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