🤖 AI Summary
This paper addresses the challenges of fragility analysis and poor constructivity in observational equivalence proofs. We propose a stepwise reasoning method based on hypergraph rewriting. Our key contributions are threefold: (1) We formally characterize robustness as a critical sufficient condition for observational equivalence—a novel formulation; (2) We establish a neighborhood-based local reasoning framework that supports generalized observational equivalence definitions and verification under syntactically restricted contexts and quantitative step bounds; (3) Leveraging a hypergraph rewriting abstract machine inspired by geometric interaction, we structurally model function abstraction and application in higher-order stateful lambda calculus. Experimental evaluation on call-by-value lambda calculus demonstrates that our approach significantly improves modularity, constructivity, and analyzability of fragility in observational equivalence proofs.
📝 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.