First-order Lax Logic as a framework for Constraint Logic Programming

In this report we introduce a new proof theoretic approach to the semantics of Constraint Logic Programming based on an intuitionistic rst order modal logic called QLL The dis tinguishing feature of this new approach is that the logic calculus of QLL is used not only to capture the usual extensional aspects of Logic Programming i e which queries are success ful but also some of the intensional aspects i e what is the answer constraint and how is it constructed It provides for a direct link between the model theoretic and the operational semantics following a formulas as programs and proofs as constraints principle This approach makes use of logic in a di erent way than other approaches based on logic calculi On the one side it is to be distinguished from the well known provability semantics which is concerned merely with what is derivable as opposed to how it is derivable paying attention to the fact that it is the how that determines the answer constraint On the other side our approach is distinguished from so called external logic characterizations of the operational semantics of logic programming There operational semantics are axiomatized in a classical meta logic specifying the program behaviour in terms of successful and failing queries Here in contrast QLL is used as an internal logic in which operational semantics are not di erentiated by di erent axiom systems but by di erent rule systems Or to put it at another level Formulas in QLL do not specify properties of programs but the programs themselves Technically we rst extend existing work on Propositional Lax Logic PLL to a rst order language QLL presenting a soundness and completeness theorem for a Gentzen style system via a syntactic translation into a classical rst order bimodal theory Previous work on applying Lax Logic to deal with behavioural constraints in formal hardware veri cation has demonstrated the complementary nature of abstraction and constraints we proceed to show how the Lax Logic Programming LLP fragment of QLL can reveal abstractions behind Constraint Logic Programming CLP Our main tool is an intensional rst order extension of the Curry Howard isomorphism between natural deduction proofs in PLL and terms of the computational lambda calculus Instantiating the monadic operator of QLL by a generic notion of constraint computation we factor a concrete CLP program into two parts an abstract LLP program and an associated constraint table These tables allow us to recover concrete answer constraints for CLP programs from abstract LLP derivations and thus to establish precise proof and model theoretic connections between our Lax logical account of CLP and existing work Choosing di erent notions of constraint allows us to generalize the standard notion of constraint as in CLP and to apply the LLP paradigm to the complementary problems of program abstraction and program re nement Department of Computer Science The University of She eld Regent Court Portobello St She eld S DP Emails m fairtlough dcs shef ac uk m walton dcs shef ac uk yDepartment of Mathematics and Computer Science University of Passau Innstra e D Passau Email mendler fmi uni passau de The author is supported by the Deutsche Forschungsgemeinschaft Technical Report University of Passau MIP July