Coinductive big-step operational semantics

TL;DR

This paper introduces coinductive big-step operational semantics for call-by-value languages, enabling the formal description of diverging evaluations and establishing their equivalence with small-step semantics, all formalized in Coq.

Contribution

It presents a novel coinductive approach to big-step semantics, formalized in Coq, that captures divergence and connects with small-step semantics.

Findings

Coinductive big-step semantics can describe diverging evaluations.

Formal proofs of equivalence between big-step and small-step semantics.

Application of coinductive semantics to proofs of type soundness.

Abstract

Using a call-by-value functional language as an example, this article illustrates the use of coinductive definitions and proofs in big-step operational semantics, enabling it to describe diverging evaluations in addition to terminating evaluations. We formalize the connections between the coinductive big-step semantics and the standard small-step semantics, proving that both semantics are equivalent. We then study the use of coinductive big-step semantics in proofs of type soundness and proofs of semantic preservation for compilers. A methodological originality of this paper is that all results have been proved using the Coq proof assistant. We explain the proof-theoretic presentation of coinductive definitions and proofs offered by Coq, and show that it facilitates the discovery and the presentation of the results.