# A robust graph-based approach to observational equivalence

## TL;DR

This paper introduces a graph-based, local reasoning approach to proving observational equivalence, including its fragility, using hypergraph-rewriting and a new robustness concept, applicable to complex language features.

## Contribution

It presents a novel step-wise proof methodology leveraging local graph reasoning and hypergraph-rewriting to establish a generalized, quantitative notion of observational equivalence.

## Key findings

- The approach effectively proves observational equivalence in lambda-calculus with state.
- It introduces a formal robustness concept as a key condition for equivalence.
- The methodology can handle syntactically restricted and quantitatively constrained contexts.

## 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.