Appendix A “Feasible Computational Methods in the Propositional Calculus”, the Seminal Report by M. Davis and H. Putnam “Our report for the NSA, entitled Feasible Computational Methods in the Propositional Calculus is dated October 1958. It emphasizes the use of conjunctive normal form for satisfiability testing (or, equivalently, the dual disjunctive normal form for tautology testing). The specific reduction methods whose use together have been linked to the names Davis- Putnam are all present in this report.” (M. Davis, this volume p. 15) “The DPLL procedure, even half a century after its introduction, remains a foundational component of modern day SAT solvers. Through SAT solvers ···, as well as through sat- isfiability modulo theory ··· and answer set programming ··· solvers that build upon SAT techniques, the DPLL procedure has had a tremendous practical impact on the field with applications in a variety of areas such as formal verification, AI planning, and mathematical discovery.” (D. Loveland et al., this volume p. 316) A research report which Martin Davis and Hilary Putnam jointly wrote in 1958 is faithfully reproduced in this appendix, where it gets published for the first time; three papers which promoted its wide influence on later research in the field of automated deduction are: [1] A Computing procedure for Quantification Theory by Davis and Putnam, 1960 (a typescript of which appears inside a research report of 1959); [2] A Machine Program for Theorem-Proving by Davis, George Logemann, and Donald W. Loveland, 1962 (preprinted as a research report in 1961); [3] Eliminating the Irrelevant from Mechanical Proofs by Davis alone, 1963 (trans- lated into Russian in 1970). The 1958 report tackles propositional calculus from a broader angle than the subsequent papers just cited. Its first part discusses the advantage of putting formulas into some normal form (such as the Gégalkine polynomial form); it notes that not all normal forms have the same properties and argues that conjunctive normal form is usually preferable to the disjunctive one for treatment by satisfiability testing methods; moreover, it adds, it is convenient to have ‘more than one such method available ···, since in case one method fails one can then always attempt the others’. © Springer International Publishing Switzerland 2016 371 E.G. Omodeo and A. Policriti (eds.), Martin Davis on Computability, Computational Logic, and Mathematical Foundations, Outstanding Contributions to Logic 10, DOI 10.1007/978-3-319-41842-1 372 Appendix A: “Feasible Computational Methods in the Propositional Calculus” … In [1], Davis and Putnam will propose a satisfiability decision algorithm for propo- sitional formulas in conjunctive normal form. This will consist of three rules drawn or adapted from the kit of rules described in the 1958 report. Two of these, named elimination of one-literal clauses and affirmative-negative rule, will be retained also in [2, 3]. The third, named rule for eliminating atomic formulas, will be replaced in [2, 3] by another one called, ever since, the splitting rule. The new rule is theo- retically equivalent to the superseded one but preferable, for practical reasons duly explained in [2]. The revised procedure has today acquired great renown under the name DPLL (from the initials of its inventors). The two interchanged rules were specified in [1, 2], respectively, in the following terms: Rule for Eliminating Atomic Formulas. Let the formula F [given in conjunctive normal form] be put into the form (A ∨ p) & (B ∨¯p) & R, where A, B, and R are free of p. (This can be done simply by grouping together the clauses containing p and “factoring out” occurrences of p to obtain A, grouping the clauses containing p¯ and “factoring out” p¯ to obtain B, and grouping the remaining clauses to obtain R.) Then F is inconsistent if and only if (A ∨ B) & R is inconsistent. Splitting Rule. Let the given formula F be put in the form (A ∨ p) & (B ∨¯p) & R, where A, B, and R arefreeofp. Then F is inconsistent if and only if A & R and B & R are both inconsistent. In the 1958 report as reproduced below, these rules are specified in dual form (that is, they refer to tautology testing of an F which is given in disjunctive normal form) and they bear the respective names ‘rule for eliminating variables’ and ‘rule of case analysis’; the latter, in particular, reads: Let F and F be obtained from F by substituting 0 and 1 respectively for all occurrences of p in F and making the obvious simplifications. Then F is a tautology if and only if F and F both are. ———— The second part of the 1958 Davis-Putnam report presents a Gentzen-type sys- tem for a version of propositional calculus embodying the exclusive disjunction connective. It shows that such a system is complete and that it remains such if the cut inference rule is withdrawn; through this system, another decision procedure alternative to truth-table methods is obtained. ———— Davis and Putnam were showing so much interest in feasible computational meth- ods for propositional calculus ten years before the notion of NP-completeness began to emerge and Stephen Cook brought the propositional satisfiability problem under the spotlight of the very challenging question as to whether P = NP. Appendix A: “Feasible Computational Methods in the Propositional Calculus” … 373 374 Appendix A: “Feasible Computational Methods in the Propositional Calculus” … Feasible Computational Methods in the Propositional Calculus Submitted by Martin Davis Associated Professor of Mathematics Rensselaer Polytechnic Institute Hartford Graduate Division and Hilary Putnam Assistant Professor of Philosophy Princeton University October 1958 It is a consequence of a well-known result of Church and Turing that there is no effective calculating procedure for determining whether or not a given schema of elementary quantification theory is or is not valid. But if we restrict ourselves to schemata of the propositional calculus, then, as has long been known (since Post’s thesis of 1921, cf. Post [1]), an effective calculating procedure does exist, namely the method of truth-tables. Unfortunately, this solution of the decision problem for the propositional calculus is mainly of theoretical interest. Since the truth-table for a formula containing n variables must contain 2n lines, truth-table methods are quite unfeasible for use in practical computation once the number of variables becomes at all large. The present report summarizes our investigations of alternative computational methods which could be employed where truth table methods are inapplicable. Part I of this report presents several techniques, none of which is superior to truth-table methods in all cases, but which are nonetheless vastly better than truth-tables in the sorts of cases which can be expected to arise in practice. In Part II, a modified version of the Gentzen-type system of Kleene [1] for propositional calculus (modified to allow exclusive disjunction as a primitive connective) is discussed. Its completeness and a version of the Gentzen Hauptsatz are derived. This is shown to lead to a decision Appendix A: “Feasible Computational Methods in the Propositional Calculus” … 375 procedure for the propositional calculus, which, however, is not particularly superior to truth-table methods. Nevertheless, we feel that this represents a promising line of investigation. For, if a complete system could be constructed which has all of the present desirable properties and which enjoys one additional property to be described below, such a system could be used to provide a decision procedure which would probably be quite useful. Part I: Case Methods, Boolean Methods, and a Combinatorial Method for Propositional Calculus 1. Validity and Disjunctive Normal Form (i) An important special case of the decision problem for propositional calculus. No known method of testing formulas of the propositional calculus for consistency exists which is uniformly superior to the truth-table method. The search for feasible methods of computation in the propositional calculus when truth-table developments are unfeasible naturally concentrates, therefore, on important special cases. The most important of these special cases will now be described, since the methods to be explained below are all presented with this case in mind. We shall use ∼ for negative, · for conjunction, ∨ for inclusive disjunction, + for exclusive disjunction, ⊃ for material implication, and ≡ for material equivalence. Parentheses will be omitted wherever possible without confusion (in Part I), and where possible because of associative operations. Also, for single letters we write e.g. p¯ for ∼ p. Also we often omit the dot of conjunction, writing e.g. pqr¯ for p·q·∼r. The principal case we consider is that of a formula or system of formulas to be tested for consistency. Let us suppose that the formulas in the system have been actu- ally written down as a formalization of some practical problem. What specialization does this introduce? In the first place, we can not assume that the number of variables is small. One formula can easily contain over 20 variables without being very long, e.g.1 (pqrs ∨ tu) · (uv¯ ∨ w) · (pabc ∨ de ∨ gh) · (ijkl ∨ mno). Thus, a system of, say, 10 formulas could easily contain over a hundred distinct propositional variables. But we can assume that the formulas are relatively short. Indeed, this is already assumed when we suppose that the system is capable of being written down by a human being. A formula or system of formulas with 100 distinct propositional variables could be as long as, say, 250 symbols without having, so far as is known, any shorter normal equivalent. However, any system of formulas actually written down by human beings will presumably have far fewer than 250 symbols. We shall, therefore, concentrate on the following case: The case of a system of formulas each one of which is short enough to be “manageable” (i.e., short enough for a clerk to perform algebraic manipulations on the formula, such as putting the formula in normal form), although the number of formulas in the system may be 1In the present part formulas may be thought of as names of themselves.