From SOS rules to proof principles

Abstract

Structural Operational Semantics (SOS) is a widely used formalism for specifying the computational meaning of programs, and is commonly used in specifying the semantics of functional languages. Despite this widespread use there has been relatively little work on the for such semantics. As a consequence the operational approach to reasoning is considered ad hoc since the same basic proof techniques and reasoning tools are reestablished over and over, once for each operational semantics specification. This paper develops some metatheory for a certain class of SOS language specifications for functional languages. We define a rule format, Globally Deterministic SOS (GDSOS), and establish some proof principles for reasoning about equivalence which are sound for all languages which can be expressed in this format. More specifically, if the SOS rules for the operators of a language conform to the syntax of the GDSOS format, then - a syntactic analogy of continuity holds, which relates a recursive function to its finite unwindings, and forms the basis of a Scott-style fixed-point induction technique; - a powerful induction principle called improvement induction holds for a certain class of GDSOS semantics; the Improvement Theorem from [Sands, POPL'95] is a simple corollary; - a useful bisimulation-based coinductive proof technique for operational approximation (and its instrumented variants) is sound.