On a Dependently Typed Encoding of Matching Logic

2025-09-16

AI Summary

This paper addresses the lack of a formal type-theoretic foundation for matching μ-logic in programming language semantics. We propose the first encoding of matching μ-logic within dependent type theory: sorts are embedded as indices in types, thereby establishing a strict correspondence between well-sortedness in the object language and well-typedness in the host type system. This construction inherently excludes ill-formed syntactic terms and guarantees that the semantic interpretation of any well-sorted term resides precisely within the semantic domain of its associated type. Our key contribution is the first fully dependent-type-theoretic formulation of matching μ-logic, unifying syntactic sorting constraints with type checking. Moreover, this encoding provides a machine-verifiable foundation for meta-theoretic reasoning, significantly enhancing the rigor and automation potential of semantic models. The approach enables formal verification of logical properties directly within proof assistants supporting dependent types.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Matching logic is a general formal framework for reasoning about a wide range of theories, with particular emphasis on programming language semantics. Notably, the intermediate language of the K semantics framework is an extension of matching $μ$-logic, a sorted, polyadic variant of the logic. Metatheoretic reasoning requires the logic to be expressed within a foundational theory; opting for a dependently typed one enables well-sortedness in the object theory to correspond directly to well-typedness in the host theory. In this paper, we present the first dependently typed definition of matching $μ$-logic, ensuring well-sortedness via sorted contexts encoded in type indices. As a result, ill-sorted syntax elements are unrepresentable, and the semantics of well-sorted elements are guaranteed to lie within the domain of their associated sort.

Problem

• Defining matching logic in dependent types
• Ensuring well-sortedness through type indices
• Making ill-sorted syntax elements unrepresentable

Innovation

• Dependently typed matching logic encoding
• Sorted contexts in type indices
• Ensures well-sortedness via typing