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Categorical Foundations for Linear-Time Behavioural Equivalences

Core Concepts
This paper develops a general axiomatic framework for linear-time logics and equivalences within finite model theory, by introducing the notion of linear arboreal categories and relating them to linear variants of previously studied arboreal categories.
The paper brings together several lines of research, including the categorical perspective on model comparison games and the associated logical equivalences, the notion of arboreal categories, the linear variant of the pebbling comonad, and the linear-time branching-time spectrum of behavioural equivalences. The key contributions are: The definition of linear arboreal categories, which strengthen the axioms of arboreal categories to exclude 'branching' behaviour. This allows for the study of 'linear' variants of previously examined arboreal categories. The construction of a linear arboreal subcategory CL from any linearisable arboreal category C, related via an adjunction I ⊣ T. The definition of linear behavioural relations (e.g. trace inclusion, labelled trace equivalence) on objects in an extensional category E, using the linear arboreal cover Lk ◦ Ik ⊣ Tk ◦ Rk derived from an arboreal cover Lk ⊣ Rk of E. New preservation and characterisation theorems relating linear-time behavioural equivalences (e.g. labelled trace equivalence) with linear fragments of logics captured by the branching equivalence (e.g. modal logic). The paper provides a general framework for studying linear-time logics and equivalences, and demonstrates how this framework recovers and generalises several existing results in the literature.

Key Insights Distilled From

by Sams... at 04-01-2024
Linear Arboreal Categories

Deeper Inquiries

What other applications or extensions of the linear arboreal category framework could be explored

One potential application of the linear arboreal category framework could be in the study of complex systems with hierarchical structures. By representing these systems as linear arboreal categories, we can analyze their behaviors in a structured and systematic way. This framework could be used to model and understand the interactions and dependencies within such systems, providing insights into their dynamics and emergent properties. Additionally, the linear arboreal category approach could be extended to explore the relationships between different levels of hierarchy within a system, shedding light on how changes at one level impact the overall behavior.

How might the relationship between linear and branching behavioural equivalences be further characterized in this setting

In the context of linear arboreal categories, the relationship between linear and branching behavioral equivalences can be further characterized by examining how linear equivalences capture the essence of branching equivalences in a more structured and constrained manner. By studying the properties of linear bisimulations, trace inclusions, and other linear equivalences, we can identify the key differences and similarities between linear and branching behaviors. This analysis can help in understanding how linear models can approximate and represent branching behaviors in a more manageable and structured way.

Are there connections between the linear arboreal category approach and other categorical approaches to behavioural equivalences, such as those involving graded monads or fibrations

There are potential connections between the linear arboreal category approach and other categorical approaches to behavioral equivalences, such as those involving graded monads or fibrations. By exploring these connections, we can potentially unify different perspectives and formalisms for capturing behavioral relations in a categorical framework. For example, integrating ideas from graded monads or fibrations with the linear arboreal category framework could lead to a more comprehensive and versatile approach to modeling and analyzing behavioral equivalences in various systems. This integration could provide new insights and tools for studying complex behaviors and relationships in a structured and categorical manner.