Higher-Order Mathematical Operational Semantics (Early Ideas)

2024-05-26

Conference on Algebra and Coalgebra in Computer Science

AI Summary

This paper addresses the challenge of establishing compositional proofs for operational semantics in higher-order languages—specifically, extending the Turi-Plotkin bialgebraic GSOS framework from first-order to higher-order settings. Method: The authors introduce *higher-order GSOS laws*: a class of pointed dinatural transformations that uniformly capture dynamic behavior across systems such as the λ-calculus and combinatory logic. They develop the first abstract GSOS-style operational semantics theory for higher-order languages and prove the first general compositionality theorem within this setting. Contribution/Results: Using this framework, the paper formally establishes compositionality of combinatory logic and the λ-calculus under strong applicative bisimilarity. The resulting theory provides a rigorous, reusable, and verifiable foundation for operational semantics of higher-order languages, significantly advancing the intersection of categorical semantics and programming language theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Other Foundations of Multi Agent Systems

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
• Security and Privacy: Data transparency and provenance

Abstract

Compositionality proofs in higher-order languages are notoriously involved, and general semantic frameworks guaranteeing compositionality are hard to come by. In particular, Turi and Plotkin's bialgebraic abstract GSOS framework, which provides off-the-shelf compositionality results for first-order languages, so far does not apply to higher-order languages. In the present work, we develop a theory of abstract GSOS specifications for higher-order languages, in effect transferring the core principles of Turi and Plotkin's framework to a higher-order setting. In our theory, the operational semantics of higher-order languages is represented by certain dinatural transformations that we term emph{(pointed) higher-order GSOS laws}. We give a general compositionality result that applies to all systems specified in this way and discuss how compositionality of combinatory logics and the $lambda$-calculus w.r.t. a strong variant of Abramsky's applicative bisimilarity are obtained as instances.

Problem

• Extends bialgebraic GSOS framework to higher-order languages
• Develops higher-order GSOS laws for operational semantics
• Ensures compositionality for combinatory logics and λ-calculus

Innovation

• Extends bialgebraic GSOS to higher-order languages
• Uses dinatural transformations for operational semantics
• Provides compositionality for combinatory logics