Fast Decision Procedures Based on Congruence Closure

Abstract

ABSTRACT, The notion of the congruence closure of a relation on a graph ~s defined and several algorithms for computing it are surveyed A simple proof is given that the congruence closure algorithm provides a decision procedure for the quantifier-free theory of equality A decision procedure ts then given for the quanufier-free theory of LISP hst structure based on the congruence closure algorithm Both decision procedures determine the satisfiability of a conjunction of hterals of length n In average time O(n log n) using the fastest known congruence closure algorithm It is also shown that ff the axtomattzatton of the theory of list structure ts changed slightly, the problem of determmmg the satisfiabihty of a conjunction of hterals becomes NP-complete The decision procedures have been unplemented m the authors' simphfier for the Stanford Pascal Verifier KEY WORDS AND PHRASES' program verification, mechanical theorem proving, decision procedure, congruence closure, graph algorithms, theory of equahty, theory of recurstve data types CR CATEGORIES 5 21, 5.24, 5 25, 5.7 lntroductwnConsider the problem of verifying that one equality is a consequence of several other equalities, for example, thatf(f(a, b) All these problems reduce to the problem of constructing the "congruence closure" of a relation on a graph. In Section 2 we define th,s notion and describe a congruence closure algorithm which we implemented in 1976 for use in the theorem prover of the Stanford Pascal Verifier. Its worst-case time is O(m 2) for graphs with m edges. Downey, Seth1, and Tarjan [3] describe an algorithm with worst-case time O(m logZm), which, by using a hash table, can be made to run in average-case time O(m log m). We implemented this algorithm but did not find it faster than the simpler algorithm in our application.

Completion and Its Applications

“…This computation, by completion, of the "congruence closure" of a finite set of ground terms is reminiscent of the algorithm used in Nelson and Oppen (1980). Normal forms serve as names for classes that are eventually "merged" into congruence classes, each named by the minimal term in its class.…”

Introducing Euclidean Relations to Mizar

“…term identifications, term reductions [7], flexary connectives [8]) increasing computational power of VERIFIER (e.g. equational calculus [9], [10]). There have also been experiments on using specialized external systems to increase computational power of the Mizar system in selected domains [11], [12], [13].…”

On Rewriting Rules in Mizar

“…Effective algorithms computing congruence closure can be found in [5,10,11,14]. Congruence closure can be utilized to check if an equality of two expressions is a consequence of a set of other equalities.…”