[Paper Review] Behavioural equivalences for timed systems

Tomasz Brengos, Marco Peressotti|University of Southern Denmark Research Portal (University of Southern Denmark)|Nov 28, 2017

Formal Methods in Verification4 citations

TL;DR

This paper introduces a unified categorical framework for timed behavioural models using lax functors and general saturation, enabling a systematic derivation of behavioural equivalences such as timed bisimulation and timed language equivalence. The key contribution is the definition of q-bisimulation, which captures multiple equivalences in a spectrum ordered by discriminating power, independent of system type—valid for both non-deterministic and quantitative extensions like timed probabilistic systems.

ABSTRACT

Timed transition systems are behavioural models that include an explicit treatment of time flow and are used to formalise the semantics of several foundational process calculi and automata. Despite their relevance, a general mathematical characterisation of timed transition systems and their behavioural theory is still missing. We introduce the first uniform framework for timed behavioural models that encompasses known behavioural equivalences such as timed bisimulations, timed language equivalences as well as their weak and time-abstract counterparts. All these notions of equivalences are naturally organised by their discriminating power in a spectrum. We prove that this result does not depend on the type of the systems under scrutiny: it holds for any generalisation of timed transition system. We instantiate our framework to timed transition systems and their quantitative extensions such as timed probabilistic systems.

Motivation & Objective

• To provide a general mathematical characterisation of timed transition systems and their behavioural theory, which was previously missing.
• To unify diverse behavioural equivalences—such as timed bisimulation, language equivalence, and their weak/time-abstract variants—into a single coherent framework.
• To demonstrate that the discriminating power of these equivalences forms a spectrum independent of the underlying system type, including non-deterministic and quantitative models.
• To extend the theory of saturation to support time and time abstraction in lax functors, enabling a coalgebraic treatment of timed systems.
• To generalize existing coalgebraic approaches to timed systems by overcoming limitations of comonad-based constructions that exclude models like Timed CSP and Timed Automata.

Proposed method

• The framework is based on lax functors over order-enriched categories, particularly Kleisli categories, to model how computations are observed.
• General saturation is extended to lax functors on monoid categories to support time and time abstraction, enabling the definition of q-bisimulation.
• q-bisimulation is defined as a parametrized behavioural equivalence where parameters control observation of time duration and action types.
• The framework uses embeddings into categories satisfying left distributivity to handle monads like powerset, convex set, and quantalic monads, even when the base category does not.
• The construction is validated on timed transition systems and their quantitative extensions, such as timed probabilistic automata, via embeddings into suitable categories.
• The theory leverages prior work on saturation and weak bisimulation, extending it to support time and time abstraction in a uniform way.

Experimental results

Research questions

• RQ1 Can a single, uniform framework capture known behavioural equivalences in timed systems, including weak and time-abstract variants?
• RQ2 How can the discriminating power of behavioural equivalences be systematically compared and ordered?
• RQ3 Can the framework be applied to both non-deterministic and quantitative timed systems, such as timed probabilistic automata?
• RQ4 Is the resulting spectrum of equivalences independent of the underlying system type or computational effects?
• RQ5 Can the theory of saturation be extended to support time and time abstraction in lax functors over monoid categories?

Key findings

• The paper establishes that timed bisimulation, timed language equivalence, and their weak and time-abstract variants are all instances of q-bisimulation with different parameter choices.
• The behavioural equivalences form a spectrum ordered by discriminating power, with stronger equivalences at the bottom and weaker ones at the top, as shown in Figure 1.
• This spectrum is independent of the system type: the same ordering holds for non-deterministic timed transition systems and their quantitative extensions like timed probabilistic systems.
• The framework successfully models systems that previous comonad-based approaches could not, such as Timed CSP and Timed Automata, by avoiding the need to separate timed and discrete actions.
• The theory supports embeddings into categories satisfying left distributivity, enabling application to monads like powerset, convex set, and quantalic monads, even when the base category does not.
• The framework generalizes prior work on saturation and weak bisimulation, extending it to handle time and time abstraction in a uniform, categorical setting.