Negative Solution of the Decision Problem for Sentences True in Every Subalgebra of $<N, +>$Author(s): Ralph McKenzieSource: The Journal of Symbolic Logic, Vol. 36, No. 4 (Dec., 1971), pp. 607-609Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272464 .

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THE JOURNAL oi SYMBOuC LOGIC Volume 36, Number 4, Dec. 1971

NEGATIVE SOLUTION OF THE DECISION PROBLEM FOR SENTENCES TRUE IN EVERY SUBALGEBRA OF <N, +>

RALPH MCKENZIE1

It was shown by Taiclin (6], and independently announced by Tarski [7], that the elementary theory of commutative cancellation semigroups is hereditarily unde- cidable. In his proof Tarski exhibited a subsemigroup of <N, *>, the natural numbers with multiplication, whose theory is both hereditarily and essentially undecidable. (The details of his construction were published by V. H. Dyson [1].) In connection with these results, Tarski suggested to the author that it would be of interest to solve the decision problem for the theory K which consists of all ele- mentary sentences which are true in every subalgebra (i.e. every subsemigroup) of <N, + >. The object of this note is to prove that the theory K is hereditarily unde- cidable.

Let us note that the complement of K (in the same language) is a recursively enumerable set of sentences. In fact every semigroup A a <N, +> has for its universe an eventually periodic set of integers; thus A is a definable subalgebra of <N, +>. Consequently, to enumerate the complement of K, one may simply list the one-place predicates in the language of K, then use Presburger's decision pro- cedure for <N, + > to determine which of the predicates defines a subalgebra, and to concurrently enumerate the sentences which are false in some one of these subalgebras.

?0. The results. Each of the theories we discuss is a set of sentences formulated in some applied first-order predicate language with equality, which is closed under logical deduction in the language. If T is such a theory we denote by E(T) the set of all sentences formulated in the language of T; by L(T) the set of all such sentences which are logically valid; and by F(T) the set of all those T e E(T) which are finitely refutable in T-i.e., such that there exists a finite model of T which satisfies -'9.

We shall consider two theories: the theory K mentioned in the first paragraph (which has only one nonlogical constant, a binary operation symbol +), and the theory of bipartite graphs. This second theory, D, is formulated in a language having for its only nonlogical constant a binary predicate symbol B; it consists of all logical consequences in this language of the axiom Vx, y, z(-,Bxy V -,Byz). The undecidability of this theory was proved in [5]. A more useful property for us was proved in [3]: L(D) and F(D) are recursively inseparable sets. In other words, there is no recursive set of sentences S which simultaneously satisfies L(D) a S and S c F(D) = 0.

Received April 10, 1970. Research supported in part by NSF Grant GP-7578.

607

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608 RALPH MCKENZIE

LdmMA. There exists an effective mapping tfrom E(D) to E(K) such that

(i) f a e L(D) then t(a) e L(K); (ii) if a E F(D) then t(a) e E(K) - K.

COROLLARY. L(K) and E(K) - K are recursively inseparable. Hence K is hereditarily undecidable.

?1. Proof of the results. By "effective mapping" we mean, of course, one which becomes a partial recursive function under any standard G6del numbering of the symbols and formulas of the languages involved. Thus the corollary is an immediate consequence of the lemma.

Our proof of the lemma can be viewed as an application of a general scheme of interpretation defined by Rabin [4]. (Very similar schemes for the interpretation of one theory into another have been introduced by Ershov [2], by Scott (unpublished) and by Vaught [8]. All these schemes generalize the one introduced in Tarski- Mostowski-Robinson, Undecidable theories, North-Holland, Amsterdam, 1968.)

We begin by defining three formulas of K:

so(x, uo): Vzo, z,3z2(x + z2 o uo A -,x ~ zo + z1); ?D(x; uo, u1): (10(x, uo) V EDO(x, u1);

T(x, y; uo, u1, u2): (Fo(x, uo) A 4Do(y, u1) A 3z0(x + ZO y + U2)- In order to ensure that no undesirable clash of variables occurs in what follows,

we assume that the variables of the language of K are listed without repetition as X, y, U0, U1 U2, ZO, Z1, Z2, X0 X,1, X2, * - -. Let the variables of D be v0, vj, v2, * - .. Let now a be any member of E(D). We correlate with a a formula t(a) obtained by relativizing all quantifiers 3v. or Vv. of a to the formula (D(x.; uo, ul) and replacing all subformulas of the form Bvv,6v by the corresponding formulas

T(xag Xs; U09 U1, U2)-

We then put t(a) = Vuo, u1, u2(3x'D(x; uo, u1) -)

Given any binary algebra A = <A, +> and any triplet of elements of A- ao, al, a2-let us define A(ao, a,, a2) to be the structure <C, B> where

C = {aeA I A I ((a;ao,a,)} and B = {<a, b> e 2A I A k T(a, b; ao, a,, a2)}.

Our definitions are clearly motivated by the following fact: Let A be a binary algebra and a e E(D). Then A 1 t(a) iff for every nonempty structure A(ao, a,, a;), A(ao, a,, a2) h a. From this it is obvious that if a is logically valid then so is t(o), i.e. we have statement (i) of the lemma. To get statement (ii) it suffices to prove the following:

(iii) Let C = <C, B> be any finite model of D. Then there exists a semigroup A c <N, + > and three integers no, ni, n2 e A so that C e A(no, nj, n2).

To prove (iii); assume that C <C, B> is given. Let C1 = {d e C I 3cB(c, d)} and let C0 = C -C1. Since C is a model of D we have that C0 # 0; for any c, deC, B(c,d) implies ceCo and deC1; C C0uC1 and 0 = C0 r C1. The elements of C can be enumerated as c1,** *, Ck e C0 and d1,.* *, di E C1 where

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THE DECISION PROBLEM 609

k > 1 and I > 0 and all the listed elements are distinct. Let us choose 5 distinct prime integers p, q, P1, P2, p3 and integers aj, fl e N (1 < i ? k, 1 < j < 1) which satisfy:

(1) p, q, Pi, P2, p3 > 2k + 21 and for each i, j: (a) a < p; (b) at ip (mod pq), a, 0 (mod pl) and at 1 (mod P2P3); (c) fl jq (mod pq), P, 0 (mod P2) and ,- 1 (mod plp3).

Secondly we define

(2) (a) n2 = PqPlP2P3 * maX(a, 9); (b) et =n2+ ac,S--n2+ Pj(l < i kg 1 <j< 11); (c) no = 2 1ij~kfi;

(d) n1 = 2 llzj;z8j if I > 0, but n1 = n2 ifI = 0; (e) A = <A, +> is the subalgebra of <N, +> generated by n2, ei, Sj

(1 < i ? k, 1 < j < 1) and the set of numbers n2 + 8,- t such that C 1 B(c., dg).

Now it is an easy matter, using (1) and (2), to prove that C t A(no, ni, n2) under the map ci ->. e, di - 8j. This remark concludes the demonstration.

REFERENCES

[1] V. H. DYSON, On the decision problem for theories of finite models, Israel Journal of Mathematics, vol. 2 (1964), pp. 55-70.

[2] Y. L. ERSHOV, I. A. LAVROV, A. D. TAIMANOv and M. A. TAICLIN, Elementary theories, Russian mathematic surveys, vol. 20 (1965), pp. 35-105.

[3] I. A. LAVROV, Effective inseparability of the sets of identically true formulae and finitely refutable formulae for certain elementary theories, Algebra i Logika Seminar 2, vol. 1 (1963), pp. 5-19 [Russian].

[4] M. 0. RABIN, A simple method for undecidability proofs and some applications, Logic, methodology and philosophy of science, Proceedings of the 1964 International Congress, Bar- Hfillel ed., Amsterdam, 1965, pp. 58-68.

[5] H. ROGERS, JR., Certain logical reduction and decision problems, Annals of Mathematics, vol. 64 (1956), pp. 264-284.

[6] M. A. TAiCuN, Undecidability of the elementary theory of commutative cancellation semigroups, Sibij * Matematieeski' Zurnal, vol. 3 (1962), pp. 308-309 [Russian].

[7] A. TARSKI, Solution of the decision problem for the elementary theory of commutative semigroups, Notices of the American Mathematical Society, vol. 9 (1962), p. 205.

[8] R. L. VAUGHT, Cobham's theorem on undecidable theories, Logic methodology and philosophy of science, Proceedings of the 1960 International Congress, Nagel, Suppes and Tarski, eds., Stanford 1962, pp. 14-25.

UNIVERSITY OF CALIFORNA

BERKLEY, CALIFORNIA 94720

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