## Canonized Rewriting and Ground AC Completion Modulo Shostak Theories## Sylvain Conchon and Évelyne Contejean and Mohamed Iguernelala |

Abstract:AC-completion efficiently handles equality modulo associative and commutative function symbols. When the input is ground, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground AC-completion for deciding formulas in the combination of the theory of equality with user-defined AC symbols, uninterpreted symbols and an arbitrary signature disjoint Shostak theoryX. Our algorithm, called AC(X), is obtained by augmenting in a modular way ground AC-completion with the canonizer and solver present for the theoryX. This integration rests on canonized rewriting, a new relation reminiscent to normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is proved sound, complete and terminating, and is implemented to extend the core of the Alt-Ergo theorem prover.

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