We present an algorithm for the generation of prime implicates in equational logic, that is, of the most general consequences of formulæ containing equations and disequations between first-order terms. This algorithm is defined by a calculus that is proved to be correct and complete. We then focus on the case where the considered clause set is ground, i.e., contains no variables, and devise a specialized tree data structure that is designed to efficiently detect and delete redundant implicates. The corresponding algorithms are presented along with their termination and correctness proofs. Finally, an experimental evaluation of this prime implicate generation method is conducted in the ground case, including a comparison with state-of-the-art propositional and first-order prime implicate generation tools.

Abduction has been extensively studied in propositional logic because of its many applications in artificial intelligence. However, its intrinsic complexity has been a limitation to the implementation of abductive reasoning tools in more expressive logics. We have devised such a tool in ground flat equational logic, in which literals are equations or disequations between constants. Our tool is based on the computation of prime implicates. It uses a relaxed paramodulation calculus, designed to generate all prime implicates of a formula, together with a carefully defined data structure storing the implicates and able to efficiently detect, and remove, redundancies. In addition to a detailed description of this method, we present an analysis of some experimental results.