Category-based constraint logic

The research reported in this paper exploits the view of constraint programming as computation in a logical system, namely constraint logic. The basic ingredients of constraint logic are: constraint models for the semantics (they form a comma-category over a fixed model of ‘built-ins’); generalized polynomials in the role of basic syntactic ingredient; and a constraint satisfaction relation between semantics and syntax. Category-based constraint logic means the development of the logic is abstract categorical rather than concrete set theoretical.We show that (category-based) constraint logic is an institution, and we internalize the study of constraint logic to the abstract framework of category-based equational logic, thus opening the door for considering constraint logic programming over non-standard structures (such as CPO's, topologies, graphs, categories, etc.). By embedding category-based constraint logic into category-based equational logic, we integrate the constraint logic programming paradigm into (category-based) equational logic programming. Results include completeness of constraint logic deduction, a novel Herbrand theorem for constraint logic programming characterizing Herbrand models as initial models in constraint logic, and logical foundations for the modular combination of constraint solvers based on amalgamated sums of Herbrand models in the constraint logic institution.

The resolution principle for ω+-valued logic

A resolution-style theorem proving system for the ω+-valued Post logic is developed. The soundness and the completeness of the system are proved. The two versions of the Herbrand theorem for the logic considered are given.

Algorithmic logic and its applications in the theory of programs I

The paper presents tools for formalizing and proving properties of programs. The language of algorithmic logic constitutes an extension of a programming language by formulas that describe algorithmic properties. The paper contains two axiomatizations of algorithmic logic, which are complete. It can be proved that every valid algorithmic property possesses a formal proof. An analogue of Herbrand theorem and a theorem on the normal form of a program are proved. Results of meta-mathematical character are applied to theory of programs, e.g. Paterson’s theorem is an immediate corollary to Herbrand’s theorem.