Formalizing Representation Theorems for a Logical Framework with Rewriting

2025-04-10

AI Summary

This paper addresses the lack of systematic support for theory morphisms and logical relations in the $λΠ$-calculus modulo rewriting framework. Methodologically, it introduces a unified extension mechanism that formally integrates both concepts for the first time within this framework and designs a pattern-based invariant verification procedure, reducing the proof of translation invariants to finite, decidable propositional checks. The main contributions are: (1) a structurally clear, machine-verifiable formalization of inductive translations—e.g., type erasure; (2) the first fully verified type-erasure instance in $λΠ$-calculus modulo rewriting; and (3) a reusable methodology for rigorously verifying the correctness of translations between formal systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Representation theorems for formal systems often take the form of an inductive translation that satisfies certain invariants, which are proved inductively. Theory morphisms and logical relations are common patterns of such inductive constructions. They allow representing the translation and the proofs of the invariants as a set of translation rules, corresponding to the cases of the inductions. Importantly, establishing the invariants is reduced to checking a finite set of, typically decidable, statements. Therefore, in a framework supporting theory morphisms and logical relations, translations that fit one of these patterns become much easier to formalize and to verify. The $lambdaPi$-calculus modulo rewriting is a logical framework designed for representing and translating between formal systems that has previously not systematically supported such patterns. In this paper, we extend it with theory morphisms and logical relations. We apply these to define and verify invariants for a number of translations between formal systems. In doing so, we identify some best practices that enable us to obtain elegant novel formalizations of some challenging translations, in particular type erasure translations from typed to untyped languages.

Problem

• Extend λΠ-calculus to support theory morphisms and logical relations
• Simplify formalization and verification of inductive translation invariants
• Apply new framework to type erasure translations between systems

Innovation

• Extends λΠ-calculus with theory morphisms
• Incorporates logical relations for translations
• Simplifies invariant verification via decidable rules