Relational Connectors and Heterogeneous Bisimulations

2024-10-18

arXiv.org

AI Summary

This paper addresses the lack of a unified behavioral equivalence theory for heterogeneous systems—such as nondeterministic, probabilistic, and game-based systems. To this end, it proposes the first coalgebraic framework for heterogeneous (bi)simulation grounded in universal coalgebra. The core methodological innovation is the introduction of “relational connectors” between functors, serving as bridges for behavioral relations across system types, and systematically defining their composition, inversion, and identity operations. By integrating Kantorovich liftings and Barr extensions, the framework establishes a dual interpretation of modal logic and generalizes the Hennessy–Milner theorem, proving full correspondence between heterogeneous bisimilarity and logical equivalence on finitely branching systems. The framework uniformly characterizes behavioral consistency across labeled transition systems, probabilistic systems, and I/O conformance testing, thereby providing a foundational tool for verification of multi-paradigm systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

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

Abstract

While behavioural equivalences among systems of the same type, such as Park/Milner bisimilarity of labelled transition systems, are an established notion, a systematic treatment of relationships between systems of different type is currently missing. We provide such a treatment in the framework of universal coalgebra, in which the type of a system (nondeterministic, probabilistic, weighted, game-based etc.) is abstracted as a set functor: We introduce relational connectors among set functors, which induce notions of heterogeneous (bi)simulation among coalgebras of the respective types. We give a number of constructions on relational connectors. In particular, we identify composition and converse operations on relational connectors; we construct corresponding identity relational connectors, showing that the latter generalize the standard Barr extension of weak-pullback-preserving functors; and we introduce a Kantorovich construction in which relational connectors are induced from relations between modalities. For Kantorovich relational connectors, one has a notion of dual-purpose modal logic interpreted over both system types, and we prove a corresponding Hennessy-Milner-type theorem stating that generalized (bi)similarity coincides with theory inclusion on finitely-branching systems. We apply these results to a number of example scenarios involving labelled transition systems with different label alphabets, probabilistic systems, and input/output conformances.

Problem

• Heterogeneous system relationships
• Relational connectors in coalgebra
• Generalized bisimulation theorems

Innovation

• Relational connectors unify system types
• Kantorovich construction links modal relations
• Hennessy-Milner theorem extends to heterogeneous systems