CONTRIBUTIONS TO THE REDUCTION THEORY OF THE DECISION PROBLEM Third paper Prefix (xl)(Ex2)...(Ex~ 2)(x~-l)(x,~), a single binary predicate By LASZL0 KALMAP (Szeged), corresponding member of the Academy 1. In a previous paper, a it has been proved that any formula of the first order predicate calculus is equivalent (as to being satisfiable or not) to some binary first order formula having a GtSdel prefix2 (1) (xl) (x~) (x3) (F-x3... (Ex~) and a matrix containing a single predicate variable. In another paper, ~ I have shown that in the above statement, we can replace the prefix (1) by the prefix (2) (Xl) (X2)(Ex3)-.. (Ex~_I) (xn), containing three universal quantifiers just as (1), but having an advantage over (1) as to the order of the existential quantifiers ~ which is 3 in (1) but only 2 in (2). In the same paper, I proposed the question whether the same holds for the prefix (3) (x,) (ex2) . (ex,~_2) (x,,_l) (x~,) 1 LASZL0 KALMARand JANOSSURANYI, On the reduction of the decision problem, second paper, G6del prefix, a single binary predicate, Journal of Symbolic Logic, 12 (1947), pp. 65 - 73. See KORT G6DEL, Zum Entscheidungsproblem des logischen Funktionenkalktils, Monatshefte fitr Mathematik and Physik, 40 (1933), pp. 433-443, where an analogous reduction theorem has been proved, but saying nothing about the number of predicate variables contained in the matrix. 3 L~,szL6 KALMAn, Contributions to the reduction theory of the decision problem, first paper, prefix (xO (x2)(Ex~)...(Exn-1)(x,), a single binary predicate, Acia Math. Hung., 1 (1950), pp. 64-73. 4 I. e. the number of universal quantifiers preceding, or, the number of arguments of the descriptive functions involved by, the existential quantifiers in question. 2* 20 L. I~ALMAR too, where the order of the existential quantifiers is reduced to one. 5 Now, I shall answer this question affirmatively by THEOREM 1. To any given formula of the first order predicate calculus # is possible to construct an equivalent one with a prefix of the form (3) and a matrix containing no predicate variable other than a single binary one. 2. To prove this theorem, we cannot use the set-theoretic method adopted in the previous papers (loc. cir. 1 and ~), for it is based on the fact that for any individuals x, y, there is an ordered pair p with x and y as its first and second component, respectively, which requires, when formalized, a second order existential quantifier (Ep). Therefore, we shall employ an ar#hmetic method similar to that used in an older paper 6 where the analogous theorem for the Ackermann prefix 7 (4) (Exl) (x2) (Ex3) (x~) . (x,,~) has been proved. In that paper, the set of positive integers divisible by 3 was used for the set of individuals (this can be done by the familiar LOwenheim-- Skolem theorem); the pairs of individuals were replaced by the integers of the form 3k--1, the ordered pair (x,y), x, y=~O (rood 3), being represented by 3m(x/3, y/3)--l, ~(i,j) denoting for any positive integers 4] the ordinal number of the pair (i, j) in the Cauchy enumeration of the pairs, (1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1), (1, 4), (2, 3),... (The integers of the form 3k+ 1 were used, besides replacing the predicates contained in the original formula, as second exemplars of the individuals, in order to make it possible to express the relations corresponding to those A further reduction of the order of all existential quantifiers, viz, to zero, is impossible by CHURCH'S theorem on the unsolvability of the decision problem by any recursive algorithm (ALONZOCHURCH, A note on the Entscheidungsproblem,Journal of Symbolic Logic, 1 (1936), pp. 4(?--41 and 10i--102) and by an obviously recursive solution of the satisfiability question for first order formulae with a prefix containing no other existential quantifiers than those of order zero, see P. BeRNAYS and M. SCHONH~KEL, Zum Entscheidungs- problem der mathematischen Logikl Math. Annalen, 99 (1928), pp. 3422-372, especially pp. 359--360. However, the order of some of the existential quantifiers can be reduced to zero; hence the question arises, which is the smallest positive integer m such that to any first order formula h it is possible to construct another one with a prefix of the form (Exl)...(Ex,,) (Yl) (Ex~+~)...(Ex,+,,)(y2)(y~) and containing a single, binary, predicate variable in the matrix, n being dependent of A. By a slight modification of the argument used in the present paper, we can prove that this proposition holds for m = 50 (and indeed, as we can prove by a further device, for m = 48 too), see footnote ~; but the minimal value of m is probably much less. o L/~SZL6 KALM/~R,On the reduction of the decision problem, first paper, Ackermann prefix, a single binary predicate, Journal of Symbolic Logic, 4 (1939), pp. 1 -9. 7 See WrLHeLM ACKERMANN, Beitr~ige zum Entscheidungsproblem der mathematischen Logik, Math. Annalen, 112 (1936), pp. 419-432, where the analogous theorem but saying nothing about the number of predicate variables in the matrix has been proved. .