Mathematical Logic and Foundations of Set Theory, Proceedings of an International Colloquium Held Under the Auspices of The Israel Academy of Sciences and Humanities

149 Pages·1970·3.12 MB·English

by  Yehoshua Bar-Hillel (Eds.)

STUDIES IN LOGIC
A N D
T H E F O U N D A T I O N S OF MATHEMATICS
Editors
A. HEYTING, Amsterdam
A. MOSTOWSKI, Warszawa
A. ROBINSON, New Haven
P. SUPPES, Stanford
Advisory Editorial Board
Y. BAR-HILLEL, Jerusalem
K. L. D E B 0 UV8R E, Sanfa Clara
H. HERMES, Freiburg i/Br.
J. HINTIKKA, Helsinki
J. C. SHEPHERDSON, Brisfol
E. P. SPECKER, Zurich
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM LONDON
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MATHEMATICAL LOGIC AND
FOUNDATIONS OF SET THEORY
PROCEEDINGS OF AN INTERNATIONAL COLLOQUIUM
HELD UNDER THE AUSPICES OF
THE ISRAEL ACADEMY OF SCIENCES AND HUMANITIES
JERUSALEM, 11-14 NOVEMBER 1968
Edited by
YEH 0S H UA B A R-H I LLEL
Professor of Logic and Philosophy of Science
The Hebrew University of Jerusalem, Israel
1970
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM LONDON
0
@ NORTH-HOLLAND PUBLISHING COMPANY - 1970
All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,
recording or otherwise, without the prior permission of the Copyright owner.
Library of Congress Catalog Card Number 73-97195
ISBN 7204 2255 8
PUBLISHERS:
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
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NORTH-HOLLAND PUBLISHING COMPANY, LTD LONDON
PRINTED IN ISRAEL
PREFACE
This volume comprises seven of the eight addresses presented before the
International Colloquium on Mathematical Logic and Foundations of Set
Theory held at the Academy Building in Jerusalem, Israel, on November
11-14, 1968. The Colloquium was sponsored by the Israel Academy of
Sciences and Humanities, the Mathematical Institute of the Hebrew Uni-
versity of Jerusalem, and the International Mathematical Union, and was
dedicated to the memory of Professor Abraham A. Fraenkel, one of the
founders of axiomatic set theory, the beloved teacher of three of the eight
invited speakers and of the editor, and a founding member of the Israel
Academy.
The bulk of the support for the Colloquium was granted by the Inter-
national Mathematical Union, while additional aid came from the Israel
Academy which also served as the host of the Colloquium and whose staff
was most helpful with regard to all technical arrangements.
At the opening public session of the Colloquium, held on the evening
of November 11, at the Hebrew University of Jerusalem, Professor Y. Bar-
Hillel presented (in Hebrew) an appreciation of Professor Fraenkel’s
contribution to the Philosophy of Mathematics, and Professor Alfred
Tarski gave a lecture on Some Reflections on Recent Developments in the
Foundations of Set Theory.
No attempt was made to unify the contributors’ notation, terminology
and bibliographical style, but it is the editor’s feeling that no appreciable
harm was caused thereby; on the other hand, it facilitated the speedy
publication of this volume. Similarly, it was quickly decided that no index
was called for.
WEAKLY DEFINABLE RELATIONS AND SPECIAL AUTOMATA*
BY
MICHAEL 0. RABIN
In this paper we consider monadic second-order theories and study
problems of definability. As a by-product we obtain certain decidability
results. Let Nz= (Tyro,rl) be the structure of two successor functions
(see $1). Let L be the monadic second-order language appropriate for M2
which has individual variables x, y, z, .. . , ranging over elements of T,
finite-set variables a,p, y, ... , ranging over finite subsets of T, and set
variables A,B,C, ..., ranging over arbitrary subsets of T. A relation
H E P(T)” between subsets of T is called definable in the second-order
theory (language) of M2 if for some formula F(A,, ...,A n) of L
The relation H is weakly-definable if (1) holds for a formula F(A,, ...,A n)
containing just individual and finite-set quantifiers.
In [6] we have characterized the definable relations by means of finite
automata operating on infinite trees. This result was used to solve the
decision problem of the second-order theory of Nz. T his in turn entailed
the decidability of many theories.
Here we introduce the notion of a special automaton on infinite trees and
use it to characterize the weakly definable sets. An automaton on infinite
trees may be viewed as representing a relation H c P(T)” for some n.
It turns out that a relation H c P(T)” is weakly definable if and only if
both H and complement P(T)” - H are represented by appropriate special
automata. On the other hand, not every relation H represented by a special
automaton is weakly definable. Rather, a relation H G P(T)” is represented
by a special automaton if and only if (1) holds with a formula F(A,, ...,A ,)
in prenex form which has only existential arbitrary-set quantifiers. This
yields the following syntactical result. A formula F(A 1, .. . , A ,) is equivalent
(in MZ)w ith some formula G(A,, ..., A,) containing only finite-set quanti-
fiers, if and only if F is equivalent to some prenex formula F,(A,, ... )A,)
having only existential arbitrary-set quantifiers, and also to some prenex
,,
formula Fz(A ..., A,) having only universal arbitrary-set quantifiers.
This research was sponsored under Contract No. N00014 69 C 0192, U.S. Office
of Naval Research, Information Systems Branch, in Jerusalem.
1
2 MICHAEL 0. RABIN
As a by-product of the characterization of weakly definbale relations
we get the solution of certain decision problems. In [6] we have shown
that the weak second-order theory of a unary function, and the weak second-
order theory of linearly ordered sets (see [4]), are decidable. These results
were actually corollaries of stronger theorems concerning the correspond-
ing full monadic second-order theories. Here we deduce the same decida-
bility results using the information concerning weakly definable relations
and special automata. Also the many applications by D. M. Gabbay of
[3] to the solution of the decision problem of various logical calculi fol-
low already from Theorem 24 of the present paper.
1. Notations and basic standard definitions
We shall employ the standard notations and terminology concerning
sets, mappings, structures, and logical calculi, used in [6].
As usual, each natural number n is the set of all smaller numbers. Thus
0 = pl, 1 = {0} , 2 = (0,l }, and n = {O,l, ..., n-l} .An n-termed sequence
is a mapping x: n + A. The sequence x is also called a word on A. The
ith coordinate of the sequence is x(i), 0 5 i < n, and will sometimes be
.
denoted by xi. The length l(x) of x is l(x) = n The sequence x will also
be written as (xo, .. ., x,- l). If x = (xo, .. . , x,- and y = (yo, ..., ym- 1) then
xy will denote the sequence (xo, ...,x ,-~ , yo, ...,y,- l). We have
+
l(xy) = l(x) l(y). The sequence (xo) of length one will also be written
as xo . Thus x = xoxl ... x,- 1. The unique empty sequence of length 0
will be denoted by A.
For each i < o,t he projection pi is the function which is defined by
pi(x) = xi for x = (X~,...,X,,-~i )<, n.
The injinite binary tree is the set T = {0,1}* of all finite words on (0, l}.
The elements x E Tare the nodes of T. For x E T, the nodes x0,xl are called
the immediate successors of x. The empty word A is called the root of T.
Our language is suggested by the following picture. The lowest node of T
is the root A. The root branches up to the (say) left into the node 0 and
to the right into the node 1. The node 0 branches into 00 and 01; the node 1
branches into 10 and 11. And so on ad infinitum.
On T we define a partial-ordering by x 6 y (x is an initial of y) if and
.
only if 3z[y = XZ] If x 5 y and x # y then we shall write x < y.
1 .
For x E T, the subtree T, with roof x is defined by T, = {y y E T,x 5 y}
Thus TA = T.
A path 7c of a tree T, is a set 7c c T, satisfying 1) x ~ n2;) f or YE? L,
either ~ O E Xo r yl EX, but not both; 3) 7c is the smallest subset of T,
satisfying 1-2.
Note that if 7c c Tis a path and X , ~ E Rt,h en x 5 y or y 5 x.
WEAKLY DEFINABLE RELATIONS AND SPECIAL AUTOMATA 3
A subset F c T, is called a frontier of T, if for every path z c T, we have
c(n n F) = 1. It is readily seen that if F c Tx is a frontier then F is finite.
If F, c T, and F, c T, are frontiers we shall say that F, is bigger than
-=
F, (F, < F,) if for every y E F, there exists a x E F, such that x y . For
S E T we have c(Sn n) = w for every path n c T, if and only if
S = Un<aF,,w here F,, is a frontier of T and F,, < F,,, 1, n < w.
I
A finite (frontiered) tree is a set E = {x x 5 y for some y E F} where
F is a fixed frontier of T. For E as above, F is called the frontier of E and
.
denoted by Ft(E) By “finite tree” we shall always mean a finite frontiered
tree.
For a E (0,l } define the (immediate) successor function ra:T + T by
ra(x) = xu , x E T. The structure of two successor functions is
”% = (T,ro,r1>.
With N, we associate an appropriate (monadic) second-order language
.
L, This L, has function-constants ro,r l , to denote ro and r,; the usual
logical connectives and quantifiers; the membership symbol E; equality;
individual variables x, y, z, ... ,r anging over elements of T;f inite-set variables
a,p , y, .. . ,r anging over finite subsets of T;s et variables A,B ,C, .. . ,r anging
over arbitrary subsets of T. The atomic formulas of L, include formulas
of the form t E V where t is a term of L, and Vis a (finite or arbitrary) set
variable. Quantification is possible over all the three sorts of variables.
The second-order theory of two successor functions (S2S) is the set of
all sentences F of L, such that Jv; =! F (F is true in N,). The theory S2S
was proved decidable in [6] by means of a theory of automata on infinite
trees.
DEFINITIO1. NA n n-ary relation R E P(T)” between subsets of T is
definable in L, (S2S) if there exists a formula F(A,, ..., A,,) of L, such that
(1) R = {(A19 ***,An[) F(A1, .-.,An)}*
The relation R is weakly-definable if (1) holds for a formula containing
quantifiers only over individual and finite-set variables.
2. Special automata
As stated in the Introduction, our aim is to characterize the weakly-
defined relations. To this end we develop a theory of special automata.
In the following, X denotes a finite set called the alphabet.
DEFINITI2O. NA Z-(ualued)tree is a pair (u, T,) such that u: T, + Z. If
(u,T) is a valued tree then (u,T,) will denote the induced valued subtree
I
(u T, , T,). Theu se,t o f all C-trees (u, T,) , for a fixed x E T, will be denoted
by V&. The set V,,, of all %trees will be denoted by V..
4 MICHAEL 0. RABIN
DEFINITIO3N. A table over Z-trees is a pair (S, M )w here S is a finite
set, the set of states, and M is a function M:S x Z + P(S x S), the (non-
deterministic) table of moues (P(A) denotes the set of all subsets of A).
A special finite automaton (s.f.a.) over Z-trees (a special C-automaton)
is a system 'u = (S, M,S o, Fi) where (S, M )i s as above, So E S is the set
of initial states, F G S is the set of designated states.
DEF~N~T4I. ONA run of (S, M )o n the Z-tree t = (u, T,) is a mapping
r: T, + S such that for y E T, , (r(y0) , r(y1)) E M(r(y),~ (y)).W e also talk
about a run of an automaton 'u on a tree, meaning a run of the associated
.
table. The set of all %-runs on t is denoted by Rn('$l, t)
For a mapping +:A+B define In(+) = {bIbEB, c(+-I(b)) 2 o}.
DEFINITION5. The special automaton 2l= <S, MyS o, F ) accepts (u, T,)
if there exists an accepting %run r on (u, 7'') such that r(x)E So and for
every path n of T', In(r1 n) nF # 0. The set T(2l) of C-trees defined
by 'u is
I
a}.
T(2l) = {(u, T,) x E T, (u, T,) accepted by
A set A c V, is s.f.a. definable if for some s.f.a. 3,A = T(2l).
REMARK.I t is quite clear that the special automata introduced here
are a weaker version of the automata defined in [6]. For every special
automaton 'u there exists an automaton 23 in the sense of [6], such that
T(3)= T(23).I n fact, 23 may be taken to have the same table and initial
status as 3.T hat the converse statement is not true is shown by the example
in $3.
REMARK.A set A c V, is called invariant if for every Z-tree t = (u, T)
and every XET,t EA if and only if the tree t' = (u',T,) defined by
~'(xy=) u(y), y E T, is in A . The invariant subsets of V, are a boolean
algebra. It is clear from Definition 5 that every set T(%)i s invariant. To
prove that an invariant set A is s.f.a. definable, it suffices to construct
an automaton 'u such that (u, T)E T('1I)i f and only if (u, T)E A.
The following results are immediate.
LEMMA1 . If A E V, is s. f.a . definable, then there exists an automaton
'u = (S,M,So,F) such that So = {so}, s~ESa,n d T('1I) = A. This 'u
may be chosen so that (s1,s2)~M(s,cri)m plies s1 # so, and s2 # so.
THEOREM2. If A,B G V, are s.f.a. definable, then so are A uB and
A nB .
WEAKLY DEFINABLE RELATIONS AND SPECIAL AUTOMATA 5
Proof. Let A = T('u), B = T(B) where 'u = (S,M,so,F), B
= (S', M', do,F ' ); we assume that S n S' = !J. Construct the automaton
'uuB = (SUS', MUM', {so,s~)F, UF').
Clearly, T('u U B) = A U B.
With the above notations, define 3 x B = (S x S' x {0,1,2},
ii7,(so,sb,0),F) as follows. ((s1,si,b), (s,,s;,~))E ~@((s,s',a),c)i f and only
if (slys 2)E M(s, c) , (si, s;) E M'(s', 0);b = 1 if and only if a = 0 and s E F,
or a = 1 and s'#F'; b = 2 if and only if a = 1 and ~'EF'b; = 0 if and
only if a = 2, or a = 0 and s$F. Put F = S x S' x (2). We have
T(%xB)=AnB.
DEFINITIO6N. Let t = (u,T) be a Z, x &-tree and let po be the pro-
jection po(x,y ) = x. The projection po(t), by definition, is the Z,-tree
(Po% T)* I
The projection po(A)o f a set A c V,, x,, , is po(A) = {po(t) t E A}. The
C,-cylindrification of a set B c V,, is the largest set A c Vz,xz2s uch that
P O ( 4 = B.
THEORE3M. If A E VzIx,, is a s.f.a. definable set, then po(A) c V,,
is a s.f.a. definable set. If B E V,, is s.f.a. definable, so is its Z,-cylind-
riJcation A s V,,X,2.
Proof. Let '3 = (S, M,s o, F )b :: a Z, x C2-automaton with T(%)= A.
Define a XI-automaton by 'ul = (S,M1,so,F), where
u
M,(s, a,) = M(s,( 01, c,)), g1 E Z, 9 s E S.
UZE22
One can check that T('u,) = po(A).
The proof concerning cylindrification is left to the reader.
3. A counterexample
We wish to show that the class of s.f.a. definable sets is not closed under
complementation. We shall exhibit a s.f.a. definable set B c V, such
that A = V, - B is not s.f.a. definable.
Let Z = (0, l} and let B be the set of all Z-trees (u, T,) such that for some
I .
path n c T, we have: 1 E In(u n) It is readily seen that B is s.f.a. definable.
The set A = V, - B consists of all Z-trees (u, 7'') such that for every
I
path n c T,, 14 In(u n). We claim that A is not s.f.a. definable. To prove
this, let us assume that 'u = (S,M,s,,F) is a special automaton such
that T(3)= A and derive a contradiction Throughout this section, unless
otherwise specified, 'u will denote this particular automaton. We shall need
the following construction.
6 MICHAEL 0. RABIN
DEFINITI7O. NL et t = (u,T) and t, = (ul,Tx),X E T , be C'-trees. The
result of grafting the tree t, on t at y E T is the tree (uz, T) such that
uz(z)= u(z) for x$ Ty, and uz(yz) = u,(xz) for z E T. (Note that
I
Ty = {yz z E T},a nd similarly for T, .)
DEFINITI8O. N Let t, = (un,T ), n < w, be C'-trees. We shall say that
limn+&,= (u, T) if there exists an integral valued function N(x), x E T,
such that N(x) 5 n implies u,(x) = u(x).
LEMMA4. Let t = (0, T)E T(%)a nd let r E Rn(%, t) be an accepting
run If there exist nodes x < z < y such that r(x) = r(y) = s, s E F, and
u(z) = 1, then there exists a tree t'$A which is accepted by %.
Proof. Assume that y = xu. Let C' = S x C, and let (4,T ) be the
Z'-tree such that +(z) = (r(z),u (z)), z E T. Graft t, = (4,T ,) on (4,T )
at the node y = xu and call the resulting tree t, = (41,T ).S ince r(x) = r(y),
we have that po41 is an %-run on (plcjl,T).
Note that p04,(xu2) = s. Graft t, on rl at xu2 to obtain t2 = (4z,T).
Again p0& is an %-run on ( P ~ ~ ~and, Tpo)4z (xu3)= s. Continue this
process inductively for every n < w, where at the nth step we graft t, on
tndl at xu" to obtain t, = (4,,,7). Let limn+&,= f = ($, T), and
t' = (pl$,T). Since each p04,, is an %-run on (p14,,,T), J = po$ is an
%-run on t'.
I
We claim that for every path n c T, In(! n) n F # J! so that t' E T(%[).
Namely, if xu" E n for infinitely many (and hence all) n < w then ~(xu"=) s
and s E In(Jl n) n F. Otherwise, two cases may occur. Either n contains
no xu", then Jln = rln and In(J1n)nF = In(r(n)nF# !J. Or else
there is an n < w such that x u " ~ nx,u nfl$n. In this case there exists
a path n' c T, such that J(xu"u) = r(xu) for all X U E ~ .T hus
In(r1n)nF = In(r1n')nF #.PI.
Now, if z = xw then p,&xu"w) = 1, n < w; hence t'$ A.
Let t = (u,T) be a {O,l}-tree. We define, by induction on n, subsets
I 1
o,,(t) E T. Do(t) = {X U(X) = I}; D,, l(t) = D,,(t) n {X c(o,,(t) n T,) = w}.
LEMMA5 . Let 3 = (S,M,so,F> be a special {O,l}-automaton;
t = (u, T)E V{o,l}, A E D,, ,(t). If r E Rn(%, t) is an accepting run such
that c(r(T - {A})) n, then for some x < z < y , r(x) = r(y) = s, s EF,
and r(z) = 1.
Proof. By induction on n. For every path n, let ~ ( nb)e the first
1
A < x E n such that r(x)E F. The set {x(n) n c T}i s a frontier and hence
I
E = {y y ~ ( n )fo,r some n c T}, is finite. Since c(D,(t) n T) = w,
there must be a point z E Dn(t)s uch that z $ E and hence for some x = x(n),
x < z. Now r(x) = s, SEFa nd u(z) = 1. If for some z <y, r(y) = s.