Conditional Equational Theories and Complete Sets of Transformations

The idea to combine the advantages of function and logic programming has attracted many researches. Their work ranges from the integration of existing languages over higher-order logic to equational logic languages, where logic programs are augmented with equational theories. Recently, it has been proposed to handle those equational theories by complete sets of transformations. These transformations are extensions of the rules introduced by Herbrand and later used by Martelli and Montanari to compute the most general uniier of two expressions. We generalize this idea to complete sets of transformations for arbitrary conditional equational theories, the largest class of equational theories that admit a least Herbrand model. The completeness proof is based on the observation that each refutation with respect to linear paramodulation and reeection can be modelled by the transformations. As certain conditions imposed on an equational theory restrict the search space generated by paramodulation and reeection we can easily reene our transformations { due to the completeness proof { if the conditional equational theory is ground connuent or canonical.