Inductive and Coinductive Techniques in the Operational Analysis of Functional Programs: an Introduction

Many works have shown that operational semantics is a useful framework for the formal study of programs properties. Our investigation takes as object of study a call-by-name variant of FPC, a functional language with higher order functions and recursive types. We show that by using Plotkin’s structural operational semantics it is possible to specify formal semantics for FPC for both convergent and divergent programs. Structural operational semantics can be specified by means of rules, therefore it allows to use induction and coinduction principles, to define objects and to reason about their properties. Two approaches are well known in giving structural operational semantics for programming languages: big-step and small-step. For a language like FPC they can be related by showing that they capture the same meaning. Furthermore, we show that it is possible to translate a proof in the big-step formalism in a proof in the small-step by adding some information to rules. Each formal theory of programming languages must analyze equality in-depth. In an operational setting, Morris’s style contextual equivalence is the widely accepted natural notion of equivalence for programs. Many properties of contextual equivalence can be proved by using simple proof principles. In particular, we prove two interesting properties, syntactic continuity and rational completeness. They can thought of as the syntactical counterparts of well-known notions in domain theory. Unfortunately, contextual equivalence is not straightforward to prove. Nevertheless it is coinductive in nature, for this reason it is interesting to find different formulations which permit to coinductively reason about program equality. We show that two classical programs equivalence notions, named experimental equivalence and bisimilarity respectively, coincide with contextual equiavalence for FPC. Finally we conclude by showing an example of how the theory developed in this work can be used in programming practice.