Compositionality in Coalgebraic Trace Semantics

2026-05-18

AI Summary

This work addresses the challenge of ensuring compositionality for process languages under trace equivalence—rather than merely strong bisimilarity—by extending the abstract GSOS framework of Turi and Plotkin to Kleisli categories. It introduces a generalized notion of De Simone laws suitable for this categorical setting and proves that the induced operational semantics is compositional with respect to coalgebraic trace equivalence. Building on this foundation, the paper derives a novel De Simone format that enables compositional reasoning about trace equivalence in probabilistic systems. The approach not only recovers the classical compositionality results for De Simone rules in non-probabilistic settings but also establishes, for the first time, a rule format yielding compositional trace semantics for probabilistic concurrency.

Abstract

A key requirement on any well-behaved process language is its compositionality: behavioural equivalence of processes should be respected by the constructors of the language. Turi and Plotkin's abstract GSOS provides an elegant bialgebraic framework for modelling rule formats that guarantee compositionality from the outset. Their original results, however, are restricted to compositionality of strong bisimilarity, a rather fine-grained notion of process equivalence. In the present paper, we demonstrate that Turi and Plotkin's approach also applies to trace equivalence, which only observes external actions of processes. To this end, we revisit the general compositionality result of their original theory and present it in a refined form with regard to the required naturality conditions. This step makes abstract GSOS applicable over Kleisli categories and thereby enables reasoning about compositionality in the setting of coalgebraic trace semantics. As our main contribution, we introduce De Simone laws, a type of GSOS laws over Kleisli categories, and prove that their operational models are compositional for coalgebraic trace equivalence. This result recovers and explains compositionality of the well-known De Simone rule format for labelled transition systems in a natural categorical setting. As a further application, we derive from our general framework a novel De Simone-type format for probabilistic systems, compositional for probabilistic trace equivalence.

Problem

• compositionality
• trace equivalence
• coalgebraic semantics
• abstract GSOS
• Kleisli categories

Innovation

• abstract GSOS
• coalgebraic trace semantics
• compositionality
• De Simone laws
• Kleisli categories