First-Order Unification in Equational Theories and its Application to Logic Programming

This thesis studies first-order unification in equational theories, called E-unification, paying particular attention to complete unification algorithms for classes of equational theories. It also investigates how results and notions of E-unification can be applied to logic programming systems that support equality handling and functional notation. First-order unification in equational theories is proved to be complete. An E-unification algorithm using the narrowing process is then shown to be complete for the class of theories that can be described by a closed linear term rewriting system with the non-repetition property. This result sets the stage for investigation of applications to logic programming. An extended equational programming paradigm, referred to as equational logic programming, is proposed, which uses "narrowing" and "deletion upon successful unification" as the inference rules, and enjoys the semantic simplicity of the classical theory of equality. It is shown that the kind of programming features that were initially possessed solely by Prolog can also be provided in this extended equational programming paradigm. Finally, a computational model integrating functional programming and logic programming is described.