Modularity and Combination of Associative Commutative Congruence Closure Algorithms enriched with Semantic Properties

Algorithms for computing congruence closure of ground equations overuninterpreted symbols and interpreted symbols satisfying associativity andcommutativity (AC) properties are proposed. The algorithms are based on aframework for computing a congruence closure by abstracting nonflat terms byconstants as proposed first in Kapur's congruence closure algorithm (RTA97).The framework is general, flexible, and has been extended also to developcongruence closure algorithms for the cases when associative-commutativefunction symbols can have additional properties including idempotency,nilpotency, identities, cancellativity and group properties as well as theirvarious combinations. Algorithms are modular; their correctness and terminationproofs are simple, exploiting modularity. Unlike earlier algorithms, theproposed algorithms neither rely on complex AC compatible well-foundedorderings on nonvariable terms nor need to use the associative-commutativeunification and extension rules in completion for generating canonical rewritesystems for congruence closures. They are particularly suited for integratinginto the Satisfiability modulo Theories (SMT) solvers. A new way to viewGroebner basis algorithm for polynomial ideals with integer coefficients as acombination of the congruence closures over the AC symbol * with the identity 1and the congruence closure over an Abelian group with + is outlined