Paramodulation, superposition, and simplification

Abstract

Techniques for equational reasoning are a key component in many automated theorem provers and interactive proof and verification systems. A notable recent success in equational theorem proving has been the solution of an open problem (the “Robbins conjecture”) by William McCune with his prover Eqp [13].

Eqp is one of many equational theorem provers that use completion as the main deductive mechanism. Completion derives from the work of Knuth and Bendix [11] and is characterized by the extensive use of rewrite techniques (especially normalization by rewriting) for reasoning about equational theories. More specifically, the Knuth-Bendix procedure attempts to transform a given set of equations into a set of rewrite rules so that any two equivalent terms, and only equivalent terms, have identical normal forms. Not every equational theory can be presented as such a convergent rewrite system, but various refinements of the approach have led to the formulation of a refutationally complete method called ordered completion; the main contributions can be found in [12, 10, 8, 3, 1, 2].

The deductive inference rule used in completion procedures is superposition, which consists of first unifying one side of one equation with a subterm of another, and then applying the two possible equational replacements to this “overlapped” term. In ordered completion the selection of the two terms to be unified is guided by a given term ordering, which imposes certain restrictions on inferences (and thus usually results in a smaller search space, though also potentially longer proofs). The superposition rule is actually a restricted instance of a clausal inference rule, called paramodulation, that was proposed by Robinson and Wos [16]. (Informally, certain paramodulation inferences contain a superposition step applied to two equality literals selected from two given clauses.)

Paramodulation is often combined with resolution in clausal theorem provers and provides a refutationally complete inference system for clauses with equality. In its original form, paramodulation was not constrained by any of the restrictions that are considered to be indispensable for the efficiency of completion, but many improvements of paramodulation have been proposed since the inference rule was first introduced. In particular, orderings have been used to control the selection of the literals and subterms in them to be unified; see for instance [15, 9, 20, 19, 17, 4]. The most advanced variant of paramodulation is perhaps basic paramodulation, as described in [6, 14], where in addition to ordering restrictions one also prevents selection of subterms that have been obtained solely by instantiation of variables in previous inference steps.

Inference rules naturally form the core of any reasoning system. But the control of the proof search, and hence the theorem proving process, by a judicious use of techniques for simplifying formulas and eliminating (or avoiding) redundant formulas and inferences is often even more important. Typical simplification mechanisms are subsumption (i.e., elimination of subsumed clauses) and normalization by rewriting (of which demodulation [18] is essentially a special case). McCune [13], for instance, reports that the (successful) proof search on the Robbins problem required about eight days on a Sparc-5 class UNIX computer: less than 1% of the total search time was spent on deriving equations, while most of the time was spent on simplification. Similar observations pertain not only to equational theorem provers, but to (resolution-based) saturation methods in general (in the sense of [5]).

We will discuss (i) the fundamental ideas underlying paramodulation and superposition, (ii) a suitable notion of redundancy, (iii) specific simplification techniques, and (iv) the connection between deduction and simplification.

This research was supported in part by the National Science Foundation under grant CCR-9510072.