Martin-Löf’s intuitionistic type theory is a widely-used framework for constructive mathematics and computer programming. In its most popular form, type theory consists of a collection of inference rules inductively defining formal proofs. These rules are justified by Martin-Löf’s meaning explanations, which extend the Brouwer-Heyting-Kolmogorov interpretation of connectives to a rich collectio...

This paper describes a framework, based on Abstract Interpretation, for creating abstractions for model-checking. Specifically, we study how to abstract models of calculus and systematically derive abstractions that are constructive, sound, and precise, and apply them to abstracting Kripke structures. The overall approach is based on the use of bilattices to represent partial and inconsistent i...

A modified realisability interpretation of infinitary logic is formalised and proved sound in constructive type theory (CTT). The logic considered subsumes first order logic. The interpretation makes it possible to extract programs with simplified types and to incorporate and reason about them in CTT. 1 Modified realisability Modified realisability interpretation is a well-known method for givi...

We provide a constructive and intensional interpretation for the speciication language Z in a theory of operations and kinds T. The motivation is to facilitate the development of an integrated approach to program construction. We illustrate the new foundations for Z with examples.

In this paper we introduce a new axiom scheme, the Relation Reflection Scheme (RRS), for constructive set theory. Constructive set theory is an extensional set theoretical setting for constructive mathematics. A formal system for constructive set theory was first introduced by Myhill in [8]. In [1, 2, 3] I introduced a formal system CZF that is closely related to Myhill’s formal system and gave...

It is a tempting idea to use formal existence proofs as a means to precisely and verifiably express algorithmic ideas. This is clearly possible for “constructive” proofs, which are informally understood via the BrouwerHeyting-Kolmogorov interpretation (BHK-interpretation for short). This interpretation of intuitionistic (and minimal) logic explains what it means to prove a logically compound st...