Abstract

This chapter presents some basic issues in the type-theoretic study of programming language concepts. It presents Church's typed lambda calculus as an illustrative example. The chapter also presents a survey of typing features such as polymorphism, data abstraction, and automatic inference of type information. The main topics discussed in the study of typed lambda calculus are context-sensitive syntax and typing rules, a reduction system modeling program execution, an equational axiom system, and semantic models. Henkin models are used as the primary semantic framework with emphasis on three examples: set-theoretic, recursion-theoretic, and domain-theoretic hierarchies of functions. The chapter presents completeness theorems for arbitrary Henkin models and models without empty types, followed by an introduction to Cartesian closed categories and Kripke-style models. The technique of logical relations is used to prove a semantic completeness theorem and Church–Rosser and strong normalization properties of reduction. Logical relations also encompass basic model-theoretic constructions such as quotients.

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