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Relating Structure and Power: Comonadic Semantics for Computational Resources

Authors Samson Abramsky , Nihil Shah

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Author Details

Samson Abramsky
  • Oxford University Department of Computer Science, Wolfson Building, Parks Road, Oxford OX1 3QD, U.K.
Nihil Shah
  • Oxford University Department of Computer Science, Wolfson Building, Parks Road, Oxford OX1 3QD, U.K.

Funding

  • Abramsky, Samson: Samson Abramsky’s work was supported by EPSRC grant EP/N018745/1.

Cite AsGet BibTex

Samson Abramsky and Nihil Shah. Relating Structure and Power: Comonadic Semantics for Computational Resources. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 2:1-2:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.CSL.2018.2

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Theory of computation → Categorical semantics
Keywords
  • Finite model theory
  • combinatorial games
  • Ehrenfeucht-Fraïssé games
  • pebble games
  • bisimulation
  • comonads
  • coKleisli category
  • coalgebras of a comonad

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