Modular abstract syntax trees (MAST): substitution tensors with second-class sorts

2025-11-06

AI Summary

Conventional monadic approaches fail to modularly handle binding, substitution, and hole structures in languages with second-order syntax (e.g., call-by-value λ-calculus, call-by-value let-binding). Method: We propose the first categorical framework for abstract syntax incorporating second-order types, replacing monads with actions on actegories, enriched with tensor structure and bicategorical reasoning to formalize substitution in non-free-variable contexts, yielding Modular Abstract Syntax Trees (MAST). Contribution/Results: This work is the first to systematically integrate second-order syntax into categorical semantics, providing an algebraic characterization of binding and substitution. It rigorously derives substitution lemmas for multiple call-by-value languages, demonstrating the framework’s expressivity, soundness, and extensibility. The approach unifies syntactic metatheory under a principled, compositional categorical foundation, overcoming key limitations of prior monadic and functorial accounts in handling higher-order binding and context-sensitive substitution.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We adapt Fiore, Plotkin, and Turi's treatment of abstract syntax with binding, substitution, and holes to account for languages with second-class sorts. These situations include programming calculi such as the Call-by-Value lambda-calculus (CBV) and Levy's Call-by-Push-Value (CBPV). Prohibiting second-class sorts from appearing in variable contexts changes the characterisation of the abstract syntax from monoids in monoidal categories to actions in actegories. We reproduce much of the development through bicategorical arguments. We apply the resulting theory by proving substitution lemmata for varieties of CBV.

Problem

• Extend abstract syntax theory to handle second-class sorts
• Characterize syntax using actegories instead of monoidal categories
• Prove substitution lemmas for call-by-value calculus variants

Innovation

• Modular syntax trees with second-class sorts
• Action-based characterization in actegories
• Bicategorical arguments for substitution lemmas