David A. Plaisted

A Structure-preserving Clause Form Translation

Most resolution theorem provers convert a theorem into clause form before attempting to find a proof. The conventional translation of a first-order formula into clause form often obscures the structure of the formula, and may increase the length of the formula by an exponential amount in the worst case. We present a non-standard clause form translation that preserves more of the structure of the formula than the conventional translation. This new translation also avoids the exponential increase in size which may occur with the standard translation. We show how this idea may be combined with the idea of replacing predicates by their definitions before converting to clause form. We give a method of lock resolution which is appropriate for the non-standard clause form translation, and which has yielded a spectacular reduction in search space and time for one example. These techniques should increase the attractiveness of resolution theorem provers for program verification applications, since the theorems that arise in program verification are often simple but tedious for humans to prove.

Completion Without Failure

Publisher Summary This chapter discusses completion without failure. The design of efficient methods for dealing with the equality predicate is one of the major goals in automated theorem proving. Just adding equality axioms almost invariably leads to unacceptable inefficiencies. A number of special methods have been devised for reasoning about equality. Within resolution-based provers, demodulation, that is, using equations in only one direction to rewrite terms to a simpler form, is frequently employed. A complete method for handling equations is paramodulation in which equational consequences are generated by using all equations in both directions. Paramodulation is difficult to control and may produce hosts of irrelevant or redundant formulas. The chapter discusses the purely equational case in which a theory is presented as a set of equations and one is interested in proving a given equation to be valid in that equational theory. In important special cases, validity can be decided using canonical rewrite systems that have the property that all equal terms simplify to an identical form. Deciding validity in theories for which canonical systems are known is thus easy and reasonably efficient.Publisher Summary This chapter discusses completion without failure. The design of efficient methods for dealing with the equality predicate is one of the major goals in automated theorem proving. Just adding equality axioms almost invariably leads to unacceptable inefficiencies. A number of special methods have been devised for reasoning about equality. Within resolution-based provers, demodulation, that is, using equations in only one direction to rewrite terms to a simpler form, is frequently employed. A complete method for handling equations is paramodulation in which equational consequences are generated by using all equations in both directions. Paramodulation is difficult to control and may produce hosts of irrelevant or redundant formulas. The chapter discusses the purely equational case in which a theory is presented as a set of equations and one is interested in proving a given equation to be valid in that equational theory. In important special cases, validity can be decided using canonical rewrite systems that have the property that all equal terms simplify to an identical form. Deciding validity in theories for which canonical systems are known is thus easy and reasonably efficient.

Eliminating duplication with the hyper-linking strategy

The efficiency of almost all theorem proving methods suffers from a phenomenon called duplication of instances of clauses. In this paper, we present a novel technique, called the hyper-linking strategy, to eliminate such duplication. This strategy is complete for the full first-order predicate calculus. We show the effectiveness of this strategy by comparing it with other proving methods. We give empirical evidence that both the Davis-Putnam procedure and the hyper-linking strategy are comparable to each other and better than other common theorem proving strategies on propositional calculus problems. The fact that the Davis-Putnam procedure is faster than resolution and other common methods on propositional problems seems not to be appreciated by a large segment of the theorem proving community. Also, we give empirical evidence that the hyper-linking strategy is better than other common theorem proving methods on near-propositional problems like logic puzzles. We attempt to explain the superior behavior of the hyper-linking strategy and the Davis-Putnam procedure by examining the kinds of duplication that can occur during the search with the different methods. In addition, we show the completeness of the hyper-linking strategy combined with several support strategies.

Semantic confluence tests and completion methods

We present semantic methods for showing that a term-rewriting system is confluent. We also present methods for completing a given term-rewriting system to obtain an equivalent confluent system. These methods differ from the well-known and widely studied Knuth-Bendix method in that they emphasize semantics rather than syntax. Also, they often require more user interaction than the purely syntactic Knuth-Bendix method. The concept of “ground confluence” is discussed; methods for demonstrating ground confluence are also given. We give decision procedures for some sub-problems that arise in this method.

Non-horn clause logic programming without contrapositives

We present an extension of Prolog-style Horn clause logic programming to full first order logic. This extension has some advantages over other such extensions that have been proposed. We compare this method with the model elimination strategy which Stickel has recently implemented very efficiently, and with Lovelands extension of Prolog to near-Horn clauses. This new method is based on the authors “simplified problem reduction format” but permits a better control of the splitting rule than does the simplified problem reduction format. In contrast to model elimination, this new method does not require the use of contrapositives of clauses, permitting a better control of the search. This method has been implemented in C Prolog and has turned out to be a respectable and surprisingly compact first-order theorem prover. This implementation uses depth-first iterative deepening and caching of answers to avoid repeated solution of the same subgoal. We show that the time and space used by this method are polynomial functions of certain natural parameters of the search space, unlike other known methods. We discuss the relation of these upper bounds to Savitchs theorem relating nondeterministic time to deterministic space.

Termination orderings for associative-commutative rewriting systems

In this paper we describe a new class of orderings-associative path orderings-for proving termination of associative-commutative term rewriting systems .These orderings are based on the concept of simplification orderings and extend the well-known recursive path orderings to E - congruence classes, where E is an equational theory consisting of associativity and commutativity axioms. Associative path orderings are applicable to term rewriting systems for which a precedence ordering on the set of operator symbols can be defined that satisfies a certain condition,the associative path condition. The precedence ordering can often be derived from the structure of the reduction rules. We include termination proofs for various term rewriting systems (for rings,boolean algebra,etc.) and, in addition, point out ways to handle situations where the associative path condition is too restrictive.

Rewrite, rewrite, rewrite, rewrite, rewrite, …

Abstract We study properties of rewrite systems that are not necessarily terminating, but allow instead for transfinite derivations that have a limit. In particular, we give conditions for the existence of a limit and for its uniqueness and relate the operational and algebraic semantics of infinitary theories. We also consider sufficient completeness of hierarchical systems.

A heuristic triangulation algorithm

Abstract We propose a heuristic triangulation algorithm. The algorithm runs in polynomial time and produces a triangulation within a ratio of O (log n ) to the cost of an optimal triangulation of a set of n points in the Euclidian plane.

Ordered Semantic Hyper-Linking

The ordered semantic hyper-linking strategy is complete for first-order logic and accepts a user-specified natural semantics that guides the search for a proof. Any semantics in which the meanings of the function and predicate symbols are computable on ground terms may be used. This instance-based strategy is efficient on near-propositional problems, is goal sensitive, and has an extension to equality and term rewriting. However, it sometimes has difficulty generating large terms. We compare this strategy with some others that use semantic information, and present a proof of soundness and completeness. We also give some theoretical results about the search efficiency of the strategy. Some examples illustrate the performance of the strategy.

New NP-hard and NP-complete polynomial and integer divisibility problems

Abstract We show that some problems involving sparse polynomials are NP-hard. For example, it is NP-hard to determine if a sparse polynomial has a root of modulus 1, and it is NP-hard to decide if two sparse polynomials are not relatively prime. Also, we show that a divisibility problem involving an unbounded number of sparse polynomials is NP-complete using a theorem of Linnik concerning the distribution of primes in arithmetic sequences. From these results it follows that certain problems involving inequalities, recurrence relations, differential equations, and eigenvalues of sparse matrices are NP-hard. Problems involving divisibility properties of two sparse polynomials, divisibility of sparse binary numbers and ring homomorphisms are also NP-hard.