Towards a Higher-Order Bialgebraic Denotational Semantics

2026-02-20

AI Summary

This work addresses the absence of a unified categorical bialgebraic denotational semantics for higher-order languages that simultaneously guarantees congruence of bisimilarity and coherence of denotational equivalence. Building upon the higher-order abstract GSOS framework, it realizes— for the first time—the bialgebraic semantic vision proposed by Turi and Plotkin by employing locally final coalgebras as the semantic domain for behavioral endofunctors. The resulting syntax-agnostic, compositional denotational semantics is parametric in the construction of the semantic domain, thereby uniformly accommodating both typed and untyped higher-order languages, as well as those featuring probabilistic or nondeterministic effects. This approach not only subsumes existing models such as step-indexed semantics but also ensures the compositionality of bisimilarity and semantic consistency across language variants.

Technology Category

• Knowledge Representation and Reasoning: Knowledge Representation Languages
• Cognitive Modeling & Cognitive Systems: Agent Architectures
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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

The bialgebraic abstract GSOS framework by Turi and Plotkin provides an elegant categorical approach to modelling the operational and denotational semantics of programming and process languages. In abstract GSOS, bisimilarity is always a congruence, and it coincides with denotational equivalence. This saves the language designer from intricate, ad-hoc reasoning to establish these properties. The bialgebraic perspective on operational semantics in the style of abstract GSOS has recently been extended to higher-order languages, preserving compositionality of bisimilarity. However, a categorical understanding of bialgebraic denotational semantics according to Turi and Plotkin's original vision has so far been missing in the higher-order setting. In the present paper, we develop a theory of adequate denotational semantics in higher-order abstract GSOS. The denotational models are parametric in an appropriately chosen semantic domain in the form of a locally final coalgebra for a behaviour bifunctor, whose construction is fully decoupled from the syntax of the language. Our approach captures existing accounts of denotational semantics such as semantic domains built via general step-indexing, previously introduced on a per-language basis, and is shown to be applicable to a wide range of different higher-order languages, e.g. simply typed and untyped languages, or languages with computational effects such as probabilistic or non-deterministic branching.

Problem

• higher-order
• bialgebraic semantics
• denotational semantics
• abstract GSOS
• categorical semantics

Innovation

• higher-order abstract GSOS
• bialgebraic semantics
• locally final coalgebra
• denotational semantics
• behavior bifunctor