Fast Decision Procedures Based on Congruence Closure

The notion of the congruence closure of a relation on a graph is 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 is then given for the quantifier-free theory of LISP list structure based on the congruence closure algorithm. Both decision procedures determine the satisfiability of a conjunction of literals of length n in average time O ( n log n ) using the fastest known congruence closure algorithm. It is also shown that if the axiomatization of the theory of list structure is changed slightly, the problem of determining the satisfiability of a conjunction of literals becomes NP-complete. The decision procedures have been implemented in the authors' simplifier for the Stanford Pascal Verifier.