A Computing Procedure for Quantification Theory by Martin Davis; Hilary PutnamReview by: J. A. RobinsonThe Journal of Symbolic Logic, Vol. 31, No. 1 (Mar., 1966), pp. 125-126Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2270653 .

Accessed: 18/06/2014 00:08

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to TheJournal of Symbolic Logic.

http://www.jstor.org

This content downloaded from 62.122.77.28 on Wed, 18 Jun 2014 00:08:45 AMAll use subject to JSTOR Terms and Conditions

REVIEWS 125

is satisfiable; otherwise closed. Then S is satisfiable if there is a complete open S- chain, unsatisfiable if there is a closed S-chain; and an infinite S-chain is open if each of its finite subchains are open.

The actual procedure used by Gilmore consists of computing, given S, the successive quantifier-free sentences M1, M2, ..., of a particular complete S-chain (which is finite when no existential quantifier follows a universal one in the prefix of S). For each n > 1 the disjunctive normal form Dn of the conjunction of M1, ... , Mn is constructed as a satisfiability test. The enormous rate of growth of Dn, as n increases, is one reason why working storage is quickly exhausted in actual calculations with the procedure. The other, more important, reason lies in the nature of the particular complete S-chain used.

Let the individual constants in an S-chain be enumerated P1, P2, . in the order of their first occurrence in the chain as one reads successively across each sentence. Call an S-chain SI, S2, . . , exhaustive if, for each A, whenever Sk+I comes from an earlier universal sentence by instantiation with Pj, then if j > 1, the chain SI, . . ., Sk is complete for every Pi, 1 < i < j.

Gilmore's S-chains are exhaustive in this sense. This means that the length of the S-chain is at least hr before Pk+I is first used in a universal instantiation, where r is the number of universal quantifiers in S.

The level of a constant in an S-chain is 0 if it occurs before any existential instan- tiations are performed; it is L + 1 if the constant is introduced by existential instan- tiation and L is the highest level-of any constant on which it depends. The level of an S-chain is the highest level of any constant used for universal instantiation in the chain, and the level of an unsatisfiable S is that of the closed S-chains of lowest level.

If the level of S is L, then the length of the shortest exhaustive closed S-chain is of the order of hr(tL), where r is the number of universal quantifiers in S, t is the maxi- mum number of variables on which any existentially quantified variable in S depends, and h is a number < 1 which is approximately the total number of individual constants and existential quantifiers in S.

In the Discussion, Porte elicited the fact, stressed in Gilmore's second paper, that this procedure is a decision procedure for sentences whose prefix satisfies the condition earlier mentioned. Dag Prawitz and his colleagues announced their work, reported in the paper reviewed below. J. A. ROBINSON

MARTIN DAVIS and HILARY PUTNAM. A computing procedure for quantification theory. Journal of the Association for Computing Machinery, vol. 7 (1960), pp. 201-215.

The procedure given in this paper has the same purpose as that of Gilmore discussed in the previous review, but differs from that procedure in several respects: (1) the given sentence S is assumed to be in prenex conjunctive normal form instead of prenex disjunctive normal form; (2) function symbols, as well as predicate symbols and in- dividual constants, are allowed in the construction of S; (3) each existential quantifier (3Xj) in the prefix of S is dropped, and Xj replaced throughout by a term constructed from a function symbol Fj and arguments which are all the variables on which Xj depends; (4) a different method of testing finite sets of quantifier-free sentences for satisfiability is used.

Items (1) and (4) are related. If the matrix of S is in conjunctive normal form, then any conjunction (Ml & ... & Mn) of quantifier-free sentences of any S-chain is already in conjunctive normal form; and an elegant, efficient method of testing quantifier-free sentences in conjunctive normal form is supplied which bypasses most of the combinatorial explosion of the corresponding method of Gilmore. How-

This content downloaded from 62.122.77.28 on Wed, 18 Jun 2014 00:08:45 AMAll use subject to JSTOR Terms and Conditions

126 REVIEWS

ever, this achieves only a small improvement in the overall procedure, since the same exhaustive S-chain philosophy is followed here also.

Item (2) is a matter of extra convenience but not an essential advance on Gilmore's procedure, since function symbols can always be paraphrased away in favor of pred- icate symbols.

Item (3) means that in the Davis-Putnam procedure the prefix of S can be assumed to consist entirely of universal quantifiers, and that S-chains are grown entirely by universal instantiation. The individual constants may now be compounded from function symbols, with individual constants as arguments, to any level. The level of a constant is now one higher than the highest level of any of its immediate arguments, with uncompounded constants having level 0.

The procedure, as in the case of Gilmore, consists of generating, given S, the suc- cessive quantifier-free sentences of an exhaustive complete S-chain, and consequently the shortest closed S-chain, for an unsatisfiable S of level L, will have a length that is of the same order as that which Gilmore's procedure calls for.

Davis and Putnam argue (pp. 207, 208) that "the crucial difficulty" in designing procedures of this kind is in getting an efficient scheme (such as theirs) for testing quantifier-free sentences for satisfiability. In fact this is not really very crucial at all when exhaustive S-chains are used, since no satisfiability test, however efficient, is of any use if one cannot ever reach the point in the overall calculation when the test will be applied to a closed chain. The crucial difficulty is surely thst exhibited by the exponential growth of the number of instantiations. J. A. ROBINSON

DAG PRAWITZ, HAKON PRAWITZ, and NERI VOGHERA. A mechanical proof procedure and its realization in an electronic computer. Ibid., pp. 102-128.

The procedure described in this paper was programmed for the Swedish computer FACIT EDB and applied to a collection of very simple sentences. It differs from the Gilmore-Davis-Putnam procedures mainly in being essentially a realization of Beth's semantic tableau construction: consequently it is directly applicable to any sentence S of quantification theory, without the need for any normalization. However, the spirit of the procedure is wholly similar; in particular it is exhaustive in its universal instantiation: one instantiates a sentence with a constant P1 only if one has instantiated all sentences present at that stage with P1, ..., Pjj_. Indeed this is pointed out by the authors themselves (p. 123) as the fundamental limitation on the procedure, and they announce work in progress to develop a more efficient method. Such a method was in fact soon published by Dag Prawitz in the paper reviewed below.

J. A. ROBINSON

DAG PRAWITZ. An improved proof procedure. Theoria (Lund), vol. 26 (1960), pp. 102-139.

In view of the eliminability of existential quantifiers in favor of function symbols, the fundamental form of the problem which proof procedures are designed to solve is as follows: given an open sentence A, to find, if one exists, a sequence a, , an of substitutions (of terms for the variables in A) such that the conjunction

(1) Aa1 & ... &Aan

is truth-functionally unsatisfiable. The terms may without loss of generality be sup- posed to lie in the set H of all those that are constructed out of symbols (variables, individual constants, and function symbols) which occur in A.

The "exhaustive" methods described by Gilmore, Davis and Putnam, and Prawitz, et al. in the papers reviewed above are solutions to this problem which are built on the leading idea of enumerating all possible substitutions from H for the variables of A, and then letting al, . . ., a. simply be the first n substitutions in this enumeration,

This content downloaded from 62.122.77.28 on Wed, 18 Jun 2014 00:08:45 AMAll use subject to JSTOR Terms and Conditions