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We describe the application of proof orderings---a technique for reasoning about inference systems---to various rewrite-based theorem-proving methods, including re#nements of the standard Knuth-Bendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congruence; ordered completion #a refutationally complete extension of standard completion#; and a proof by consistency procedure for proving inductive theorems. # This is a substantially revised version of the paper, #Orderings for equational proofs," co-authored with J. Hsiang and presented at the Symp. on Logic in Computer Science #Boston, Massachusetts, June 1986#. It includes material from the paper #Proof by consistency in equational theories," by the #rst author, presented at the ThirdAnnual Symp. on Logic in Computer Science #Edinburgh, Scotland, July 1988#. This researchwas supported in part by the National Science Foundation under grants CCR-89-01322, CCR-90-07195, and CCR-90-24271. 1 ...

An equational approach to the synthesis of functional and logic program is taken. In this context, the synthesis task involves nding executable equations such that the given speci cation holds in their standard model. Hence, to synthesize such programs, induction is necessary.We formulate procedures for inductiveproof,aswell as for program synthesis, using the framework of \ordered rewriting". We also propose heuristics for generalizing from a sequence of equational consequences. These heuristics handle cases where the deductive process alone is inadequate for coming up with a program. 1.

. In an earlier paper, one of the present authors presented a preliminary account of an equational logic called PIM. PIM is intended to function as a "transformational toolkit" to be used by compilers and analysis tools for imperative languages, and has been applied to such problems as program slicing, symbolic evaluation, conditional constant propagation, and dependence analysis. PIM consists of the untyped lambda calculus extended with an algebraic rewriting system that characterizes the behavior of lazy stores and generalized conditionals. A major question left open in the earlier paper was whether there existed a complete equational axiomatization of PIM's semantics. In this paper, we answer this question in the affirmative for PIM's core algebraic component, PIM t , under the assumption of certain reasonable restrictions on term formation. We systematically derive the complete PIM logic as the culmination of a sequence of increasingly powerful equational systems starti...

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. We present a new transformation method by which a given Horn theory is transformed in such a way that resolution derivations can be carried out which are both linear (in the sense of Prologs SLD-resolution) and unit-resulting (i.e. the resolvents are unit clauses). This is not trivial since although both strategies alone are complete, their na ve combination is not. Completeness is recovered by our method through a completion procedure in the spirit of Knuth-Bendix completion, however with different ordering criteria. A powerful redundancy criterion helps to find a finite system quite often. The transformed theory can be used in combination with linear calculi such as e.g. (theory) model elimination to yield sound, complete and efficient calculi for full first order clause logic over the given Horn theory. As an example application, our method discovers a generalization of the well-known linear paramodulation calculus for the combined theory of equality and strict orderings. The met...

Abstract. We propose a method for testing satisfiability based on Boolean rings. It makes heavy use of simplification, but avoids the potential size increase associated with application of the distributive law by employing a combined linear and binomial representation. Several complexity results suggest why the method may be relatively effective in many cases. The framework is also amenable to learning from intersections, as in St˚almarck’s method. Some experiments have been undertaken. 1

There is not a person in this courtroom who has never told a lie, who has never done an immoral thing, and there is no man living who has never looked upon a woman without desire. 1 —Harper Lee: To Kill a Mockingbird Abstract. A potential advantage of using a Boolean-ring formalism for propositional formulæ is the large measure of simplification it facilitates. We propose a combined linear and binomial representation for Booleanring polynomials with which one can easily apply Gaussian elimination and Horn-clause methods to advantage. We demonstrate that this framework, with its enhanced simplification, is especially amenable to intersection-based learning, as in recursive learning and the method of St˚almarck. Experiments support the idea that problem variables can be eliminated and search trees can be shrunk by incorporating learning in the form of Boolean-ring saturation. 1

Given a Boolean formula A, the satisfiability problem is concerned with finding an assignment of truth values (0 and 1) to the propositional variables in A that gives the formula the value 1 (true), or—when there is no such satisfying assignment—proving that the A is equivalent to the constant 0 (false). Alternatively, to establish validity of A, one can refute its negation B by inferring a contradiction 1 = 0 from it. In general, a proof will require some form of case splitting. Suppose B contains one or more occurrences of a propositional variable x. From B, one may infer the disjunction B0 ∨ B1, where B0 and B1 denote the formula B after making the assignment x = 0 or x = 1, respectively. If both alternatives lead to a contradiction, then B is unsatisfiable and A is a tautology. Tracking “necessary ” assignments is crucial for reducing the number of backtracks that must be performed during the search. Boolean rings are an algebraic structure that provides an alternative to Boolean algebra. By using exclusive-or (+) instead of (∨), there is no need for a negation operator (x ′ is x + 1). The Boolean-ring (exclusive-or) normal-form is a sum of monomials, called a Zhegalkin polynomial. The normal form of tautologies is 1, and is 0 for contradictions. The use of this representation in automated deduction was first suggested in [1]. We explore the range of simplifications that can be employed efficiently (that is, polynomially) on Boolean ring formulæ, and show how such simplifications can be used to boost performance of satisfiability algorithms. We avoid the potential exponential size increase associated with the use of the distributive law needed to compute Boolean-ring normal forms, splitting on variables instead. We have implemented a mixed binary-linear representation of Zhegalkin polynomials [4], which can help optimize operations like unit propagation and hypothesis testing. Extra variables are used to decompose each polynomial equation M0 + · · · + Mn = 0 (n> 1) into a linear equation y0 + · · · + yn = 0 and a set of binomial equations ∗ Also with Intel, Haifa, leading a development team for formal verification tools.

Abstract. A potential advantage of using a Boolean-ring formalism for propositional formulæ is the large measure of simplification it facilitates. We propose a combined linear and binomial representation for Boolean ring polynomials, with which one can easily apply Gaussian elimination and Horn-clause methods to advantage. The framework is especially amenable to intersection-based learning. Experiments support the idea that problem variables can be eliminated and search trees can be shrunk by incorporating a measure of Boolean-ring saturation. Several complexity results suggest why the method may be relatively effective in many cases. 1