A robust graph-based approach to observational equivalence

TL;DR

A new step-wise approach to proving observational equivalence, and in particular reasoning about fragility of observational equivalence, based on what is called local reasoning, which is capable of proving a generalised notion of observational equivalence.

Abstract

We propose a new step-wise approach to proving observational equivalence, and in particular reasoning about fragility of observational equivalence. Our approach is based on what we call local reasoning. The local reasoning exploits the graphical concept of neighbourhood, and it extracts a new, formal, concept of robustness as a key sufficient condition of observational equivalence. Moreover, our proof methodology is capable of proving a generalised notion of observational equivalence. The generalised notion can be quantified over syntactically restricted contexts instead of all contexts, and also quantitatively constrained in terms of the number of reduction steps. The operational machinery we use is given by a hypergraph-rewriting abstract machine inspired by Girard's Geometry of Interaction. The behaviour of language features, including function abstraction and application, is provided by hypergraph-rewriting rules. We demonstrate our proof methodology using the call-by-value lambda-calculus equipped with (higher-order) state.