Relators and Notions of Simulation Revisited

TL;DR

This paper explores the theoretical foundations of simulation and bisimulation in various system types using universal coalgebra, establishing conditions for soundness, completeness, and closure properties of relators, and introducing a new notion of twisted bisimulation.

Contribution

It provides new theoretical results on relators and (bi)simulation, including conditions for their properties and a novel twisted bisimulation concept.

Findings

Every functor preserving 1/4-iso pullbacks admits a sound and complete bisimulation.

Full closure properties of (bi)simulation require relators to be lax extensions.

Inverse image preserving functors admit a greatest lax extension.

Abstract

Simulations and bisimulations are ubiquitous in the study of concurrent systems and modal logics of various types. Besides classical relational transition systems, relevant system types include, for instance, probabilistic, weighted, neighbourhood-based, and game-based systems. Universal coalgebra abstracts system types in this sense as set functors. Notions of (bi)simulation then arise by extending the functor to act on relations in a suitable manner, turning it into what may be termed a relator. We contribute to the study of relators in the broadest possible sense, in particular in relation to their induced notions of (bi)similarity. Specifically, (i) we show that every functor that preserves a very restricted type of pullbacks (termed 1/4-iso pullbacks) admits a sound and complete notion of bisimulation induced by the coBarr relator; (ii) we establish equivalences between properties of relators and closure properties of the induced notion of (bi)simulation, showing in particular that the full set of expected closure properties requires the relator to be a lax extension, and that soundness of (bi)simulations requires preservation of diagonals; and (iii) we show that functors preserving inverse images admit a greatest lax extension. In a concluding case study, we apply (iii) to obtain a novel highly permissive notion of twisted bisimulation on labelled transition systems.