A Framework for Coalgebraic Reward-Sensitive Bisimulation (Extended Version)

2026-04-01

AI Summary

This work addresses the challenge of uniformly characterizing both qualitative behaviors and quantitative discrepancies—such as accumulated rewards—in system equivalence analysis. It proposes a unified framework grounded in fibrational coalgebra, which for the first time incorporates reward sensitivity into coalgebraic bisimulation. By leveraging categorical gluing techniques, the approach seamlessly integrates graded (metric) and ungraded (relational) bisimulations. The framework demonstrates broad applicability across diverse system models, subsuming relational bisimulation for reward-augmented automata and metric bisimulation for labeled Markov processes, thereby validating its generality and expressiveness.

Abstract

In this paper we present a framework for modelling \emph{reward-sensitive bisimulations}, that is, bisimulations that account for quantitative differences such as accumulated rewards. To capture both qualitative and quantitative aspects uniformly, we consider two interacting notions of bisimulation: a graded variant that tracks bounded reward differences, and an ungraded one that abstracts from them. Our characterization of these notions is done in the fibrational and coalgebraic approach to (bi)simulation initiated by Hermida and Jacobs. To formally relate the graded and ungraded notions, we deploy categorical gluing, a standard technique in categorical logic. Furthermore, we show that this construction interacts well with standard coalgebra concepts, such as final coalgebras, and that it yields a unified characterization in terms of combined notions of bisimulations under mild assumptions. In order to demonstrate the versatility of our approach, we show how it encompasses various bisimulation notions for different kinds of systems, including relation-based bisimulations for automata with rewards and metric-based notions of bisimulations for labelled Markov processes.

Problem

• reward-sensitive bisimulation
• coalgebraic
• graded bisimulation
• quantitative reasoning
• categorical gluing

Innovation

• reward-sensitive bisimulation
• graded bisimulation
• coalgebraic approach
• categorical gluing
• fibrational semantics