MARTIN-LÖF'S TYPE THEORY AS AN OPEN-ENDED FRAMEWORK

Abstract

This paper treats Martin-Löf's type theory as an open-ended framework composed of (i) flexibly extensible languages into which various forms of objects and types can be incorporated, (ii) their uniform, effectively given semantics, and (iii) persistently valid inference rules. The class of expression systems is introduced here to define an open-ended body of languages underlying the theory. Each expression system consists of two parts: the computational part is a structured lazy evaluation system with a bisimulation-like program equivalence; the structural part is a system of strictly positive inductive definitions for type constructors in terms of partial equivalence relations. Types and their objects are uniformly and inductively constructed from a given expression system as a type system, which can provide a semantics of the theory. Building on these concepts, this paper presents two main results. First, all the inference rules of the theory are sound; that is, they remain valid in every type system built from an extension of an initial expression system. This result gives a characterization of the class of types that can be introduced into the theory. Second, each type system is complete with respect to the underlying bisimulation-like program equivalence. This result provides a useful form of type-free equational reasoning in the theory.