New Foundations for Fixpoint Computations: FIX-Hyperdoctrines and the FIX-Logic

Abstract This paper introduces a new higher-order typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi (in “Proceedings, 4th Annual Symposium on Logic in Computer Science,” pp. 14–23, IEEE Comput. Soc. Press, Washington, 1989) and contains a version of Martin-Lof's “iteration type” (in “Proceedings, Workshop on Semantics in Programming Laguages,” Chalmers University, 1983) . The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial and total correctness statements about evaluation of computations to values. The constructive nature of the logic is witnessed by strong metalogical properties which are proved using a category-theoretic version of the “logical relations” method ( Plotkin, unpublished lecture notes from CSLI Summer School, 1985 ).