Category-based Semantics for Equational and Constraint Logic Programming

This thesis proposes a general framework for equational logic programming, called catf:gory­ based equational logic by placing the general principles underlying the design of the pro­ gramming language Eqlog and formulated by Goguen and Meseguer into an abstract form. This framework generalises equational deduction to an arbitrary category satisfy­ ing certain natural conditions; completeness is proved under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as modcls rather than variables, and regards valuations as model morprusms rather than functions. This is used as a basis for a model theoretic category-based approach to a paramodulation-based op­ erationa.l semantics for equational logic programming languages. Category-based equational logic in conjunction with the theory of institutions is used to give mathematical fonndations for modularisation in equational logic programming. We study the soundness and completeness problem for module imports in the coutext of a category-based semantics for solutions to equational logic programming queries. Constraint logic programming is integrated into the equational logic programming paradigm by showing tha.t constraint logics are a particular case of category-based equa­ tionallogic. This follows the methodology of free expansions of models for built-ins along signature inclusions as sketched by Goguen and Meseguer in their papers on Eqlog. The mathematical foundations of constraint logic programming are based on a Herbrand The­ orem for constraint logics; this is obtained as an instance of a more general category-based version of Herbrand's Theorem. The results in this tbesis apply to equational and constraint logic programming lan­ guages that a.re based on a variety of equational logical systems including many and order sorted equational logics, Horn clause logic, equational logic modulo a theory, con­ straint logics, and more, as well as any possible combination between them. More impor­ tantly, this thesis gives tbe possibility for developing the equational logic (programming) paradigm over non-conventional structnres and thus significantly extending it beyond its tradition.