Proving Behavioural Apartness: A Coalgebraic Approach to Distinguishing States

Core Concepts

This paper proposes behavioural apartness, defined by dualising behavioural equivalence, as an alternative to cobisimilarity for proving distinguishability of states in coalgebras. The authors provide a sound and complete proof system for behavioural apartness that allows for finite proofs, in contrast to the universal quantification required by the cobisimilarity approach.

Abstract

The paper starts by reviewing the notion of cobisimilarity, which is defined by dualising coalgebraic bisimulations. The authors show that for certain functors, such as the subdistribution functor, the cobisimilarity proof system requires reasoning about an infinite set of couplings, making it less practical.

To address this issue, the authors introduce behavioural apartness, which is defined by dualising behavioural equivalence rather than bisimulations. They provide a proof system for behavioural apartness, prove it to be sound and complete, and show how it can be optimised to only require reasoning about states that are reachable in one step.

The paper demonstrates the benefits of the behavioural apartness proof system through several examples, including labelled Markov processes and stream systems. In the subdistribution functor example, the behavioural apartness proof system allows for finite proofs of distinguishability, in contrast to the cobisimilarity approach.

The authors also discuss potential future work, such as investigating the connection between proofs of behavioural apartness and distinguishing formulas in a corresponding modal logic, as well as exploring the notion of codensity apartness and its applications to quantitative settings.

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Proving Behavioural Apartness

by Ruben Turken... at arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16588.pdf

Deeper Inquiries

How can the connection between proofs of behavioural apartness and distinguishing formulas in a corresponding modal logic be further explored

The connection between proofs of behavioural apartness and distinguishing formulas in a corresponding modal logic can be further explored by investigating how the proofs of apartness can be translated into logical formulas. By identifying the patterns and structures in the proofs that lead to the determination of apartness between states, we can potentially derive logical expressions that capture the essence of this distinction. This exploration can involve mapping the steps of the proof process to the construction of formulas in the modal logic, highlighting the relationships between the two. Additionally, analyzing the properties of apartness and how they align with the semantics of the modal logic can provide insights into the expressive power of the logic in capturing behavioural distinctions.

What are the potential benefits of investigating the notion of codensity apartness and its applications to quantitative settings

Investigating the notion of codensity apartness and its applications to quantitative settings can offer several potential benefits. Firstly, codensity apartness may provide a more refined and nuanced way of capturing distinctions between states in coalgebraic systems, especially in scenarios where quantitative measures play a crucial role. By incorporating quantitative aspects into the notion of apartness, we can potentially develop more sophisticated and precise models for analyzing system behaviors. This can lead to the development of quantitative logics that can handle probabilistic or quantitative aspects of systems more effectively. Furthermore, exploring codensity apartness in quantitative settings can enhance our understanding of the relationships between coalgebraic structures and quantitative logics, paving the way for the development of novel approaches to reasoning about system behaviors.

Are there other coalgebraic notions of inequivalence beyond cobisimilarity and behavioural apartness that could be worth exploring, and how might they relate to the development of expressive logics

There are several other coalgebraic notions of inequivalence beyond cobisimilarity and behavioural apartness that could be worth exploring. One such notion is simulation, which captures a weaker form of behavioural equivalence where one system can simulate the behavior of another. Investigating simulation as a form of inequivalence can provide insights into the hierarchical relationships between systems and how one system can represent or mimic the behavior of another. Additionally, exploring notions such as bisimilarity up-to techniques, which allow for more efficient proofs by considering a coarser equivalence relation, can be valuable in developing proof systems and logics that balance precision and computational complexity. These alternative notions of inequivalence can contribute to the development of more versatile and expressive logics for reasoning about complex system behaviors.