Samson Abramsky ; Luca Reggio - Arboreal Categories: An Axiomatic Theory of Resources

lmcs:9839 - Logical Methods in Computer Science, August 10, 2023, Volume 19, Issue 3 -
Arboreal Categories: An Axiomatic Theory of ResourcesArticle

Authors: Samson Abramsky ; Luca Reggio

    Game comonads provide a categorical syntax-free approach to finite model theory, and their Eilenberg-Moore coalgebras typically encode important combinatorial parameters of structures. In this paper, we develop a framework whereby the essential properties of these categories of coalgebras are captured in a purely axiomatic fashion. To this end, we introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of a game, to be defined. These are related to extensional or "static" structures via arboreal covers, which are resource-indexed comonadic adjunctions. These ideas are developed in a general, axiomatic setting, and applied to relational structures, where the comonadic constructions for pebbling, Ehrenfeucht-Fra\"issé and modal bisimulation games recently introduced by Abramsky et al. are recovered, showing that many of the fundamental notions of finite model theory and descriptive complexity arise from instances of arboreal covers.

    Volume: Volume 19, Issue 3
    Published on: August 10, 2023
    Accepted on: July 1, 2023
    Submitted on: July 27, 2022
    Keywords: Computer Science - Logic in Computer Science,Mathematics - Category Theory,Mathematics - Logic


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