Nominal Matching Logic

TL;DR

Nominal Matching Logic (NML) extends Matching Logic with names and binding, enabling inductive reasoning over syntax with binding, such as $b5$-terms, which was not possible with previous encoding approaches.

Contribution

NML introduces a formal system combining Matching Logic with nominal techniques, supporting inductive reasoning over syntax with binding, and providing a foundation for reasoning about languages like the lambda calculus.

Findings

Proves meta-theoretical properties of NML.

Demonstrates inductive reasoning over lambda calculus terms.

Shows NML's expressive power with examples.

Abstract

We introduce Nominal Matching Logic (NML) as an extension of Matching Logic with names and binding following the Gabbay-Pitts nominal approach. Matching logic is the foundation of the $\mathbb{K}$ framework, used to specify programming languages and automatically derive associated tools (compilers, debuggers, model checkers, program verifiers). Matching logic does not include a primitive notion of name binding, though binding operators can be represented via an encoding that internalises the graph of a function from bound names to expressions containing bound names. This approach is sufficient to represent computations involving binding operators, but has not been reconciled with support for inductive reasoning over syntax with binding (e.g., reasoning over $\lambda$-terms). Nominal logic is a formal system for reasoning about names and binding, which provides well-behaved and powerful principles for inductive reasoning over syntax with binding, and NML inherits these principles. We discuss design alternatives for the syntax and the semantics of NML, prove meta-theoretical properties and give examples to illustrate its expressive power. In particular, we show how induction principles for $\lambda$-terms ($\alpha$-structural induction) can be defined and used to prove standard properties of the $\lambda$-calculus.