Relating Structure and Power: Comonadic Semantics for Computational Resources [chapter]

Samson Abramsky, Nihil Shah
2018 Lecture Notes in Computer Science  
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-Fraïssé 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
more » ... 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-Fraïssé 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. ACM Subject Classification Theory of computation → Finite Model Theory, Theory of computation → Categorical semantics In this paper, we develop a novel approach to relating categorical semantics, which exemplifies the first strand, to finite model theory, which exemplifies the second. It builds on the ideas introduced in [2], but goes much further, showing clearly that there is a strong and robust connection, which can serve as a basis for many further developments. C S L 2 0 1 8
doi:10.1007/978-3-030-00389-0_1 fatcat:kuwar27movhtroa6ztlz57vdke