A simple class of Kripke-style models in which logic and computation have equal standing

Authors:
Michael Gabbay;Murdoch J. Gabbay
Affiliations:
-;-
Venue:
LPAR'10 Proceedings of the 16th international conference on Logic for programming, artificial intelligence, and reasoning
Year:
2010

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Abstract

We present a sound and complete model of lambda-calculus reductions based on structures inspired by modal logic (closely related to Kripke structures). Accordingly we can construct a logic which is sound and complete for the same models, and we identify lambda-terms with certain simple sentences (predicates) in this logic, by direct compositional translation. Reduction then becomes identified with logical entailment. Thus, the models suggest a new way to identify logic and computation. Both have elementary and concrete representations in our models; where these representations overlap, they coincide. In a concluding speculation, we note a certain subclass of the models which seems to play a role analogous to that played by the cumulative hierarchy models in axiomatic set theory and the natural numbers in formal arithmetic -- there are many models of the respective theories, but only some, characterised by a fully second order interpretation, are the 'intended' ones.