Relators and Notions of Simulation Revisited

2025-02-03

AI Summary

This paper addresses the lack of semantic unification for simulation and bisimulation across heterogeneous concurrent systems—including probabilistic, weighted, neighborhood, and game-based models. We propose a functor-category-theoretic universal algebraic framework: system types are abstracted as set functors, and (bi)similarity is derived via relation liftings (relators). Our contributions are threefold: (1) We prove, for the first time, that 1/4-iso pullbacks preserving functors admit complete bisimulations induced by coBarr relators; (2) We establish precise equivalences between key relator properties—e.g., monotonicity and diagonal preservation—and closure properties of simulations; (3) We construct the maximal lax extension of inverse-image-preserving functors and introduce “twisted bisimulation”, yielding a strictly coarser, yet sound and complete, behavioral equivalence on labeled transition systems—thereby extending the expressive boundaries of simulation theory.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Agent-Based Simulation and Emergent Behavior

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Explore relators' role in bisimulation
• Establish relator properties and bisimulation closure
• Develop novel twisted bisimulation concept

Innovation

• Universal coalgebra abstracts systems
• Relators induce (bi)simulation notions
• Greatest lax extension for functors