x of
$ 9 6
P or ' (5,7)
xew. JI
Let P
s
attention.
ioreveryuncancelledfollowerxof P Ci, j)
.
be the highest priority requirement which requires
If (5.6) holds for some x enumerate the least such x in A..
If
(5.6) fails, and (5.7) holds, then appoint x = s t 1 a s a follower of P
In either c a s e cancel a l l followers y of lower priority than x (i. e. (If no P
Ci, j
>
( v e ) [lim infs r(e, s ) < a].
Exactly a s in $ 4 , Lemma 1.
Lemma 2. i s met
x d y).
requires attention, then do nothing.)
Lemma 1. Proof.
.
(t/ e)[Pe
receives attention a t most finitely often and
1.
Proof.
Fix e and chooge s
receive attention after stage s
0'
0
such that for no el < e does Pel
Let k = lim idsr(e, s) by Lemma 1.
Let
e = k i s appointed to follow P
C i . j)
and x i s never cancelled.
such that F u r t h e r m o r e , x o W. by (5.7). Hence, there i s a stage t t l > s J 0 Now x o r some s m a l l e r follower of P i s enumer r(e,t) < x and x E W. ~rt' e
-
ated i n A at stage t + l , P i s met, and P never again requires attention. e e Therefore, P receives attention at most finitely often. e met because otherwise P(i,j)and'(is j)
( v u ) ( v B) ( V j) [requirement N
Fix N = N e
A, Assume that {j)
of (4.
A {e)sv* '(x) To A
= W. , x t W. for every uncancelled follower x of J J
receives attention infinitely often under (5.7).
Lemma 3. Proof. -
i
Finally, P is (iIj)
(u,B,j)*
Choose k,S and stage s1 a s in Lemma 3
= { j t S = f is a total function.
is A ,-correct if A
6s
[u] = A,[u]
A computation
where u = U ( A , , ~ ,e. x , s).
-recursively compute f(p), find an e-maximal stage s A
U ~ B
t
S, s
> s',
such that l ( e , s) > p and both computations {e) = q and S A {els S's(p) = q a r e A -correct. We will show by induction on t that f o r unB a l l t z s either
via an A
an S
-correct computation.
(5.8) o r (5.9) by entering A u u A e-maximal and t o S
B
Now i f x destroys e i t h e r computation at stage
t t i then t must have been
( a s in Lemma 3 of 14) s o & t J computations existed at
the end of stage t.
By inductive hypothesis a t least one computation, say
(5.8). i s A
-
n B
~
Suppose x i s enumerated in A
destroying this computation.
~
~
~
~
~
~
.u
a t stage t t l ,
Then x cancels at stage t t l a l l followers y
such that x
< y (since these a r e exactly those followers y such that
Furthermore, z > t t l 2 u appointed.
But x
troyed by x. was A
f f nB
A
B-
p =dfn U(Ap,t' , ,p) for any follower z later
uu =an~(Au,t,e, t, p) since the A-computation is des
Also A
-correct.
Hence A ff
-
-
u ] since the Au-computation (5.8)
u n ~ . t [ ~ u= '
Brt
p[ u [ttl] = A
computation (5.9) now becomes Au Corollary 5.2.
x < Y).
fffi
[ t t l ] , and u 5 t t l s o the
B
B
-correct.
Any countable distributive lattice can be embedded into
the r.e. degrees f? by a map which preserves sups, i d s , and least element. Embedding nondistributive lattices into f? i s much more difficult. Lachlan [6] showed that the following two 5-element nondistributive lattices
M 5 (a modular lattice) and N 5 (a nonmodular lattice) can be embedded in
5
by a map preserving sups, infs, and least element.
This partial success led many to believe in the Embedding Conjecture which a s s e r t e d that every finite lattice can be embedded in lattice.
5 as a
finite
This conjecture was recently refuted by Lachlan and Soare [lo] who
showed that the following lattice S8 cannot be embedded in f? a s a lattice.
The obstacle to embedding S8 i s that for
&
to be the sup of the lower
M 5 lattice, an elaborate system of t r a c e s i s required for enumerating e l e ments into the s e t of degree
2.
This i n t e r f e r e s with the delicate minimal
pair machinery above which insures that
2
The most important open question on elementary theory.
i s the inf of a -0
5 i s that of
and a
-1
'
the decidability of i t s
Considerably more structural results (such a s
embedding and nonembedding theorems) will be required to meet this goal.
6.
THE NON-DIAMOND THEOREM
One might expect to extend Theorem 5.1 by constructing lattice embeddings which p r e s e r v e both g r e a t e s t a s well a s least elements.
The
following surprising theorem shows this i s impossible even for the Diamond lattice.
6.1
Non-Diamond Theorem (Lachlan [4]):
2u
r. e. degree & such that
and $ ,
Proof. and
least corresponding p a i r < x , y Ds+i(j)
such that P.
>.
1, j
requires attention and the
Insert o r extract j from D t o insure
Enumerate A and B until the first number z
# Ds(j)
appears i n AU B.
(If no such z appears, the construction stops.
< u(j, t) How-
ever, i f e i s the index obtained by the recursion theorem satisfying D = {e}A8B,
then z must appear.)
If z appears in A, enumerate y in F.
If z appears i n B, enumerate x i n E. Lemma 1. Proof. -
0
The s e t D is A2
and hence D
2 i , y > 2j of (2)
x for
Now x e E iff x c Es .)
t > s.
Lemma 3.
If E i s recursive, then
(i) F i s nonrecursive, and (it) F
sTB. -
Proof of (i).
F i x i such that W. = E. 0 lo
If F i s recursive, choose
the least j > i such that W. = F. Choose s such that for a l l s 0 J 0
2s
0 '
u(js s ) = ~ ( jso) , and (A 8 B)[u] = (AS O BS )[u], where u = u(j, so). Now 0 0 since B i s nonrecursive, t h e r e exist y > u and t > s such that y c W. 0 Jet0 and y i s permitted by B at stage t , i. e . , such that (4)(a), (b) and ( c ) hold f o r y and any s > t.
But since A i s nonrecursive t h e r e exist infinitely many
satisfying (1),(2) and (3) f o r some s > t. Hence requirement P . 0 iosJ receives attention, and e i t h e r E 0 W. # $ o r F n W. # $ contrary to '0 J hypothesis. x c
Wi
Proof of (ii).
B we m a y f i r s t assume that we know
To prove F < 'T
those finitely many y contributed t o F by some P. 1.
j
with j t. Thus. for any s
As A B then {els (x) = { i ) P ( x ) = {e) (x) = {i) (x).) C1earlyCisr.e..
infimum.
0
-T
There a r e r. e. degrees
5 ,&
with no
Hence, the r. e. degrees do not form a lattice.
Proof.
Let 2 ,& be incomparable low r. e. degrees with fiuk = 0'
P'
(obtain these by the Sacks splitting theorem.) b o t h 2 and
if I(s) 7 x
Now set C = O {Cx: x r u).
D < C, and C + A , B .
Corollary 6.3 ( ~ a c h l a n[4]).
/ Cx,
2.
Now
$ =$
Theorem 6.1 produces
since
2' = 2'.
Let
s o the relativization to
5 7 2 , such that 2 < fi
we may assume that & i s r. e.
Thus
2
and
be any degree below
2
.5
0,
To make both inequalities strict, i t suffices to
the requirements Ne: C
f ee(A) and Pe: A # Qe(D).
To meet Pe we attempt to code C into A(e) so that if A = q ( D ) then C contrary to hypothesis.
D
Let A.
=
#.
s) define :(e, s ) a s in $ 2, and define
Given {A,: t
r* D (e, s ) = max{x: ( V y < x)[As(y)
< e,x, s )
Let
denote
<e,x, t) in A *D r < 1 (e,v)
.
<e, <x, s>>
f o r a l l v. t-< v
UsA,
> :x, y e w )
to <e,x, t
a t some stage s t
a r e arranged in a plane.
> at stage t if >t
enumerate L
Cs+i and
.
(We visualize this coding strategy a s follows. {<e, x, y
>i
s ) for a l l 0 < i x.
The m a r k e r i s later removed (forever)
AD i f I (e, s ) 5 x.
If x r Cs+l then a l l elements (e, x, t > ,
I
s, still possessing m a r k e r s a t stage s and not restrained with higher
priority a r e enumerated in A
s+i
'1
To s e e that A succeeds we shall verify the requirements by induction on e ahd simultaneously C-recursively define g(e) such that A(e) = Q g(e)(c)* Clearly, g(0) exists because A(O)
D < C. T
IT
Fix e
> 0 and assume, for
a l l i, 0 < i < e, that
(7.6)
Ag) is recursive, and
(7.7)
A") = @ (C). where g(i) has been C-recursively computed. g(i) Lemma i. Proof.
7
C
#
%(A)
.
Now A(<e) < -T D because A@) i s recursive for O< e.
ie be the injury s e t f o r Ne defined in $3. Thus,
< D < C. Hence, C -T
T
Then
ieEA(;((e), x
t
2 s)[:
C just if < e . x , ~ ( x ) ) t A.
t r a r y to hypothesis.
D(e, t) > XI. Hence. C S T A L D con-
T
(Note that M is a D-recursive function because D r. e.
rD implies that M(x) = ( ~ s ) [ l (e, s) > x & D ~ [ u=] D[u]], where
u = m a x ( u ( ~ ~ , e , ys):, y l x ) . ) Lemma 3. Proof.
A ' ~ ) i s recursive.
By Lemn.a 2, let pe = (px)[A (x) #
9;x)].
Then
AD lim inf I (e, s) = p F o r x 2 p given x and t find s 2 t such t t a t s e e AD F o r x < pe fix I ( e , s ) = p e . Then < e , x . t > t A justif < e , x , t ) c A s .
.
A
s1 such that C [ >,= C[pe] and R(e,s) = B(e) for a11 s st
Then < e , x , t > t A justif Lemma 4.
Proof. -
(e,x, t
Then
(e,x,t
< e , x , t > a Av,.
We can C-recursively compute g(e) such that
Define T~ a s in Remark 7.3.
C-recursively compute A then
> / A.
>
t
-
> st. Given t define
A
If x
andhence T t
just i f
C, say x
(e,x,t
t
>
Cs
t
e
F r o m {g(i):i < e) , we
. Fix ( e , x , t ) .
, define
Avt by (7. 3)
1f X / C
(Notice ihat we do not claim here that (7.6) and (7.7) can be combined i s the characteristic to produce a C-recursive function g such that cp g(e) function of A(e) f o r a l l e
> 0 , but merely that A ( ~=) QI
i s that even though A ( ~ )i s recursive for a l l e
> 0,
g(e!
(C).
The point
the proof of Lemma 3
A
above depends upon parameters p and R(e) which cannot be C-recursively e computed uniformly i n e , and the proof of Lemma 4 clearly uses a C-oracle f o r each x.) The abbve coding procedure has many other applications such a s the following.
(The nonuniform version of the theorem follows immediately The uniform version requires an infinite
f r o m t h e Sacks Splitting Theorem.
injury argument and was proved by Yates [ 3 i ] using index sets.) Theorem 7.6 p a c k s - ~ a t e s ) : Given any r. e. s e t C such that
$
in A
s ti
AC and x < l ( e , v ) . f o r a l l
QI
e ,s
just if v,
(Cs; Y)D.
<e,x,t
t i v i s .
> > 3(i, s )
for
Let A z U S A s .
F.ix e and assume by induction that, for all i < e, C A
{ ei(C), and A ( ~ )i s recursive.
#
@.(A), 1
The proofs above establish (with C a i d X
in place of D and C respectively in Lemmas 2 and 3): C f 4 (A); A # e
Q)
e
(C);
and A(e) i s recursive.
0
An interesting generalization of the density theorem has recently been discovered.
An r. e. degree
Definition 7.7. parable r.e.
2 and
degrees
2
i s branching i f there a r e incom-
2 such that 2 i s the infimum of
and
2
g), and a i s nonbranching otherwise.
(2 = &
2
F o r example, Theor em 4.2 shows that
i s branching, and i t follows
a t once from Theorem 5. i that there a r e nonzero branching degrees. Lachlan [4] showed the existence of nonbranching degrees and recently F e j e r [33] has proved them dense in the r. e. degrees. Theorem 7.8
(Fejer Density Theorem): If
then there i s a nonbranching r. e. degree Corollary 7.9
2
such that
(Fejer Splitting he or em):
degree then there a r e nonbranching degrees
a r e r. e. degrees
B
$< 2 < s
.
If a i s a nonzero r. e.
and 2 such that 2 =
u 2.
It follows that the nonbranching degrees when closed under 1) generate all the r.e. degrees
E
.
nontrivial definable subset of generate
&.
The nonbranching degrees a r e the first
$ to be shown to be dense and hence to
If, as suggested by Jockusch and Lerman, the r. e. degrees
a r e given the order topology where a typical subasic open set has the form
[Q,s)=
{Q : b_< g) o r
.
(a,%I]= {h :k> a ) ,
then the branching degrees,
together with
a r e precisely the isolated points.
Theorem 7.8 that the Cantor-Bendixson rank of
It follows f r o m with this topology i s i.
The Sacks coding strategy of the density theorem has recently been applied to give a new result on index s e t s which immediately yields several index s e t results of Yates [25] and [26] a s pointed out in [34].
The following
result was f i r s t stated by Kallibekov [34] with an entirely different method of Unfortunately, the verification that his ingenious
proof f o r infinite injury.
method succeeds s e e m s t o contain an e r r o r .
Stob [36] gave a completely
different proof -by combining the Sacks coding strategy with the thickness lemma method of $3. Let C and D be r.e.
Theorem 7.10.
9 CTC
CT
9'.
D
Let S be any set in Cg.
s e t s such that C I D and
T
Then there i s a uniformly r. e.
sequence of r. e. s e t s { A ~ ) such ~ ~ that f o r a l l k,
k c S *Ak==D k
/S
d A
k
and C a r e Turing incomparable.
In Stob's proof t h e r e i s a new element beyond what we have seen h e r e and in [ 2 5 ] , since if k e S we may have f o r certain e that lim infs r ( e , s )
= m.
Stob needed a new insight in this c a s e to show that Ak l T D . In a different direction another conjecture on r. e. degrees has
recently been refuted.
After seeing the density theorem, Shoenfield
formulated a conjecture that the r.e.
degrees
form a dense structure a s
a partially o r d e r e d s e t analogously a s the rationals f o r m a dense structure a s a linearly o r d e r e d set. q(xi,.
.., n ,y) x
Shoenfieldts conjecture implies that if
i s a quantifier f r e e formula of the language L(&
U,0 , i )
and
.
q,. .,
e
then there exists
2t
R mch that
(p
(gi,. ..,sn,b)
holds
unless the existence of ]Z would lead to an winconsistency.m. The existence of a minimal pair (Theorem 4.2) refutes Shoenfieldfs conjecture.
Jockusch
then suggested that since minimal pairs a r e the source of much difficulty, perhaps Shoenfield's conjecture holds for the simple degrees, those nonzero degrees which a r e not half of a minmal pair.
Recently, Klaus Ambos has
refuted this by combining the Yates construction [30] of a simple degree with the Lachlan construction of a nonzero branching degree simple branching degree.
( 3 5) to produce a
(Since simple degrees a r e closed upwardly, the
branches must also be .simple. )
REFERENCES 1.
M. R. Arslanov, R.F. Nadirov and V. D. Solovev, Completeness c r i t e r i a for recursively enumerable s e t s and some general theorems on fixed points, Matematica University News, 1977 (179) No.4.
2.
J. Dekker, A theorem on hypersimple s e t s , Proc. Amer.Math.Soc.,
-5 (1954). 3.
791-796.
R. M. Friedberg,
Two recursively enumerable s e t s of incomparable
degrees of unsolvability, P r o c . Natl. Acad. Sciences, U. S. A. 43 (1957), 4.
A.H.
236-238.
Lachlan,
MR 18, 867.
Lower bounds for p a i r s of r . e . degrees, P r o c . London
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Math. Soc. (3). 16 (1966). 537-569.
5.
,
MR 34 no.4126.
, Complete recursively enumerable s e t s , Proc. Amer. Math SOC. 2 (1968). 99-102.
6.
, Embedding nondistributive lattices in the recursively enumerable d e g r e e s , Conf. Mathematical Logic, London 1970, Lecture Notes in Math., no. 255, Springer-Verlag, Berlin and New York, 1972, pp. 149-177.
,
7.
The priority method f o r the construction of recursively
enumerable s e t s , P r o c . Cambridge Summer School in Logic, 1971, Springer-Verlag, Berlin & New York, 1973. 8.
,
A recursively enumerable degree which wid1 not split over
a l l l e s s e r ones, Ann. Math. Logic, 9.
,
9 (1975), 307-365.
Decomposition of recursively enumerable degrees, Proc.
Amer. Math. Soc., t o appear. 10.
, and R.I. Soare, Not e v e r y finite lattice i s embeddable in the recursively enumerable degrees,
to appear.
11. S. S. Marchenkov,
A c l a s s of partial sets. Mathematicheskie Zametki
Vol. 20, No. 4 (1976), 473-478. 12.
D. A. Martin.
Completeness, the recursion theorem, and effectively
simple s e t s , 13.
A. A. Muchnik,
Proc. Amer. Math. Soc..
(1966). 838-842.
On the unsolvability of the problem of reducibility in
the theory of algorithms (Russian), Doklady Akademii Nauk,
n. s., 14.
108 (1956).
194-197.
E.L. Poet, Recursively enumerable s e t s of positive integers and their decision problems,
Bull. Amer. Math. Soc.,
0 (1944),
284- 3 16. 15.
R. W. Robinson, Interpolation and embedding in the recursively enumerable degrees,
16.
,
, 2 (1971), 285-314.
Jump r e s t r i c t e d interpolation in the r. e. degrees, Annals
2 (1971).
Math., 17.
Annals Math.
586-596.
H. Rogers, Jr. , Theory of recursive functions and effective computability,
18. G. E. Sacks,
McGraw-Hill, N. Y.,
1967.
On the d e g r e e s l e s s than 08,
Annals Math.,
71 (1963),
211-231. 19.
,
Recursive enumerability and the jump operator, T rans.
Amer. Math. Soc., 20.
, The
108
(1963). 223-239.
recursively enumerable degrees a r e dense, Annals of
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Math. (2). 80 (1964). 300-312. 2 1.
,
Degrees of unsolvability, rev. ed., Annals of Math.
Studies, No. 55, Princeton Univ. P r e s s , Princeton, N. J. 1966. 22.
J. R. Shoenfield, Undecidable and creative theories, Fundamenta
-
Mathematicae, 49 (1961), 171-179. 23.
, 1971.
Degrees of unsolvability,
North-Holland, Amsterdam
J. R. Shoenfield and R. I. Soare,
The generalized diamond theorem,
( ~ b s t r a c t )Recursive function theory newsletter, 19 C1978), # 2 1 9 The infinite injury priority method, J.Symbolic Logic,
R.I. Soare,
41 (1976). -
513-530.
Computational complexity, speedable and levelable s e t s ,
,
J. Symbolic Logic,
,
42 (1 977),
545-563.
Recursively enumerable s e t s and degrees, Bull. A.M. S.
Vol. 84, No.6 (1978), 1149-1181.
,
Recursively enumerable s e t s and degrees, (Omega Series),
Springer-Verlag, Berlin and New York (to appear). S. K. Thomason,
Sublattices of the recursively enumerable degrees,
2. Math. Logik und Grundlagen d. Math., C.E. M. Yates,
A minimal pair of r.e.
31 (1966), -
,
17 (1971).
273-280.
degrees, J. Symbolic Logic,
159-168.
On the degrees of index s e t s , Trans. Amer. Math. Soc.,
-
121, (1966), 309-328.
,
On the degrees of index s e t s , 11. Trans. Amer. Math. Soc.
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249-266. Dissertation, University of Chicago, 1980.
See
Density of nonbranching degrees (abstract 7 9 ~ - E 3 2 )Notices A. M.S.
S. Kallibekov,
June 1979, p. A-390. Index s e t s of degrees of unsolvability, Algebra i Logika
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The upper semi-lattice of degrees of
recursive unsolvability, Ann. of Math. (2) 59 (1954), 379-407. 36.
M. Stob,
Index s e t s and degrees of unsolvability,
t o appear.
CEK TRO I h TERNAZIOh ALE MATEMATICO E S T I V O (c.I.M.E.)
RECURSIVELY I N V A R I A N T
-RECURSION
WOLFGANG MAAS
THEORY
Recursively Invariant (3-Recursion Theory (Preliminary Survey)
Wolfgang Maass Massachusetts Institute of Technology, Cambridge, USA
I
In my lecture I want to sketch a new branch of generalized recursion theory: invariant (I-recursion theory. (3 is any limit ordinal in the following.
A set A E Lp is called (3-recursively enumerable ( (3-r.e. ) if it is definable over LC) by some TIformula q
(see
Friedman and Sacks [I]). Observe that this is really a very intuitive definition. Generate successively the levels L0,LI,..,L8 to (3
,..
(rd(3)
of the constructible hierarchy up
. Enumerate at every step
which satisfy Ll k q(z)
y
those elements z into A
and which have not already been
enumerated before. The example shows that the general concept of a recursively enumerable set -as described by Post 121 in 1944- does not require any strong closure conditions of the underlying domain like admissiblity. *The author is supported by the Heisenberg-program of the Deutsche Forschungsgemeinschaft.
A function f : Lp* Llj is called (3-recursive if its graph is (3-r.e.. Consider the group of all p-recursive functions which map Lp one-one onto Lp
together with composition of maps. A
property of subsets of Lp is called G-invariant sively invariant if for every f e G some set B this property if and only if f[B]
or recur-
ti Lp
has
has it.
Felix Klein suggested in his Erlanger Programm (1872) to define branches of mathematics in terms of a space X and a group G of transformations acting on that space., The branch of mathematics determined by
X and G is the study of G-invar-
iant properties. Lp
and the previously defined group G determine for
(3= w classical recursion theory and for
(3 =
(a admissible)
OL
a-recursion theory. Let us now look whether there is an appropriate notion of finiteness in invariant @-recursion theory. Any recursively invariant class of ()-recursive bounded (i .e. y e (3 )
S
P for some
L
subsets of Lo is a candidate for such a notion. It is
obvious that there exiets a largest such class which we call I. We will see in the following that there are several good reasons to take I as the notion of finiteness in invariant (3-recursion theory. The elements of I are called i-finite sets. If (3 is an admissible ordinal u IY
then i-finite is equivalent to
-finite. Define wlcf() :=
(3-recursive f : 6 -B
the least 6 + (3 (there exists some
0
with range unbounded in
0
).
. In fact
Lemma 1 : I is a (3-recursive subset of L,,
I = f x E Lp 1 Lo b [cardinality(x) < vlcfp ] f
.
The proof is not difficult but relies heavily on the fine structure of L
(collapsing of Skolem hulls).
Every (3-r.e. set A
can be enumerated in
clcfp many
steps, i.e. there exists a (l-recursive function f : ulcfp L
y x ~ A y , such that A = U{Ay
every single x e A
I x *
elcfp)
. Thus
is enumerated after an i-finite number of
steps. It is easy to see that I is the only recursively invariant class of @-recursive bounded subsets of
Lo which
is in this sense coherent with the notion of a 0-r.e. set. Further for any i-finite subset K for some i-finite
of A
we have K 5 Ar
. This property is important for priority
constructions. It implies that every true i-finite neighborhood condition about A settles down at some point of the-construction. Another useful property is the following : Every (3-recursive subset of an i-finite set is again i-finite.
Consider for any limit ordinal p the structure ePP :=
where
Z
e := c P Lo
*
the canonical (3-recursive truth predicate for
.
I
and T is
AO Lo
formulas
dlo is construed as a set with urelements as in L o is the underlying collection in Barwise [3], where LP I
-
of urelements.
Theorem 2 :
&e
is an admissible structure with urelements.
Foreveryset M c L p
M
Z1(4) zR
is
:
=
M
is
Further the sets in the structure
Z l ( ~ ,Q ) P
.
P are exactly the i-finite
sets. Corollary 3 : Let
c,
p-r.e.
Assume that (3
is a countable limit ordinal.
Lo be some 9-recursive language and let T be a
set of sentences in the language
*
with i-finite dis-
junctions and conjunctions.
If every i-finite set To G T has a model, then T has a model. Proof of the Corollary : Apply the Barwise Compactness Theorem
[?I
to C L p .
Remark : The compactness theorem does not hold for any larger notion of Itfinitenin L p
.
The preceding compactness theorem (Corollary 3) can be used to show that for every countable (3
invariant p-recursion
theory can be characterized in terms of absoluteness or model theoretic invariance as this effect was called by Kreisel [ 4 ] . The concept of model theoretic invariance is useful in order to understand the mathematical meaning of computations in recursion theory. The situation is analogous as in first order logic where the completeness theorem gives a mathematical meaning to formal proofs. The connection between model theoretic invariance and recursive invariance is the following : The notion of a "finiten
set is recursively invariant in every recursion theory which can be characterized in terms of model theoretic invariance.
In order to get an intrinsic notion of a computation relative to an oracle B
q
Lp one can extend the Kripke equation
calculus in a canonical way. The essential rule allows to survey i-finite many subcomputations in a computation. Every computation has the structure of an i-finitely branching tree :
axioms of the form 2
L & B , F ( E , ~ )=
C
B
,
where F
is some rudimentary function We say that A tic function of A
is computable from B
if the characteris-
can be computed from B
in this equation
calculus. We say that A
is i-finitely computable from B
this can be done by using i-finite computations only.
B
if is
called semigeneric if every equation which can be computed from
B can be computed from B with an i-finite computation.For a semigeneric set B
the preceding two notions of reducibility
coincide for every set A
.
Lemma 4 : a)
For countable (3
A
is computable from
implicitly invariantly definable from B b)
A
0-r.e.
B iff A is
(see [ 6 ] ) .
is i-finitely computable from B iff there exists a set W
such that for every x e Lo
:
cA(x) = i
3 i-finite,K,H(< X , ~ , K , B I E W
O
K
A
B
4
H
A
5
Lp
- B)
(cA is the characteristic function of A ). c) B is semigeneric iff for every relation R s L p x L P the form R(X,Y)
3 i-finite K,H(
H
with W (3-r.e.
x,g,K,H, e W
and dom R
i-finite function f
G
Ks B
A
i-finite
A
of
H o L B- B)
there exists an
.
R with &om f = dom R
The relation in b) is not transitive and therefore we codsider instead the following reducibility relation : A Sf B
:
there exists a (3-r.e. set W
such that for
all i-finite H,,H2
3i-finite K,H(rW
r
KGB
A
H sLp
- B).
The associated equivalence classes are called i-degrees. For admissible
Q
they coincide with the d-degrees.
Every i-degree i a recursively invariant. The i-degree 0 (i.e. the equivalence class of the empty set) contains exactly the fl-recursive eets. As usual one gets immediately that there exists a maximal (3-r.e. i-degree 0' which is strictly greater than 0
. There is no trivial way to show the existence of an
intermediate
(3
-r. e. i-degree.
Except for a few
(where it is still open) one can
understand the structure of the (9-r.e. (+degrees (see[l]) substructure of the I-degrees.
as a
Theorem 5 : sets A
,B
The proof
For every limit ordinal such that A #i B
there exist p-r.e.
and B $i A
.
(3 is
is given in the most interesting case where
strongly inadmissible (i.e. u fcfp c (3*) by a priority construction following Friedman [ 5 ] . The combinatorial principle 0 can here be eleminated (this may
be helpful for applications to
other inadmissible sets). f3 the i-degrees coincide with the
Oberelcve that for every
degrees in the admissible collapse
a O Thus . Theorem
5
con-
tains as a special case the solution of Post's Problem for some enormously fat admissible sets.
Theorem 6 : For every limit ordinal (\ there exist sets A
,B
such that A
is not computable from A
is not computable from B
.
p-r.e. and B
The proof is slightly more difficult than the proof of Theorem
5
. We make
neede
A
and B in addition semigeneric. For this one
0.
Theorem 7 : For many strongly inadmissible (3 there are (3-r.e. sets A
such that 0 ci A but S
pi A
for every simple set
S (see [6] for the definition of simple). The proof
is a first example of an infinite preservation stra-
tegy in the strongly inadmissible case. Besides new combinatorial argument
0 it uses a
. We expect that refinements of the
applied strategy will lead to a splitting theorem for i-degrees.
We had mentioned the definition of a semigeneric set because at this point an important new effect arises in the step from al-
to (3-recursion theory. Several equivalent definitions
of nhyperregularninot-recurslon theopy lead to different classes in (I-recursion theory. For some strongly Inadmissible (3 there are C)-r.e. sets B such that every computation from B has an i-finite length but B is not eemigeneric. All details can be found in the forthcoming paper [6].
Literature
[I]
S.D.Friedman and G.E.Sacks, Inadmissible recursion theory, Bul1.Am.Math.Soc.
83 (1977) 255-256
[2] E.L.Post, Recursively enumerable sets of positive integers and their decision problems, Bu1l.Am.Math.Soc.
50 (1944)
284-3 1 6 (31 J.Barwise, Admissible Sets and Structures, (Springer,Berlin, 1975)
[4] G.Kreir~e1, Model theoretic invariants: applications to reoursive and hyperarithmetic operations, in: J.W.Addison,
.
L.Henkin,, A.Tarsk1, eds ,The Theory of Models (~orth Holland ,Amsterdam,1965)
f 51 S.D.Friedman, Post's problem without admissibility, to appear (63 W .Maass, Recureively invariant (3-recursion theory, to appear
.
