INTRODUCTION TO T H E THEORY OF FINITE AUTOMATA
N. E. KOBRINSKII AND
B. A. TRAKHTENBROT T R A N S L A T E D FROM THE R...
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INTRODUCTION TO T H E THEORY OF FINITE AUTOMATA
N. E. KOBRINSKII AND
B. A. TRAKHTENBROT T R A N S L A T E D FROM THE R U S S I A N
Translation edited by . I C. . Shepherdson University of Bristol
1965
NORTH -H 0LLAND PUBLISHING COMPANY AMSTERDAM
1965 N O R T H - H O L L A N D P U B L I S H I N G COMPANY
Nopart of this book may be reproduced iii any form by print, photoprint, microfilm or any other means without written permission .from the publisher
PREFACE
K
on
book
V
PREFACE
VI
by
j
by
abstract synthesis;
structural synthesis
do
PREFACE
on
by by book, on no
book. book
book book 5), 111 V
by 5
by book
A.
by S.
0. B. book. S.
on
Yu.
A.
on by
E. KOBRINSKII B. A. TRAKHTENBROT
INTRODUCTION
1.
for book, by
on
A a
by
At,
by
by
by
of
on
t,
on -
-
t
for This term must be distinguished from the term “determinate”, which will be used in
a different sense. 1
2
INTRODUCTION
by
by
internal state on memoryless automata. by
0.1. second
on dt=O),
on by
l
i
l
l
0.1 TABLE 0.1
0 0
0 1
1 1
0
0 0 0 0
0 0 0 0 1 1 1 1
1
0
0
1
0
0
0 1 2
0 0 0
3
0
4 5
6
7 8 9
0 - non-excited output, 1 -excited output.
0 0 1 1
1 0
1 0
1 0 1
3
INTRODUCTION
by
by
of
A
on 100 100 100 ....
by
2.
by by
0.2
4
INTRODUCTION
A *D
-
q
B
E
A,
C, D,
I
Z.
0.2b Z
{A,
{B, C } ,
(A, E, C}, (B,
no
loops
do is analogue automata1.
0.3
for
Fig. 0.3 See E. Kobrinskii, Matematicheskie mashiny nepreryvnogo deystviya (Mathematical analogue machines), Gostekhizdat, Moscow 1954.
5
INTRODUCTION
:
a l l x l - %2XZ
= bl,
+ %2X2
= b,,
- %1X1
0.3b to
3. logical nets. book.
by C . E.
by V. by
on by by V.
1
S. V. B.
V.
A. W. Burks and G. B. Wright, Theory of logical nets, Proc. IRE (1953) 41, no. 10.
6
INTRODUCTION
by L.
on 1953
by
W.
by A
1956
by C . 1956
on 3.
analysis of an automaton
4.
by
on p. 5 .
1
by
2
no.
1956.
3
1956; (1956) 261, no. 3, 4; (1956) 6, no. 2, (1959)3, no. 3 ; (1959)3, no. 6;
by
3,
1960;
1.
(1960)21, no. 2.
11,
(1960) 21, no. 3; by 21, no. 2; Yu.
(1960) 1958; 12,2.
M.
7
INTRODUCTION
by
by by
by E. F. A.
by
on by
by This approach is typical the “abstract” theory automata. This approach is typical of the “structural” theory automata. E. F. Moore, Gedanken experiments on sequential machines, Automata Studies, ed. by C. E. Shannon and J. MacCarthy (Annals of Mathematics Studies no. 34) Princeton Univ. Press 1956. 4 B. A. Trakhtenbrot, Operators which can be realised in logical nets, DAN SSSR (1957) 112, no. 6.
8
JNTRODUCTION
5.
This -
by 0.1
p.
by by no
no
by
A by
on
Known as inclusion tables. See M. A. Gavrilov, The theory of realy-contact circuits, Izd. AN SSSR, Moscow Leningrad 1950.
9
INTRODUCTION
1.
0 t
o 5),
1.1 by
1.3). 1
0
1.1 f ( x l , x2,.... x),
xi As
i = 1,2, .... m).
14
[I.1
THE ELEMENTS OF LOGIC
:
1. 2
complement
negation
x,
x”. A
x,
X
1);
x x x,&x2)’,
2
11. x 1 - x 2
conjunction
x2”. 111. x 1V x 2 ,
x1 x2 logical addition,
disjunction
x1 x2 IV. x 1 + x 2 , implication,
xl,
“x1
x2”
x2”.
xl,
x2”.
x1 x 2 is V . xlszx2, equivalence, x2”. x1 x2
“xl
x2”
“xl
TABLE1.2
0 .
1
1
0 0 1 1 . .
_
-
-
0 1 0 1
0 0 0 1
0
1 1
1
0
~
1 0 0
I
- _ _ _ ~
-.
We shall henceforth be indicating this operation by using either notation. The point will generally be omitted.
_
1.11
15
GENERAL REMARKS
by
...,
...,”
on,
by
EXAMPLE.
A
p. 4, B
C,
A,
C,
B,
: :
Z = ( A - D V B * CV A - E * CV B . E * D ) . on the algebra of logic,
logic of propositions mathematical logic.
logic of predicates.
on
16
II.2
THE ELEMENTS OF LOGIC
by 0. on
1.2.
of
of
:
q,
I. pl, q 2 , ..., x l , x2, ..., zl, z 2 , ._.
1
0
(%)
'u 91
2.
..., x,y , z,
for
23
(aV w , (%&B),
(a+%), (?I = 93)
3. on
1
2.
EXAMPLE.
((@&Y>
-+
(x v Y,)
v
x,y (x&y) ((x&y)-+(xV y ) )
by ((x&y)+(xVy))
1
(xVy)
z
(((x&y)-(xVy))z)
speaking "&" 0. B. Lupanov, A fizika (1958) 1, pp. 120-140.
"-"
1.1,
of circuit synthesis, Izv. vyssh. uchebn. zavedenii, Radio-
1.21
17
THE ELEMENTS OF THE ALGEBRA OF LOGIC
‘u
B
(2l.B) “&’
“*”
‘u
(b&Y>
+
.( &Y
z)
-+
9
z),
x&y+z,
(x &Z)
(P
-
“V”
“&’
(4
v
(x &Y>
f>>9
v
(P 4 71, (x &Y p-qV?,
v z,
(x & y )
+
-+
x
(x
v Y>)
3
v Y),
x&y+xVy.
v (x &z(2 v y)
X Z V Z , xyVxz(2V y).
‘u
B
equivdent
‘u
B
by 0
‘u by
1
23 “3
B”
‘21
B
18
[I.2
THE ELEMENTS OP LOGIC
TABLE1.4
I
__
x2
0 0 0
0 0
0
0
1 1 1 1
0 0
0
0 1 ___-
._
EXAMPLES. (xl &x2) &x,
x1 &(x2 8 ~ x 3 ) . 1.4; (x1&xz)&x3
x1&(xz&x3)). “0” “1” (x1&xz)&x3 x1&(xz&x3), ( X I &x2)
&x,
x1 &(xz
(x1&xz)&x3
XVy
&x3)
&(xZ&xj).
x&y. 1.5). TABLE1.5
in
1.4
1.21
19
THE ELEMENTS OF THE ALGEBRA OF LOGIC
“0” 1.5
xXy,
2Vy
1.
associativity, (xl (xl
= X I &(x, &x3),
&X,)&X,
v x,) v x3 = x1 v (xz v x3),
(x1 = x z) = x3
= x1
= (x, = x3).
: (xl &xz) &x3 = x1 &(x2 &x3) = x1 &x, &x3, (xl (XI
v x,) v x3 = x1 v (x, v x3) = x1 v x, v x3, = xz) = x3 = X I = (xz = x3) = x, = x,
= x3.
2. commutativity : x1 &x, = xz &xl, x1 XI
v x,
= x,
= x,
= x,
v
XI,
= x, .
3.
by
of idempotence
for 4.
(x1 ‘ j .2) (XI &xz)
&x3
= (x1 &x3)
v x3 = (xl v
with respect to disjunction.
v
&x3) x3) &(xz xg). (XZ
v
distributivity of conjunction dis~ribu~ivity
When referring to functions in the algebra of logic, we shall often use expressions of the typef(x,y) (“a function f in the algebra of logic, of the variables x and y”), g ( X I , x2, ..., xm) (“a function g in the algebra of logic, of the variables XI, XZ, ..., x,”) etc. When some function in the algebra of logic is referred to, one may imagine it to be specified by a tabulation defined by some formula in the logic of propositions.
20
11.2
THE ELEMENTS OF LOGIC
of disjunction with respect to conjunction. “&’,
V
“*”
V
(XI ~ 2 ) * ~=3 ~ 1 . ~ ~ 3 2 . ~ 3 ,
X1 ’ X 2
v
x3
= (XI
v X3)*(X2 v XJ.
5. :
x&O=O,
x&l=x, xvo=x, xVl=l. 1
0
x x x
x 1
x
x 6.
x&X=O,
xVX=l, -
x
x&X”. law of the excluded middle
X
x ‘‘x
X
7. xy,
xj
aj
0
1
x0 = X ,
x 1 =x.
x;’.xy ... X,bk
1.21
21
THE ELEMENTS OF THE ALGEBRA OF LOGIC
x 1 = a, , x 2 = a2, ..., xk = a,. xj aj) 5 x1x2x3x4 aj= xj= 1. 21xZx324 a1=04=0, a2=a3= 1 ) x1=x4=0, x2=x3=l. x , x l , x2,...
by 1 . Every function in the algebra of logic can be represented in
the f o r m : f ( X 1 , X 2 , X 3 7**.,Xk,Xk+lr '.-,x,)
( m 2 1)
where the symbol
V
x;'x;' ... xpf (alya2,..., a,, xk+ 1, ...,xm),
(1.1)
0 1 9 . . .,bk
denotes the logical sum over all alya2,03,..., a,. expansion of a,function with respect to k variables. ol,02,..., a,.
x 1= a l , x 2 = o 2 , ..., xk=ok. x k + l , ..., x,,,), 3, 5 7,
f(a,, a2,..., a,,
by az, ..., ok, xk+l,..., x,).
x1
f ( x l ,x2,x 3 )
x2.
By
f (XI,x 2 x3) ~
= 2122-f
OYx3)
v
21x2f
(O, x3) V xlfzf
v X3)
V xixzf
X3).
oj
a, =0
a2= 1.
x1
x2) 7 5,
on f(xl,x 2 , x 3 )
f(0, 1 , x 3 ) ,
1, x3).
2. Every function in the algebra of logic can be expressed by means of conjunction, disjunction and negation.
22
D.2
T H E ELEMENTS OF LOGIC
by
f (x1,x2,...)x,)
=
v...,
02 x,0 1x2
... xZm,”f(a1,a2, ...,a,).
a2,..., a,)
0
by
1,
v
- * * , x m= )
x;lx?
~ ( u I , u ~ , . . . , u=~1)
5
...x;m ,
(1.3)
a2,..., a,,)= 1 . principal disjunctive normal form
a,, a2,..., a, (1.3)
on
(1.2)
am
01,
x2, ...,
(1.3)
a (a,, ..., a ,) ...,x,) 1.
by
by EXAMPLE. x2,
1.6.
on
Solution.
x2,
V
V
V
-xy disjunctive terms
TABLE 1.6 Xl
X2
x3
1.21
23
THE ELEMENTS OF THE ALGEBRA OF LOGIC
(x;'
V xy V ...V xirn) (xzl V xy V ...V xzm)
1.
by
g
< g < 2").
(a1,az,..., a ,) g
g,
m
g = i= 1
g g(ol, az,..., a , ) [a,(g), az(g),
(a1,a2,..., a , ) g
..., om(g)].
m=4,
5.
13
rn Pj = X;'X~'
... x i m
C j = x;' V
x y V ... V xz'"
az,..., a,,,), 1 With the provisos noted above, relating to functions identically equal to 1 and those identically equal to 0.
24
II.2
THE ELEMENTS OF LOGIC
o1,Q,,
<j&(f v
x2)
=
f v (x1 &x,), do
on by by
by
pi
DEFINITION. pip =0.
pj
(o=1),
(o=
p'
= xlxz
p" = Z 1 x g ;
p'p" = 0.
DEFINITION. A absorbs
p 1V p , V ...V p , p
pj p+pl v p , v ... v p k
absorption p 1 V p 2 V...V p k p V p , V p , V ...vpk = p i V p , V ...v p k
p,
p + p 1 V p 2 V ... V p k = I , p
34
[I.3
THE ELEMENTS OF LOGIC
o n p (f f x V f =f )
f
p,
p 1 V p , V ...Vpk p 1 V p 2V.. . vpk).
fx,
by
p (fx
v...
THEOREM 4. For a disjunction p 1 v p , vpk to absorb the elementary conjunction p , which is not orthogonal to each elementary conjunction out of p 1Vp,V ...v p k , it is necessary and suficient that each conjunction pi can be represented in the ,form pi=p,!pr (the conjunctions p : and pi' can degenerate to satisfying thefollowing conditions: k
1.
VpY=l,
2.
p+p;p; . . . p ; = 1 .
i= 1
p;
pi
pi'
p,
ofp,. EXAMPLE.
D
V
= 2 1 x ' 2 ~ 3 X I X Z X ~V
2 1 ~ 2 x 3V
~2~3x4.
Df
4
V p2 V p 3
= R 1 R 2 ~ 3V ~ 1 ~ 2 V x f3 1 ~ 3 ~ 4 Df
p
= x2x3x4. Df
pl Vp, V p 3
3
1 See Yu. Zhuravlev, The separability of subsets of an n-dimensional unit cube, Trudy Yablonskiy, Functional constructions Matem. in-ta im. V. A. Steklova (1958) 51; S. in k-valued logic, &id.
1.31
35
THE MINIMISATION OF FORMULAE
p
+ p ; p ; p ; = x2x3xq -+ x2x3xq =
1.
4 p1 V p , V p 3
D
= 2122x3 V X ~ X Z XV~2 1 ~ 2 x 3 .
pi pr = p'-'x;
A
by
p,
is
p
adjacent
p>
x,
p? = p r - ' z z .
isolated no
p.
pi, p i ,
..., p;.
cohesion: pi
v p: = pr-Ixp v pr-12;
= pr-1.
- 1.
by -1
by s
v p; v ... v p;) v (p;-I v p;-l v ... v p ; - ' ) v ... V (p;-" v p;-" v ... v p;-) v ... v (p:-"" v p;-=+' v ... v p;-"")'
(p;
1..
"blindalley"
a
36
[I.3
THE ELEMENTS OF LOGIC
no
EXAMPLE.
by
f(xl, x2,x3)
V
~12223
V x ~ X Z XV~~
2122x3
1 ~ 2 x 3 .
no
2nd
X1X2,
(212223
9
2122x3)
9
(xlf2x3
3
xlxZx3)
9
(Xlf2x3
9
x122x3)
*
x ~ x ~ , X Z X ~ .
XiX2
V
V
~ 1 x 3 22x3.
By
is iiX2 V
{XlXZ.f3,
RlX2~3)
( ~ 1 2 2
~ 1 x 3 .
by ~ x1x2x3) 3 ,
by
by A
EXAMPLE.
A
=(al V x2 V x3)&(x1V Z 2 V Df
A = 2132 V
X1X3
V
V
~ 2 2 . 3 22x3
V
V
~ 1 x 2 ~ 1 x 3 ,
1.31
31
THE MINIMISATION OF FORMULAE
2nd
S
= 2182
V~
V
,
2 2 3 ~1x3
Df
by p3 = 1 &(Il &I2);
p1 = pip; = Il&I2; p2 = p;p; = I
3
&x,;
= ~2
p3 = p i p ; = 1 &(XI & x3);
p2 = XI
p1
= I2& I 1
,
1 &(x2&%),
&23;
P3 =
& x3;
p2 = x3 &XI.
V p i = 1. I
i= 1
p ; p ; p ; =I1I3,
x1x2
22x3.
S
A
this
A, :
-
-
permutation of different variants. A indeterminate coeficientsl.
2nd
1954
by
38
THE ELEMENTS OF LOGIC
n.3
is
(1.7)
by EXAMPLE. x 2 , x3),
by by
1.9.
1.31
39
MMIMJSATION OF FORMULAE
TABLE 1.9 x1
X2
0 0 0 0
0 1 0 1 0 1
1 1 1
0 0 0
0 1 1 -
AO-AO-AO-AoO-12 - A13 00 00 000 = A23 = A123 1 - 2 - 3 -A1 -A01 - 1 3 - AZ3 11 = A011 1 2 3 = A' - A:; = -
1
= A2 = A:: = AiQ = A'''1 2 3 -
1 -
= A'''-
123
l1
-
=
O1
=
- A101 1 2 3 = O* :
A11 12 All 12
v v
A001 123"I? A110 123 = l 7 A111 1 2 3 = 1.
do
we 110 A123
= A::: = 0 ,
40
12.3
THE ELEMENTS OF LOGIC
A:::
= 1,
A:
= 1.
f ( X I , x 2 , x 3 ) = XlXZ v 21xZ.,x3.
on n n
by up
TABLE1.10 X324j5
~
i i x324x5
x3x425
x3x4x5
I l l ~-
m' m"
f(xl, x2,..., xm)= 1
x;',
x y , ..., xLm,
1
all = 1.
by l. ~
This situation is similar to that which arises, e.g. in the binary coding of some alphabet, when the number of distinguishablecode groups is greater than that of the distinguishable symbols the alphabet. Thus, in coding the decimal numbers, 0, 1, 2, 3, 4, 5 , 6, 7, 8, 9, by uniform code groups of binary digits, the minimum number necessary in each group is four. Only ten of the sixteen possible code groups are used, and the remainder are unused.
1.31
41
THE MINIMISATION OF FORMULAE
formula
by
by
all :
no
by 01 1
1.11.
I11 1.12. TABLE 1.11 X2
x1
x3 ~
TABLE I. 12 x1 ~
0 0 0 0 1 1
1 1
____
x2
O 0 1
1 0 0 1 1
I
x3
i
0l
0 1
0
1 0 1
42
THE ELEMENTS OF
:
1.4.
2
k 2k
f(ul, u2, ..., ui,..., u,) essentially depends on
vi,
CONSEQUENCE. f(al,o ~..., , vi, ..., a,,)
aj ( j # i ) , ui
iji:
(1.10)
Z= u1 V u2V ... V uiV ...V v,, on ui. uiifvj=aj=O(j#i),andthefunctionZ’isequal EXAMPLES.
Z‘ = v 1 v 2 - ~ q - ~ u , 2 is
(j#i).
-
~~~
by
1
A
by on A
i (1957) 18, no. 2;V. 1955 ; I. Coll. by 1959; A. 1951 ; 1954.
1.41
EXPANSION OF FUNCTIONS IN
43
ALGEBRA OF LDGIC
TABLE 1.13
f(vl, v2, v3, v4)=Vl V O2 V O3 V V4
on
v2, v3, v4) v1 ci2 = ci3 = a, = 1,
f
(vl, u2, O3, u4)
= vl
.
u2,
cil =u3 =a4=
1,
v3
a l = a 4 = 1 , a2=0.
o l = a 2 = 0 , u4=1; ul=O, u 2 = u 4 = 1
u2, v3, a4)=v3.
v4, cil = u2= 1,
f('17
O29 O39 u4)
= '4
*
u3= 0 ,
44
[I.4
THE ELEMENTS OF LOGIC
TABLE 1.14
1
03 ~
04
~ _ _
I
~~
51
v v2 v 53 v 54 ~
61
i 1
1
03 03 03
04 u4 04
04 04
0 0 1
0 1 0
0 0 1 1 1
0 1 0 1 1
-
1
~
03
__
_
_
1 1.
03 03
v4
0 0 O 1
274
v4
~.
54
~.
1.14.
m =k + I f(x1, x2,
...)x k , yl, y2, ..., y,) > (xl, x, x g , ..., x k )
by y,, ..., y,) by ( x l , x2,..., 2k
1.10 xk)
by
fj(nl,n2?
3). nl, n,,
...)nk,
y1, YZ,
yl) 7
..., g k ,
j < 2k*
by 0.
k 5. Let aJunction P(w,,w l , ..., w n - J (where ~ = 2 ~ ) ~ - _ _ _.-
0 . B. Lupanov, A method synthesising circuits, Izv. vyssh. uchebn. zavedenii, Radiofizika (1958) 1, p. 120-140 (see Sec. 5).
f(X1,X2,
...,xk,y1,y2,
=
{$O
. * . , y l )= Cx1, x2,
.
.... x k , f (O, O, .... O, O, y l , y 2 ? .... Y l ) ]
$1 [ X l , X Z , . . - , X k , f ( O , o ,
.
.
. . . , O , l,yl,YZ? . . * ? Y l ) ]
....xk, .... 1, O, YZ?.... YL)] . . . . . . . . . . . . . . . . . . . . $2 [XI
$2k-1
(1.11)
x2,
[x1,X2,..-,xk,f(1,1,...,1,
l,Yl,yZ, . . . , y i ) ] } .
by on
1 Note that, as follows from the definition of the functionQ(correspondingto the sense of a strict "or" in natural language), if q j = 1, then w j @ q j = w j @ 1 = Wf;and if C i j = 0, then w j @ c j j = wj@O = w j .
46
[I.4
THE ELEMENTS OF LOGIC
by by EXAMPLES 1. (
on
isf(xlx2, y ) n = 2k= 4,
x2.
xI
F(w0, ~
1 ~ , 2~ , 3
F(wO,
=) W O
V W I V w2 V w 3 . < j
(T)
on
f(t),
on
on
DEFINITION. An g ( t )= 8 [ f ( t ) ]is non-anticipatory operator, t,
T
t (condition of determinacy), on T < t (absence of memory).
I
by on (7)
by
11.21
69
THE DETERMINACY OF OPERATORS
A :
0
1
1
0 crl o2 ... om
rl r2
by
... r,,
n 1'
=
x2?
*..7
xm),
z2 =fZ(x1,~2,..*,xm),
. . . . . . . zn
= fn
( x 1 7 x2,
*.*7
xm).
by (m=n = 11.1:
by
: z =2.
23
3, q2(xl,x2, x3)
0 0 0 1 1 1
ql(xl,x2,x 3 )
by
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1
0 0 1
0 1 1
1 0 0
70
m.2
OPERATORS. LOGICAL NETS
TABLE11.2 x1
22
___-
0 0
0
0 1 1 (1 1 0 0 1 (0)
1 0 1 0
1 1 0 0 1 1
1
0 1
-_-
by
11.2
11.1
by
V 2 1 x 2 ~ 3V 2 1 x 2 ~ 3V XV~ 2 1 x 2 2 3 V 2 1 ~ 2 x 3V
ZI = 2 1 2 2 x 3 ~2
=X ~ X ,
ZI = 2 1 x 3 z 2
V~
= 2, ( x 2
2 2 3 ,
of
...f ( t ) ...
f(
... g ( t ) ...
~ 1 2 2 2 3 .
v x3)v X ] (x2 v x 3 ) .
2. Constant operators.
on
x I x ~ X ~ ,
t
on
t
z>t by
A g (1) g
generated by the constant operator. autonomous devices, by
on
no
.. g ( t ) ...,
11.21
71
THE DETERMINACY OF OPERATORS
by A
by fix
trees, m
infnite information tree a$rst-rank node root of the tree, call first-rank branches.
0 ni
Fig. 2.4
Fig. 2.3
-
-
0, 1, ..., m- 1
2.3 m = 3;
0, second-rank nodes; m
- second-rank
branches -
m on
m
(i = 1, 2, 3, ...), i-th level of the tree.
mi
2.4
f ( t ) E A‘. 0
on. by
on on A.
101 ... .
on on i >1
2.4
12
LI1.2
OPERATORS. LQGICAL NETS
i, f ( l ) f ( 2 )...f( t ) ...
g(1) g(2)
...
t=t,,
do
t,,
z >t ,
on g(z)
A
B.
T,
A,
loaded tree TA-B,
B
Fig. 2.5
T.,
on
T,
B, B.
A
2.5,
2010.. .
1011 ...
by
g ( t )= 0 [ f ( t ) ] , t 6 t,
by g(t)
B
g’(t)
z1
y
by
T,
>j .
y
T, f ( l I f ( 2 )...f ( j - 1 ) (j-
bough of rank j
T,
a
T. T
T
y.
T,
8,
0
a residual operator of rank j by 0, single residual operator of rank 1).
DEFINITION. distinguishable by indistinguislzable.
0
1
11.31
75
BOUNDED-DETERMINATEOPERATORS
trees
boughs
on EXAMPLES , 1.
truth operator
2.
constant operator
j 101001000100001 ... by 3.
(j-
bough bough
0 0
f
f
..f ( j -
Or
1)
8,
Or. afinite number of (functionally) distinguishable states.
DEFINITION. determinate operator) it
bounded ( a boundedoperator with Jinite memory
unbounded.
DEFINITION. weight k of a bounded-determinate operator
76
tII.3
OPERATORS. LOGICAL NETS
1,
CONSEQUENCE. truth operator. REMARK. a.
2
by basis of a deternzinate operator
K
K,
on
K
K K
Canonical equations of a bounded-determinate operator. 8 K. Q= q l , ..., g K internal alphubet alphabet of states), 6' Q,
Q, 1)
f(t) g(t),
by q(t)
by
on Q
11.31
I1
BOUNDED-DETERMINATE OPERATORS
on
( a sequence of states). q(
EXAMPLE.
2.8. 1 1 10... on
1001 ... qoqlqlql ...
by
... z ( l ) z ( 2 )...
...,
40
Fig. 2.5
..
x'(
.... If 8
2'(1)2'(2)...
z(t)=z'(z) 5.
x(t)=x'(z) t
is x(t) =
.'(.),I
4 ( t ) = 4'(2)
z(i) = ~ ' ( 7 )
q(t
\
+ 1) = q ' ( 2 + [ x ( t ) = x'(z)]
[ q ( t )= q'(t)], [z(t)=z'(z)] [ q ( t + I)=q'(z+ q(t+l) We shall declare that the state qo is ascribed to the
z(t)
the tree.
78
[II.3
OPERATORS. LOGICAL NETS
@(x,q )
Y (x,q )
A
Q
x, z , q
is 11.2
p.
by
EXAMPLE. 0 q1
2.8,
11.3. TABLE11.3 II
:
z ( t ) = x ( t ) * q ( t )v a ( t ) y ( t ) ,
q(t
+ 1) = x ( t ) * q ( t )v
z(f>=x(t)Oy(t), q(t+ I)=x(t)@q(t) q( 1)=0.
NOTE.
x ( t ) , q(t), z ( t )
p. 66);
zl( t ) , z2(I),z3( t ) , ..., z n ( t )
z(t)
z 2 , ..., z,,
n
qo
11.31
79
BOUNDED-DETERMINATE OPERATORS
. . . . . . . . . . . . . . . . . . . . . . . I
. . . . . . . . . . . . . . . . . . . . . . .
xl,
I
.... xm,zl, .... zn, q l , q2, .... qk
@
Y
is
bound
@, Yy
the canonical equations (functions1, tables) on the alphabets X , 2, Q 2 Q is ~~
The following terms are also used in the literature: “internal ( Y ) transfer function” and “external (@) transfer function”.
80
[II.3
OPERATORS. LOGICAL NETS
TABLE 11.4
0 1 0 1 0 1 0 1
0 1 1 0 1 0 0 1 -
z(t) x(t)
8
q(t+ q(t);
q(1)
q o = q l ; Fig. 2.9
q,, 00,01, 10, 11.
\ 0
Fig. 2.9
Fig. 2.10
q1
11.31
81
BOUNDED-DETERMINATE OPERATORS
11.4.
2.10
8,
by by
2.9. 8, q( = q,,,
by
by place-by-
place sequential summing operator,
If
q1 as
by
O(8,
x(t)
z(t) q ( t + 1) q(t)):
z ( t ) = z,x,q
v x,z,q v 2,R,q v x,x,q = XI 0 x, 0 q , q ( t + 1) = XlXZ v x,q v x,q . by
11.5 q
00,01, 10, 11; y1
y,
q.
TABLE11.5
8, 11.4
82
q(
tII.4
OPERATORS. LOGICAL NETS
10, (K=2),
00
10
01
qo);
11
qi,qj indistinguishable with respect to the Y,
canonical functions @
q( 1) =qi, q(
=q j ;
distinguishable. A
q,, q j Y
@
‘uGy
K
K
u ‘,
the common bounded-determinate operator of the pair of canonicalfunctions SP, Y, weight of the common bounded-determinate operator, partial bounded-determinate operator fixing the initial condition. no
do
(e,}
‘uQY 8
do
‘u @, Y
11.4. Elements (elementary automata) and circuits
A
K
11.41
83
ELEMENTS A N D CIRCUITS
by on
by
by
circuit.
complex automaton elements,
elementary automata,
-
by
by junctional algorithm of the circuit). element
algorithm
circuit
by Q , m input channels X , (m=O, 1,2, ...), 3) n output channels Zn(n= 1,2, 3, ...), 4) element.
cell
zl(t)=
[~l(t),xz(f), -*-,xrn(t),q(t)]
. . . . . . . . . . . . . . .
7
z n ( t ) = G n [ ~ l ( t ) , ~ Z ( t ) , . . . Y ~ , ( f ) , q ( ‘,) ]
q(t
+
= !P [ ~ l ( t ) , ~ z ,...,xm(t),q(t)], (t)
xi(t),z j ( t ) ,q ( t ) X i , Zi
Q.
Our definition of an elementary automaton coincides in the main with that of a finite automaton, used by the authors of studies in which the behaviour of automata is considered abstraction from their design.
84
p1.4
OPERATORS. LOGICAL NETS
by poles (input
output by
by on zj=Qj(xl, ..., x,, q ) Y(xl, x 2 , ..., x,, q )
q(t).
t = 1,2, 3, ..., internal memory, x l , xz,..., x, q,
on external alphabets.
k volume of the internal memory Q zero. z i = D i ( x l , ..., xi, ..., x,, q l , ..., qk), on xi.
Qji
subordination
xi causally determines xi
precede
zi
imaginary variable
xi;
zi
Qi.
by by
xj
x,
no xi Qi on
xi
(2.11).
{‘3JIi}.
. .-
~
A. W. Burks 41, no. 10.
G.
Wright, Theory of logical nets. Proceedings of IRE, 1953,
11.41
85
ELEMENTS AND CIRCLTTS
(fm,}
DEFINITION. A circuit
(aj}
set of poZes of the circuit set
of cells
set of channels
graph nodes
branches (channels) poles
cells
DEFINITION. A
no
input
p++p
p
1. If a
a
g
a)
p
p), 2. 3.
p
CI
a>
a>
-p.
-+p, b>
+y,
a>
+.y.
by
EXAMPLE.
'#Ix, 1.
fm,, '%Re, fmd
%Ix
2. fm,
q(
=0
:
z(t)=x(t)m(t), q(t+ 1)=x(t)@q(t). 1
A group may contain many specimens of the same element out of the set
{mm,}.
86
tII.4
OPERATORS. LOGICAL NETS
3. $'T,I
2 z(t) =xl(t)@xz(t).
4. % I,
z ( 0 = 4 (4
q(t
9
+ 1) = x ( t ) .
z(t)
md,
i
1 Fig. 2.1 1
3,
Fig. 2.12
Fig. 2.1 3
r-I
I I I I I
I
I I
I I 1
I
I I L- -------
Fig. 2.1 5 x(t)
2.15, 2.16, 2.17
{mE, m, ma,
2.18
md>.
Fig. 2.14
11.41
87
ELEMENTS AND CIRCUITS X
$4
Fig. 2.16
Fig. 2.17
88
D1.4
OPERATORS. LOGICAL NETS
no
no 2.16b, ‘%I?,
z4
!Illd. z1
2.15, by 2.17
z3. A
no
by 2.18.
2.15
2.17
+z4
Fig. 2.18
z4
z3.
proper improper a vicious circle arises in the transmission of infornzation
11.41
89
ELEMENTS AND CIRCUITS
function of the circuit.
{YJli}. on :
I.
of
no of
by :
by
2. by logical net.
11.
zp
z,
90
[II.5
OPERATORS. LOGICAL NETS
21
. .. ... “” .. .
YZ”
t
111, -
z,,
z,
logical net,
11.5. Logical nets
N
E(N)
Note that Burks and Wright here use the term “completely proper logical net”; their term “logical net” corresponds to our term “circuit”.
11.51
91
LOGICAL NElS
z1(t)= Z 4 ( t ) O X ( t ) O 4 ( t ) ,
4(t
+ 1) = Z 4 ( t ) X ( t ) v 4(t)z4(t) v 4 ( t ) x ( t ) , (4 = 24 ( t ) 0 P (4
22
P(t
+ 1) = Z 4 ( t )
3
0P(t), ~ ~ ( t ) = ~ ~ ( t ) ~ z ~ ( t r)( ,t )z, r~( t( +t l)) = z , ( t ) . ,
*
(2.12)
92
[IIS
OPERATORS. LOGICAL NETS
E(N) by E(N)
5 by
A A
or
E(N) A
C
B
0 the system E(N) is proper the proper organisation of the circuit. A,
A, B
C,
C by SZ, n, v
m,p
9,
E(SZ2)
i . . . . . . . . . . . . . . .
f2
m+p
2.20b.
That is, 0 is an operator with anticipation.
n+v
I
11.51
93
LOGICAL NETS
zn+l
zn+v
Fig. 2.20
E(Q,),
1
by
EXAMPLE. A
S
2, S
by
do no
94
nI.5
OPERATORS. LOGICAL NETS
Q
xi
xi
4 of
,I. J
.
.
.I
. .....
21
2"
Fig. 2.21
by
by
f2 by xi
xi)
xi.
EXAMPLE. by 2.
zi xj
feedback loop
~
111,
Q
11.51
95
LOGICAL NETS
zi(t) xj.
B
zi
zi
xi
zi(t)
xj(t).
non-subordination zi
zi
xj
d by
by [xl( t ) , . . . , ~ ~ - ~
Gi As
zi
....
xi. Q, Q,
no
Q,
Q,.
52,
Q, Q,, Q,
9, cascade.
96
[US
OPERATORS. LOGICAL NETS
A z
by
x l , x2.
I
z to x1
zn (0)
Fig. 2.22
(a)
Fig.
I"' Fig. 2.23
z by
Fig.
is
loop.
x2)
11.51
91
LOGICAL NETS
%TIi) DEFINITION 1.
{mil. 2. An 3.
xi, xj
4.
xi,
zi
5.
$
identification of the inputs
3 4 coupling to the out-
put. cascade loop,
2.24
2.25. 2.23).
For any logical net N , the corresponding systenz of equations E(N) is correct, and consequently speciJies a boundeddeterminate operator 0,.
Fig. 2.25
N
DEFINITION. A
%,
implement
0,
8,. by
1-4
1.
8, 2.
ON,
0,. %*, ON2,
SN, 8,. 1
In the following exposition, the words “embodying the set 9Jb” arc often omitted.
98
tII.6
OPERATOKS. LO(;ICAL NETS
XN
3. x(t)
xi(t)
xj(t),
illN, 4. x,(t) zi(t)= ai[xl (t),
@ i [ x , ( t ) ,. . . ] ,
illN,
...I,
XNp.
N k,, ..., k , k=k,-k,..-k,
v
k
of volume of the
illN. internal store
(1)
N.
11.6. The realisability of bounded-determinate operators. Problems in the
analysis and synthesis of automata
of a 3: :
any bounded-determinate operator is
physically realisable. “t-ealisability”.
.
. . . . . .
(2.15)
11.61
99
BOUNDEDDETERMINATE OPERATORS
qi,,qi2,..., q i ,
no
% WL) ,%TII; z,
%TI:
EXAMPLE. %TI,
2.26 on
z1
+ 22 Fig. 2.26
4's
z2
4 z3 x1, x2, x3, x4.
z2
4. z3
4, 6. on
100
[II.6
OPERATORS. LOGICAL NETS
m
n n
m ~1
=
(XI, ..*,xm),
. . . . . . .
z, = @, (x,,
N, f2(x1,.... xm), fi[x,, .... y,, ....
..., x,).
z , =f , ( y , , ....
N2
z2= z2=
.... x m ] ) , N,
N, is
complete
:
by
(2.16)
. . . . . . . . . . . . . . . *I q k ( t + I ) = Y k [ X 1 ( t ) ....tX,(f),41(t),...,qk(t)], J Y,, .... Uk
....
z,, .... z,, q1 in the
.... q k ( t ) of
@,,
.... @,
11.61
BOUNDED-DETERMINATEOPERATORS
~ ( t=) q ( t ) , q ( t
101
+ 1) = x ( t ) .
2.28,
'i
.....
; i : , i t
,I..
.*[,
.
I
Fig. 2.28
4z2
Fig. 2.27
9X= {%Il, ..., %In} 0,
8.
3
I
by V.
1 V. B. Kudryavtsev, Questions of completeness for automata systems, DAN SSSR (1960) 130, no. 6; id., A completeness theorem for a class of automata without feedback, DAN SSSR (1960) 132, no. 2.
102
“1.6
OPFRATORS. LOGICAL NETS
111.
on
Analysis of an automaton.
Synthesis of an automaton.
for
!Dl
9X
so,
IV-
VII, by
0’
0
1. The
of .y(I),
_..
DEFINITION 2.
z(t)=O
z-shift (z
[x(t)]
z’=O’[.x(t)],
t : z(t+z)=z’(t).
z(t)=U [ ~ ( t ) ]
xo
z’(t)=U’[x(t)],
...
by
0‘
11.61
103
BOUNDED-DETERMINATE OPERATORS
2’(2), z’(3), ...,
t(l)x0 ... X O ,
t(2)x,
... xo, ...
-.(a- 1) times
by
...)
(a- 1) letters
...)
z’
(a- 1 ) letters
...)
z’(3),
...
(a- 1 ) letters
DEFINITION 3. 0 with a shift
Y
T
synthesis with a shift (in a time-scale)
N realises (in a time-scale 8.
N
C H A P T E R 111
O P E R A T O R S OF P H Y S I C A L ELEMENTS
111.1. Introductory remarks
on
fast operation
reliability,
cost, on on
book
a
on
2
1. valve transistorised flip-flops,
104
3 magnetic
III.21
105
VALVE AND TRANSISTORISED ELEMENTS
111.2. Valve and transistorised elements 111.2.1. PHYSICAL PRINCIPLES
on triodes
3.1) R,
E,
3.2).
pentodes
Egi
on
f+
€+
Fig. 3.1
Fig. 3.2
on
by by rheostat coupling 3.3). by capacitive coupling
is
3.4).
by
See A. M. Bonch-Bruevich, The use of electronic valves in experimental physics, Gostekhizdat, Moscow 1956.
106
[111.2
OPERATORS OF PHYSICAL ELEMENTS
Ri
by C
ORa 4
-
ii T---
l
A
6.1
TC
OR2 I
i -
d EFig. 3.3
Fig. 3.4
C R, R,R,/(R,
+
is
R, R,))
C R , R 2 / ( R l+ R,).
R, E,
R,Ri/(R, + Ri) is
for
t
f+
+ E- - R , / ( R , + R2).
ia
ORa
r
0.
4---A
k Fig. 3.5
E.3
Fig. 3.6
111.21
107
VALVE AND I’RANSISTORISED ELEMENTS
on by by
negation
4
inverter,
i9
Fig. 3.1
3.5 V.
-
by i, on E,
E, i,=f(E,) i, on
3.7 ig = q (E,). 3.5.
A 1
E, EL
A
up on by
T, 3.8.
Ea=E+,
108
[III.2
OPERATORS OF PHYSICAL ELEMENTS
on
on c1
Eg=Eeq
3.9)
by 1
b
E,.
of
;
-- -.1
OR.
'9
/€+ Fig. 3.10
Fig. 3.9
T is
U,
on
by V
3.10).
A
U,,
111.2)
109
VALVE AND TRANSISTORISED ELEMENTS
EL
A As
up on
U, =E ,
T,
1 pentodes
Z -
9
4. 3.12
Fig. 3.11
no by
3.11.
A no Z
R, 1
E,,
U,.
cathode follower U,, E, U. If
3.13
double cathode follower
R,. E, 1
If
I10
DII.2
OPERATORS OF PHYSICAL ELEMENTS
R,,
on
power amplifiers.
diodes
Fig. 3.13
--..-
........ ..... ..... ..... . 1 €2
Fig. 3.14
conjunction
diodes disjunction m
on
3.14.
R R,,), 1, 2, 3, ..., m :
A
m
R & Rfd. Two E, E,
1
111.21
111
VALVE AND TRANSISTORISED ELEMENTS
up
R
(Uh=E+),
1 E , - E,,
R.
't
Fig. 3.15
R 9 R,,
U.
0
E,,
by is
on
i,
V
I) 3.15.
R,,# 0,
by
E, :
U
=
U,
+ A'U .
on
A'U
R,,
(3.5)
m,
A'u= f (Rfd,
mo)
mo on A'U by 3.16
112
OPERATORS OF PHYSICAL ELEMENTS
tIII.2
3), a=
1/R
E , - E,
A‘U
As
on
Fig. 3.16
no of
(Rout)
-g_,FE’ R
I
A
E,,
Fig. 3.17
Uzz E , .
A
U,, U = U,+A‘U+A”U,
(3.6)
111.21
113
VALVE AND TRANSISTORISED ELEMENTS
R, -gR by
A"U R,,,IR.
on
Rout Rout
by
R.
by
E r n * - E - T 2
. . .... .... m
&
Fig. 3.18
Fig. 3.19
C,
R
E,.
'c
3RC, < 2 ,
(3.8) R.
by
R,,
on
R :
(3.9)
114
IIII.2
OPERATORS OF PHYSICAL ELEMENTS
3.19
As
disjunction
0, 1,
El,
E,.
R $ R,,
E, on on E+ : U
= E+ - A’U.
(3.10)
Rout.
(3.11) AU
on by
AU
by by
point-contact
junction transistors.
111.21
115
A N D TRANSISTORISED ELEMENTS
by by
A book, :
f
E+
Fig. 3.21
collector, emitter, base
3.21,
C
E
116
DII.2
OPERATORS OF PHYSICAL ELEMENTS
a
_L Fig. 3.22
by :
of -
-
by by on 3.22
by
111.21
117
VALVE A N D TRANSISTORISED ELEMENTS
TI
T,. i, =f( U,) ib= 'p( Ub)
(ib=
U, =
'C
Fig. 3.23
3.23.
three
cut-oflregion,
I1 I11
linear amplifring condition,
saturation
region,
I
111.
vice versa. ib = cp
(Ub)
U, =
3.24.
We
TI E-
3.25) by
a R,
A UL = U,
TI.
1
118
[III.2
OPERATORS OF PHYSICAL ELEMENTS
T,.
i , = q (U,)
3.26),
do by
U, a= R,R,/(R,+
R,),
R,
ub
U, > 0 5 V u, =0.1 v
I ib Fig. 3.24
Fig. 3.25
3.25)
Fig. 3.26
T,.
UL
TI
T, 1.
11r.21
119
VALVE AND TRANSISTORISED ELEMENTS
111.2.2. THE REALISATION OF ELEMENTARY LOGICAL OPERATORS
on
0
I
Z = I
z,
Y
TX2
(a)
Fig. 3.27
Fig. 3.28
on on -
-
of
negation.
1,
high
1.
T ( x ) = x.
(3.12)
120
[III.2
OPERATORS OF PHYSICAL ELEMENTS
1, disjunction
K , (XI,
x2)
= x1
v x2
9
K(x)=x.
= F, "4
Fig. 3.29
b.
3.28
by
A
3.29.
by 111.1. of P ( x , , x2) = 21 v
by
22
3
0 1 0
1 1 1 0
111.21
VALVE AND TRANSISTORISED mwmrs
121
3.30 on As by 3.31, 3.32).
by by
-p(b)
Fig. 3.30
t Fig. 3.31
i-
jz=x1vx2'x3
(b)
(a)
Fig. 3.32
122
[III.2
OPERATORS OF PHYSICAL ELEMENTS
Dd(x1,x2,..., x,) = XIv ~2 v ... v x,, D, (x1,x2, ...,x,) = xlxZ ... x,. by
3.28
3.30,
on on on It by by Fc by by
by
Fd by F,
by Fd
by
3.33 by F, z=
xi=O. R,
z=o.
U,,
111.21
123
VALVE A N D TRANSISTORISED ELEMENTS
on
3 2
1
P E+
Fig. 3 . 3 3
z.
z = T (XI) T(
T
~ 2 ) ( ~ 3 )= 21Z223
.
(3.16)
by T, (xl, x2, ..., x,). 3.34
A
if is
by
124
[III.2
OPERATORS OF PHYSICAL ELEMENTS
( V xi), n
E-
by
- ibR,n
2 U,.
A, (U,)
3.35
n
by
9‘
ORC
Q. Fig. 3.34
?
Fig. 3.35
by
of of
z = T ( x , ) V T ( X , ) V ... V T(x,,)=Z;-,Vi!zV
... Vi!”.
(3.17)
A i (i= 1, 2, ..., n) x,=O.
P,,(xl, x2,..., x,,). As by by by
F:
F:
rn
111.21
125
VALVE A N D TRANSISTORISED ELEMENTS
z’ = x;x;
...x; ... xj, ... x,,,
z” = xnxn 1 2 ... Xk
by
by Fk
z’
... xi”....:x
x;
xj”
F:.
Fi u‘
u”.
3.36b. I
,I
z = z z .
Fi
Fd,
v x; v ... v x; v ... v x; v ... v xi, zn = x; v x; v ... v xk“v ... v xj”v ... v x; z’ = x;
z = Z‘V
F,
Zl’.
Fd z’ = x’ x’x’ 1 2 39 z” = x; x;
v v x;
FZ
126
pII.3
OPERATORS OF PHYSICAL ELEMENTS
F,
Fd
versa
vice u’
Fig. 3.37b.
u”,
(jj
(jji
OR1 OR2
111.3. Flip-flops 111.3.1. THE PHYSICAL PRINCIPLES OF FLIP-FLOP OPERATION’
AJEip-flop by
A See L. A. Meyerovich, L. G . Zelichenko, Pulse Technique, Sovetskoe Radio, Moscow A. M. Bonch-Bruyevich, The Use of Electronic Valves in Experimental Physics, Gostekhizdat, Moscow 1956.
111.31
127
FLIP-FLOPS
on
3.38,
Fig. 3.38
Fig. 3.39
E,
R,.
R U
i
3.39
R, i
by U = E - iR, i = f (U)
(3.18)
128
pII.3
OPERATORS OF PHYSICAL ELEMENTS
by
do 1/R
PI, P,
P,
P,. by U,, by i,.
by
R
i, by by Pik)
of by
on
E,,
b on on El.
on
by by
3.40;
U' =f (El),
111.31
129
FZIP-FLOPS
3.41.
by
T', on
on
T",
by
J
Fig. 3.40
Fig. 3.41
Fig. 3.42
no
on on
by
on
on T'
A
U'
T"
130
pII.3
OPERATORS OF PHYSICAL ELEMENTS
T'
T" (A)
3.40), e. :
U'
=f
U'
= E'
(3.19)
(E'),
-e.
(3.20)
(3.20)
(3.19)
3.41.
e
45"
on
Pik) e@)
(U,e ) a
e@)
U
3.43).
11,111). I I1 U
m\ II-
I-
__
Fig. 3.43
by for
III.31
131
FLIP-FLDPS
I11
by by 3.42, on staticflip-flop,
(a)
Fig. 3.44
by
3.44b
3.38.
R
R", by
3.45
I11
E,
a=
(l/R;)
132
[III.3
OPERATORS OF PHYSICAL ELEMENTS
PI,, PI,I).
PII
Pill on
E,
by
\m Fig. 3.45
A
B
(T'
A B (T'
A
T"
T" on
3.44
by
50 -
-
by
III.31
133
FLIP-FLOPS
3.44,
by
3.46). 3.44
3.45)
of
t
U'
U"
Fig. 3.46
A
A
T",
T'
B
A
3.45
PI T' T" 3.46
of on
A
a
134
rIII.3
OPERATORS OF PHYSICAL ELEMENTS
by
R,
R,
C, by by
of
3.48
I
PE,
1.-
i
ORC
p. Fig. 3.48
I ,
111.31
135
FLIP-FLOPS
by on
Re Re
111.3.2. LOGICAL OPERATORS OF FLIP-FLOP CIRCUITS]
by
Fig. 3.49
qA,
q B= FA.
v‘
v“ cpA=l,
qA=0.
d’
0
1.
v’
pA(t)= 1
qA(t)=O
qA(t+
no
qA(t+
=O
on
See M. L. Tsetlin, Non-primitive circuits, Collection “Problemy kibernetiki” no. 1, Fizmatgiz, Moscow 1958, pp. 2 3 4 5 .
136
[III.3
OPERATORS OF PHYSICAL ELEMENTS
d(t)= 1
d’(t)= 1
has : TABLE 111.2 UU ( t ) -
0 0 1 1 0 0 1 1
0 0 0 0 1 1 1 1
0 1 0 1 0
1 0 1
~
: VA(z
+ l) = ll’ ( t ) q A ( t ) v v ” ( t ) ( P A ( t ) ,
(3.21)
z ( t ) = v”(t)(PA(t).
(3.22) (3.22) 3.50.
(3.21)
(3.22)
qB(t+
zB(t). ‘PB(t
+
= v”(t)’PB(t)
v
v’(t)(PB (t),
(3.23) (3.24)
111.31
137
FLIP-FLOPS
v‘(t)=v“(t) = v ( t )
: qA(t+
I)=u(t)@-,(t)V
~(t>(~A(t)=v(t)0(~A(t),
(t> = ( t ) q A ( t ) .
FL-fl
Fl -
+%
+PA
Fig. 3.51
3.51
Fig. 3.52
3.52 by
If
T, T’
zA(t), qA(t)=O
x”(t)= 1.
qA(t+
+
(PA(t
ZA(t)
v (PA(t)-’f’(t),
= @A(t)x”(t)
= @A(t)x“(t).
cpA=O.
x’(t)
x”(t) = 5’(t),
x‘(t) = v”(t),
+ l) = @A(t)5’(t) v P A ( t ) 0 ” ( t ) 7
qA(t
ZA(t) = @A(t)V’(t).
v’
U”
by
a
dynamic flip-flop.
a z(t+
1, z(t+2)= 1, ...,
138
[111.4
OPERATORS OF PHYSICAL ELEMENTS
on u‘= 1
u”(t+s)=O.
of
t, t + s
3.54. z ( t ) = v’(t)V z ( t - 1)6”(t - l ) u a ( t -
ZA
ZE
3.53
v,(t)
u,(t)
= Z(t)
= u’(t)
v z(t -
- 1).
111.4. Ferromagnetic elements 1 111.4.1.
~~
on no. 2, 1959, pp. 1958.
by
111.41
139
FERROMAGNETIC ELEMENTS
on by by
Fig. 3.55
I
Fig. 3.56
B B=B(H),
H.
on on H, B =B ( H )
by
140
[II1.4
OPERATORS OF PHYSICAL ELEMENTS
+B,
-Br.
+B, 1.
by
by
transformer
A
I) by
1,
11) 0.
1, +B,.
+B,.
+B,, (3.30) -B,,
-Br. pulse
d,B
= B,
+ B,.
(3.31)
A 0
by
u = - wsW
(3.32) S
121.41
141
FERROMAGNETIC ELEMENTS
:
AB u = - ws-10-8,
(3.33)
z
on
t
(Br,Bm),
(3.35) on U,,/U, on
a = Br/B,,,,
coeficient of rectangularity 0.9-0.98.
a
no on
by by write-on read-out. 0
1,
0
142
[IIIA
OPERATORS OF PHYSICAL ELEMENTS
Fig. 3.57
m
by A, 1
0
a 1
0
by no A
, T,
(C).
1, A
0
A,
no 1
0 on
111.41
143
FERROMAGNETIC ELEMEN
(B). A,,
no A,,
on
on
For
by
by by
In is
bound
on
0
In by
by
Devices which ensure that pulses are of the specified duration. Strobing consists of “excising” narrow sections from the output pulses by means of synchronising pulses at strictly determined moments. This produces an accurate synchronisation of the pulses which are fed to the cores in subsequent stages.
144
G
[III.4
OPERATORS OF PHYSICAL ELEMENTS
H
single-crystal circuits
n t
Fig. 3.59
(D) pulses
td
A,
3.60
0 by
A,. A
A,
,
A is
by
zr,
by A,
A,
z,
by
by Two-cycle circuits,
odd
111.41
145
FERROMAGNETIC ELEMENTS
A
odd
on. Three-cycle transformer circuits
T,
146
rIII.4
OPERATORS OF PHYSICAL ELEMENTS
supby
good ferrodiode choke elements1 by 3.63
Win
C (0,
W,,,.
D,) by
E
B. (11)
do holding duty
See L. M. Shekhtman, An algebraicmethod of synthesisinga ferrodiode choke circuit, in collection “Voprosy radioelektroniki” series 12, no. 18, 1959.
111.41
147
FERROMAGNETIC ELEMENIS
A on
3.64),
S
3.63). AQPA
( Wout)
on
Z
Fig. 3.64
iOut = E/Z
no
ACDEF on
S
EGQA A
Z no t (t+
148
[III.4
OPERATORS OF PHYSICAL ELEMENTS
(t) ( t + 1).
3.65 by C,
A
3.64).
C2 by
A on C, C,
A on
no
C,
by
C,
A
C,
by
on
t, (t+l),
vice
C,
versa.
( W,.), a
( Wi). A
III.41
1 49
FERROMAGNETIC ELEMENTS
A
A
ih
ih Hrh
Hrh
- Hih < H O by
Hih Ho ih
A on
4 a 111.4.2. OPERATORS OF FERROMAGNETIC ELEMENTS
3.55,
I1
(b)
(C)
Fig. 3.66
b
by by 1
A
by
by 1 1
0 0,
0 0
1.
-
~-
-
~-
0 0 0 0
0 1 0 1 0
0 0 1 1 0 0 1 1
1 1 1 1
0 0 0 0 0
0 0 1
0 1 0
1
0 1
1
1
0
0
1
'
_ _ x z.
q,
y,
111.3. 111.3 :
by
(4 = 4 (t>Y (4 4(t+
9
i
(3.36)
[4(t)Vx(t)]Y(t).)
1
m
by on
be
on
3.67 t
111.41
151
FERROMAGNETIC ELEMENTS
1
q (t)
y
u ( t ) =0 z(t) = 0 ,
q (t
+ 1) = j j ( t ) .
(t+ u= 1 z(t
q(t
+ 1) = q ( t + 1 ) U ( t + 1) = j j ( t ) , +
=0.
*-z=g
Fig. 3.67
Fig. 3.68
H y)
H
y);
(3.38)
G(1,Y) =
y ( t )=0 z(t
+ 1) = x ( t ) ,
x ( t ) = 1, G(x,O) = x,
by 3.68). u ( t )=0),
q(t
+ 1) = XI ( t ) v x2(t) v v * a .
q (t)=O, Xk(t)
on z(t
+ 1) = 4 ( t + 1) =
(t)
v x2 ( t ) v ... v Xk(t)
m
152
DII.4
OPERATORS OF PHYSICAL ELEMENTS
I
m m =1= 1, z(t
+ 1) = x(t)Jqt). by
G (w) H (x, v). by by z = x l x 2 . . . x k = % l v d f 2 v
...v%k.
%F- 1I k y - y ~
xp-
U
1
-
... . .....
X
(‘
Fig. 3.69
x l , x 2 , ..., x k .
+ 1) = d f j ( t ) , xi”(f+ 2 ) = v X S ( t + 1) = v Z j ( t ) ,
XJ(t
z(t
+
k
k
j= 1
j= 1
k
=
v j= 1
%j((t) =
xl(t)xZ(t)
.a.
Xk(t).
III.41
153
FERROMAGNETIC ELEMENTS
LT=
3.
by
by 3.70).
Fig. 3.70
’k
Fig. 3.71
on
3.71):
3.72
.’
( k = 2 ) by
(3.40);
z(t
+ 2) = x i (t)xz ( t ) = G [G(xi, O), H(1, xz)]
(3.41)
Here, and in certain cases later on, we shall omit the time parameter ( t ) in the operator notation to make the formulae more easily comprehensible.
154
[II1.4
OPERATORS OF PHYSICAL ELEMENTS
1
by
on
Fig. 3.73 3.72
a=2.
by
~ = 2 ,
on
3.73; b
is A on
q 1,
0.
go by x
by z
y
i,
~-
by
p. 1 4 6 .
111.41
155
FERROMAGNETIC ELEMENTS
TABLE111.4 4 (t>
1 0
1 1 1 0
1
111.4 4(t
+
= 4(02(0
v Y (9.
(3.42)
z (t)=O,
z(t+l)=q(t+l),)
4(t
+ 2) = 4 ( t ) .
(3.43)
) on
q (t)= 1,
t y (t)=O.
by (3.42)
(3.43), z(t
+ 1) = x ( t ) .
R(x,O) = 2 . q ( t )= 1 z(t
+ 1) = y ( t )
(3.44)
x ( t )= O =y.
by by
by
156
[III. 4
OPERATORS OF PHYSICAL ELEMENTS
by
on
A
3.74,
Fig. 3.14
q ( t ) = 1, z(t
+
= x(t)
v y(t)= by
v R(1,y).
u=2:
CHAPTER IV
A N A L Y S I S O F AUTOMATA
IV.l. General remarks 11,
on
on
{%R,},
{‘$Xi} 2.16b
zl,
This 11,
us
~
_
_
Here, and very often later, the term ‘net’ will refer to a logical net. 157
158
[IV. 1
ANALYSIS OF AUTOMATA
K.
minimising
1.
2.
by
3. 5.
In 11
%Ii
by
by EXAMPLE.
Wd).
of
3; k.
DEFINITION 2. 81+k8,)
k-distinguishable by k; 8, Wke2). 81 “ k 8 2 )
8, 81”k82
8,
8, strictly k-distinguishable 81-k-,8,. k,
E. F, Moore, Gedanken experiments with sequential machines. Automata Studies, ed. C. E. Shannon and J. McCarthy, Princeton 1956. 2 This statement is true in general for determinate operators.
166
[IV.3
ANALYSIS OF AUTOMATA
8,
8,a 02,
02,
strictly k-distinguishes qi
qj
0;
-
-
0. A, A.
k
A
3. degree of distinguishability
0
xi, x2,..., xk x2,x 3 , ..., xk qi qj
qj; qJ. xi.
qi q: by
x2,xg,..., x k
on qi
x,. c,, ... ti,
q:
qJ,
by
qj
qi y < k - 1, qi qj.
qJ
xl, c,,, ..., ti,
by y+ 1 < k
y 1, z ( t ) = 1
x3
t
x
t = 1, z(
1.2
1
z
x , ( t ) , x 2 ( t ) ,x 3 ( t ) . 0
z
by
w. z (t)= (t
>
[x(t)0 x (t -
v (t =
w.
(5.11)
v.41
205
THE LANGUAGE OF PROPOSITIONAL LOGIC
internal 41(t
+ 1) = x ( t ) j
(5.12)
41 (1) = u
q1
=u
0, I).
q1
1)
> 1) [ X ( t ) 0 41 (t)]
z ( t ) = (t
v ( t = 1) w . t> 1 q2(t),
t = 1.
by
t> 1
42(t+1)=1, 4 2 (1) = 0.
l), : (t) = 42
(9 (0 0 41 v q 2 ( t ) w 41(t + 1) = x(t),
7
42(t+1)=1, 41 (1) = u
9
4 2 0 ) = 0.
~ ( t )ql(t+ , I), q 2 ( t + x ( t ) , q1( t ) , q 2 ( t ) z ( t ) = w , ql(t+l)=O, q , ( t + I ) = l j .
23 x=q1=q2=0 V. 10 3,
00
w.
01
do
206
w.4
METHODS OF SPECIFYING OPERATORS
3. t.
wi
by
x(t)
t 3
qz, q3, t
V.11.
t3.
v.41
207
THE LANGUAGE OF PROPOSITIONAL LOGIC
'42(1) = 0 , 43(1) = 1 .
Z
( t ) = q 2 ( t )9 3 (4 [x
v
0 41
q2
(t) 4 3
( t )w 1
v
v 42 (t> (0 v 42 (4 j.3
w2
43
(t>w 3 >
41(t+ l ) = x ( t ) , 42 (t
+ 1 ) = 4 2 ( t ) 0 43 ( 9
43(t
+ 1) = & ) q ' 3 ( 4 = w +,
q1
q2
7
q3( 1) = 1.
=0
by z ( t ) = @ [ x l ( t ) , x l ( t - 1 ) , x 1 ( t - 2 ))"., x , ( t > , x , ( l t>v,
by
,...]
t = 1, 2, ..., v
z(t)
t = 1, t = 2 , ..., t = v , t > v , xi t-s, 41 ( t
+ 1) = x ( t ) ,
42(t
+ 1) = 41(%
.
. . . . . .
4 s (t
+ 1) = qs-l(t).
NOTE. 1)
q(t+l)=x(t),
2)
t= 1
2, ..., v, V.9),
4.
by z (t)
= @ [ x (~t ) , x1( t
-
...,z ( t - l),z ( t -
w is an arbitrary constant (0 or 1). [a]denotes the least of the whole numbers m, for which a
< m.
...]
v
208
[V.4
METHODS OF SPECIFYING OPERATORS
q l ( t + l)=x(t), q2(t+ l)=ql(t), p l ( t + l)=z(t),p2(t+ 1) = P I ( $ ... . (5.14)
z ( t ) = @ 1x1 ( t ) ,41 (0,42 (0,...
( 0 9
...
P2
9
+ 1) = x ( t ) , q 2 ( t + 1) = 41 ( t ) , 41 ( t
. . . . . . .
Pi ( t
+ 1) = @1.t
(t>,41 (Q, q.2
( 0 7
.*-,PI (t>,P2(&
P2 ( t + 1) = P l ( 9
7
7
by z(t)=[z(t-l)@x(t)](t> p l ( t + 1) =z(t)
p l ( l ) @ x ( l ) =x(l).
p,(l)
0x(t), P1 ( t + 1) = P1 ( t )0 x ( t ) z ( t ) = P1(t)
p,(l)
no
in t - 1 , t -2 z(tz(t-2),
...
(5.14)
VI.
z(t) quantijiers)
by
v.41
209
LANGUAGE OF PROPOSITIONAL LOGIC
EXAMPLEI. as B,
A
AB, z1
