Relating Structure and Power: Comonadic Semantics for Computational Resources

TL;DR

This paper introduces a categorical framework using comonads to unify and characterize various logical equivalences and combinatorial parameters in finite model theory, linking semantics and algorithmic theory.

Contribution

It develops a novel comonadic semantics for combinatorial games, connecting logical equivalences with categorical structures and combinatorial parameters.

Findings

Comonadic descriptions of Ehrenfeucht-Fraisse, pebble, and bisimulation games.

Characterization of tree-depth, tree-width, and synchronization-tree depth via coalgebras.

Syntax-free categorical characterizations of logical equivalences.

Abstract

Combinatorial games are widely used in finite model theory, constraint satisfaction, modal logic and concurrency theory to characterize logical equivalences between structures. In particular, Ehrenfeucht-Fraisse games, pebble games, and bisimulation games play a central role. We show how each of these types of games can be described in terms of an indexed family of comonads on the category of relational structures and homomorphisms. The index k is a resource parameter which bounds the degree of access to the underlying structure. The coKleisli categories for these comonads can be used to give syntax-free characterizations of a wide range of important logical equivalences. Moreover, the coalgebras for these indexed comonads can be used to characterize key combinatorial parameters: tree-depth for the Ehrenfeucht-Fraisse comonad, tree-width for the pebbling comonad, and synchronization-tree depth for the modal unfolding comonad. These results pave the way for systematic connections between two major branches of the field of logic in computer science which hitherto have been almost disjoint: categorical semantics, and finite and algorithmic model theory.