Big Steps in Higher-Order Mathematical Operational Semantics

2025-06-01

AI Summary

Proving consistency between big-step and small-step operational semantics for higher-order programming languages traditionally requires ad hoc, case-by-case reasoning. Method: We introduce the first category-theoretic framework for abstract big-step semantics. By generalizing Generalized Structural Operational Semantics (GSOS) to the higher-order structural operational semantics (HOSS) setting, we construct a generic, rule-driven procedure that automatically derives big-step semantics from higher-order GSOS rules, and formally establish its equivalence to the small-step semantics at the abstract categorical level. Contribution/Results: Our work provides the first axiomatization of big-step semantics in category theory, unifying equivalence proofs under a single, syntax-independent paradigm. It enables formulaic, reusable verification of semantic equivalence—free from syntactic commitments—thereby furnishing a robust mathematical foundation for programming language design, algebraic effects modeling, and formal verification.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Natural Language Processing: Sentence-level Semantics, Textual Inference, etc.

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: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Small-step and big-step operational semantics are two fundamental styles of structural operational semantics (SOS), extensively used in practice. The former one is more fine-grained and is usually regarded as primitive, as it only defines a one-step reduction relation between a given program and its direct descendant under an ambient evaluation strategy. The latter one implements, in a self-contained manner, such a strategy directly by relating a program to the net result of the evaluation process. The agreement between these two styles of semantics is one of the key pillars in operational reasoning on programs; however, such agreement is typically proven from scratch every time on a case-by-case basis. A general, abstract mathematical argument behind this agreement is up till now missing. We cope with this issue within the framework of higher-order mathematical operational semantics by providing an abstract categorical notion of big-step SOS, complementing the existing notion of abstract higher-order GSOS. Moreover, we introduce a general construction for deriving the former from the latter, and prove an abstract equivalence result between the two.

Problem

• Lack of general mathematical proof for small-step and big-step semantics agreement
• Need for abstract categorical notion of big-step operational semantics
• Requirement to derive big-step from higher-order GSOS and prove equivalence

Innovation

• Abstract categorical notion of big-step SOS
• General construction from higher-order GSOS
• Proving abstract equivalence between semantics