Relating Structure and Power: Comonadic Semantics for Computational Resources

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-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 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.

A Duality Theoretic View on Limits of Finite Structures

“…Further, it would be interesting to investigate whether the limits for schema mappings introduced by Kolaitis et al [13] may be seen also as a type-theoretic construction. Finally, we would want to explore the connections with other semantically inspired approaches to finite model theory, such as those recently put forward by Abramsky, Dawar et al [2,3].…”

“…That is, for all x, y ∈ X such that x ≤ y, there is a clopen (i.e. simultaneously closed and open) C ⊆ X which is an up-set for ≤, and satisfies x ∈ C but y / ∈ C. We recall the construction of the Priestley space of a distributive lattice D. 3 A non-empty proper subset F ⊂ D is a prime filter if it is (i) upward closed (in the natural order of D), (ii) closed under finite meets, and (iii) if a ∨ b ∈ F , either a ∈ F or b ∈ F . Denote by X D the set of all prime filters of D. By Stone's Prime Filter Theorem, the map…”

“…In [4], it was shown that this theory may be understood as an application of Stone duality, thus making a bridge between semantics and more algorithmically oriented work. Bridging this semantics-versus-algorithmics gap in theoretical computer science has since gained quite some momentum, notably with the recent strand of research by Abramsky, Dawar and co-workers [2,3]. In this spirit, a natural question is whether the structural limits of Nešetřil and Ossona de Mendez also can be understood semantically, and in particular whether the topological component may be seen as an application of Stone duality.…”

Whither semantics?

“…As a case study for this theme, we shall give a brief overview of some recent work on relating categorical semantics, which exemplifies Structure, to finite model theory, which exemplifies Power. This is based on the papers [ADW17,AS18], and ongoing work with Nihil Shah and Tom Paine. Readers wishing to see a more detailed account are referred to [ADW17,AS18].…”

“…This is based on the papers [ADW17,AS18], and ongoing work with Nihil Shah and Tom Paine. Readers wishing to see a more detailed account are referred to [ADW17,AS18]. In particular, the presentation in [ADW17], while detailed, should be accessible to readers with minimal background in category theory.…”