Metacat: a categorical framework for formal systems

2026-04-09

AI Summary

This work unifies the modeling of inference rules and proof structures of formal systems within a categorical framework. It introduces a met-variable context representation based on Cartesian PROPs, encodes assumptions and conclusions using spans, and constructs a symmetric monoidal category of proofs with met-variable substitution as the sole primitive operation. This approach is the first to uniformly embed both inference rules and proof structures into the semantics of symmetric monoidal categories, thereby supporting compositional and reusable handling of hypotheses. The authors implement an open-source verification algorithm and surface syntax, successfully encoding formulas, axioms, and representative derivations of first-order logic, and release a functional proof checker.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

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
• Security and Privacy: Data transparency and provenance

Abstract

We present a categorical framework for formal systems in which inference rules with $m$ metavariables over a category of syntax $\mathscr{S}$, taken to be a cartesian PROP, are represented by operations of arity $k \to n$ equipped with spans $k \leftarrow m \to n$ in $\mathscr{S}$, encoding the hypotheses and conclusions in a common metavariable context. Composition is by substitution of metavariables, which is the sole primitive operation, as in Metamath. Proofs in this setting form a symmetric monoidal category whose monoidal structure encodes the combination and reuse of hypotheses. This structure admits a proof-checking algorithm; we provide an open-source implementation together with a surface syntax for defining formal systems. As a demonstration, we encode the formulae and inference rules of first-order logic in Metacat, and give axioms and representative derivations as examples.

Problem

• formal systems
• categorical framework
• inference rules
• metavariables
• proof checking

Innovation

• categorical logic
• metavariable substitution
• symmetric monoidal category
• formal proof systems
• cartesian PROP