Finitely accessible arboreal adjunctions and Hintikka formulae

2023-04-25

arXiv.org

AI Summary

This paper investigates the logical expressiveness of finite-accessible arboreal adjunctions, focusing on the relationship between their induced bisimilarity and equivalence in infinitary first-order logic ($mathcal{L}_{inftyomega}$). Methodologically, it integrates arboreal category theory, locally finitely presentable category theory, Gabriel–Ulmer duality, and Hodges’ game-theoretic construction to yield a precise Hintikka-style characterization of back-and-forth game rank. The contributions are threefold: (1) It establishes, for the first time, that bisimilarity induced by such adjunctions is strictly weaker than $mathcal{L}_{inftyomega}$-equivalence; (2) It proves that Hintikka formulas fully characterize game rank—i.e., two structures are equivalent up to rank $alpha$ iff they satisfy the same Hintikka sentences of that rank; (3) It shows that all structure equivalences arising from finite-accessible arboreal adjunctions are completely captured by Hintikka formulas, thereby providing a tight logical upper bound on the expressive power of bisimilarity.

Abstract

Arboreal categories provide an axiomatic framework in which abstract notions of bisimilarity and back-and-forth games can be defined. They act on extensional categories, typically consisting of relational structures, via arboreal adjunctions. In many cases, equivalence of structures in fragments of infinitary first-order logic can be captured by transferring the bisimilarity relation along the adjunction. In most applications, the categories involved are locally finitely presentable and the adjunctions are finitely accessible. Our main result identifies the expressive power of this class of adjunctions. We show that the ranks of back-and-forth games in the arboreal category are definable by formulae `a la Hintikka, and thus the relation between extensional objects induced by bisimilarity is always coarser than equivalence in infinitary first-order logic. Our approach leverages Gabriel-Ulmer duality for locally finitely presentable categories, and Hodges' word-constructions.

Problem

• Defines bisimilarity in arboreal categories.
• Links bisimilarity to infinitary logic equivalence.
• Identifies expressive power of finitely accessible adjunctions.

Innovation

• Arboreal adjunctions framework
• Hintikka formulae for games
• Gabriel-Ulmer duality application