Substructural Abstract Syntax with Variable Binding and Single-Variable Substitution

2025-05-30

AI Summary

This paper addresses the problem of modeling abstract syntax with variable binding and capture-avoiding single-variable substitution under four substructural contexts: linear, affine, relevant, and Cartesian. Methodologically, it introduces the first unified categorical framework that simultaneously defines binding signature functors, free algebras equipped with substitution axioms, and generalized structural recursion—all within each of the four substructural constraints. The theoretical core establishes the initiality of substitution algebras as their essential categorical property. Contributions include: (1) provably correct single-variable substitution operations for each substructural type; (2) a complete algebraic characterization of such syntax; and (3) foundational support for modular, semantics-preserving syntactic modeling of formal languages and type systems. The framework ensures syntactic well-formedness and semantic safety across diverse substructural disciplines, bridging categorical semantics with practical language design.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Structured Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

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: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT

Abstract

We develop a unified categorical theory of substructural abstract syntax with variable binding and single-variable (capture-avoiding) substitution. This is done for the gamut of context structural rules given by exchange (linear theory) with weakening (affine theory) or with contraction (relevant theory) and with both (cartesian theory). Specifically, in all four scenarios, we uniformly: define abstract syntax with variable binding as free algebras for binding-signature endofunctors over variables; provide finitary algebraic axiomatisations of the laws of substitution; construct single-variable substitution operations by generalised structural recursion; and prove their correctness, establishing their universal abstract character as initial substitution algebras.

Problem

• Develop unified categorical theory for substructural abstract syntax
• Define abstract syntax with variable binding as free algebras
• Construct and prove correctness of single-variable substitution operations

Innovation

• Unified categorical theory for substructural syntax
• Free algebras define abstract binding syntax
• Generalized recursion constructs substitution operations