Wayne Snyder

Basic paramodulation

We introduce a class of restrictions for the ordered paramodulation and superposition calculi (inspired by the basic strategy for narrowing), in which paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. In addition we introduce restrictions based on term selection rules and redex orderings, which are general criteria for delimiting the terms which are available for inferences. These refinements are compatible with standard ordering restrictions and are complete without paramodulation into variables or using functional reflexivity axioms. We prove refutational completeness in the context of deletion rules, such as simplification by rewriting (demodulation) and subsumption, and of techniques for eliminating redundant inferences.

Higher-order unification revisited: Complete sets of transformations

In this paper, we reexamine the problem of general higher-order unification and develop an approach based on the method of transformations on systems of terms which has its roots in Herbrands thesis, and which was developed by Martelli and Montanari in the context of first-order unification. This method provides an abstract and mathematically elegant means of analyzing the invariant properties of unification in various settings by providing a clean separation of the logical issues from the specification of procedural information. Our major contribution is three-fold. First, we have extended the Herbrand-Martelli-Montanari method of transformations on systems to higher-order unification and pre-unification; second, we have used this formalism to provide a more direct proof of the completeness of a method for higher-order unification than has previously been available; and, finally, we have shown the completeness of the strategy of eager variable elimination. In addition, this analysis provides another justification of the design of Huets procedure, and shows how its basic principles work in a more general setting. Finally, it is hoped that this presentation might form a good introduction to higher-order unification for those readers unfamiliar with the field.

Basic Paramodulation and Superposition

We introduce a class of restrictions for the ordered paramodulation and superposition calculi (inspired by the basic strategy for narrowing), in which paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. These refinements are compatible with standard ordering restrictions and are complete without paramodulation into variables or using functional reflexivity axioms. We prove refutational completeness in the context of deletion rules, such as simplification by rewriting (demodulation) and subsumption, and of techniques for eliminating redundant inferences. Finally, we discuss experimental data obtained from a modification of Otter.

A proof theory for general unification

1: Introduction.- 2: Preview.- 3: Preliminaries.- 3.1 Algebraic Background.- 3.2 Substitutions.- 3.3 Unification by Transformations on Systems.- 3.4 Equational Logic.- 3.5 Term Rewriting.- 3.5.1 Termination Orderings.- 3.5.2 Confluence.- 3.6 Completion of Equational Theories.- 4: E-Unification.- 4.1 Basic Definitions and Results.- 4.2 Methods for E-Unification.- 5: E-Unification via Transformations.- 5.1 The Set of Transformations BT.- 5.2 Soundness of the Set BT.- 5.3 Completeness of the Set BT.- 6: An Improved Set of Transformations.- 6.1 Ground Church-Rosser Systems.- 6.2 Completeness of the Set T.- 6.3 Surreduction.- 6.4 Completeness of the Set T Revisited.- 6.5 Relaxed Paramodulation.- 6.6 Previous Work.- 6.7 Eager Variable Elimination.- 6.8 Current and Future Work.- 6.9 Conclusion.- 7: Higher Order Unification.- 7.1 Preliminaries.- 7.2 Higher Order Unification via Transformations.- 7.2.1 Transformations for Higher Order Unification.- 7.2.2 Soundness of the Transformations.- 7.2.3 Completeness of the Transformations.- 7.3 Huets Procedure Revisited.- 7.4 Conclusion.- 8: Conclusion.- Appendices.

A Fast Algorithm for Generating Reduced Ground Rewriting Systems from a Set of Ground Equations

In this paper we give a fast method for generating reduced sets of rewrite rules equivalent to a given set of ground equations. Since reduced sets of ground rewrite rules are in fact canonical, this is an efficient Knuth-Bendix procedure for the ground case. The dominant cost of the algorithm is for congruence closure, so the method runs in O(n log n) time and quadratic space, or in O(n (log n)2) time and linear space, where n is the number of occurrences of symbols in the original set of ground equations E. We also show how our method provides a precise characterization of the (finite) collection of all reduced sets of rewrite rules equivalent to a given ground set of equations E, and prove that our algorithm is complete in the sense that it can enumerate every member of this collection. Finally, we show that a modified version of this procedure can produce a reduced ground rewriting system contained in the lexicographic path ordering generated by a given total precedence ordering on the symbols of E , still in a worst-case time of O(n log n).

An algorithm for finding canonical sets of ground rewrite rules in polynomial time

In this paper, it is shown that there is an algorithm that, given by finite set <italic>E</italic> of ground equations, produces a reduced canonical rewriting system <italic>R</italic> equivalent to <italic>E</italic> in polynomial time. This algorithm based on congruence closure performs simplification steps guided by a total simplification ordering on ground terms, and it runs in time <italic>O(n<supscrpt>3</supscrpt>)</italic>.

Rigid E -unification: NP-completeness and applications to equational matings

Abstract Rigid E -unification is a restricted kind of unification modulo equational theories, or E -unification, that arises naturally in extending Andrews theorem proving method of matings to first-order languages with equality. This extension was first presented by J. H. Gallier, S. Raatz, and W. Snyder, who conjectured that rigid E -unification is decidable. In this paper, it is shown that rigid E -unification is NP-complete and that finite complete sets of rigid E -unifiers always exist. As a consequence, deciding whether a family of mated sets is an equational mating is an NP-complete problem. Some implications of this result regarding the complexity of theorem proving in first-order logic with equality are also discussed.

Higher Order E-Unification

In this extended abstract we report on an investigation of Higher-Order E-Unification, which consists of unifying typed lambda terms in the context of a first-order set of equations E. This problem subsumes both higher-order unification and general E-unification, and provides a theoretical background for reasoning systems which incorporate both algebraic and higher-order logic. The problem and its properties are discussed, a set of transformations (inference rules) extending those of Martelli-Montanari for standard unification is given, and then we prove the completeness of this non-deterministic algorithm. The completeness of restrictions of these rules for higher-order pre-E-unification and higher-order narrowing are corollaries of these results. Finally, we connect these results with previous work, and conclude with future directions and open problems. The major result is a set of inference rules for higher-order E-unification and a proof of its soundness and completeness (wrt complete sets of unifiers).

Finding Canonical Rewriting Systems Equivalent to a Finite Set of Ground Equations in Polynomial Time

In this paper, it is shown that there is an algorithm which, given any finite set E of ground equations, produces a reduced canonical rewriting system R equivalent to E in polynomial time. This algorithm based on congruence closure performs simplification steps guided by a total simplification ordering on ground terms, and it runs in time O(n3).

Goal directed strategies for paramodulation

It is well-known that the set of support strategy is incomplete in paramodulation theorem provers if paramodulation into variables is forbidden. In this paper, we present a paramodulation calculus for which the combination of these two restrictions is complete, based on a lazy form of the paramodulation rule which delays parts of the unification step. The refutational completeness of this method is proved by transforming proofs given by other paramodulation strategies into set of support proofs using this new inference rule. Finally, we consider the completeness of various refinements of the method, and conclude by discussing related work and future directions.