Relating Structure and Power: Comonadic Semantics for Computational Resources

@inproceedings{Abramsky2018RelatingSA,
  title={Relating Structure and Power: Comonadic Semantics for Computational Resources},
  author={Samson Abramsky and Nihil Shah},
  booktitle={CSL},
  year={2018}
}
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… Expand
Relating structure and power: Comonadic semantics for computational resources
  • S. Abramsky, Nihil Shah
  • Computer Science
  • J. Log. Comput.
  • 2021
TLDR
The 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. Expand
Comonadic semantics for guarded fragments
  • S. Abramsky, Dan Marsden
  • Computer Science, Mathematics
  • 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
  • 2021
TLDR
A systematic account of how a range of model comparison games which play a central role in finite model theory can be captured in terms of resource-indexed comonads on the category of relational structures, including Ehrenfeucht-Fraïssé, pebbling, and bisimulation games, is extended to quantifier-guarded fragments of first-order logic. Expand
Arboreal Categories and Resources
We introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of aExpand
Arboreal Categories: An Axiomatic Theory of Resources
We introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of aExpand
Lovász-Type Theorems and Game Comonads
TLDR
This work proposes a new categorical formulation, which applies to any locally finite category with pushouts and a proper factorisation system, and presents a novel application to homomorphism counts in modal logic. Expand
A Pebbling Comonad for Finite Rank and Variable Logic, and an Application to the Equirank-variable Homomorphism Preservation Theorem
  • T. Paine
  • Computer Science, Mathematics
  • MFPS
  • 2020
Abstract In this paper we recast the proof of Rossman's equirank homomorphism preservation theorem using comonadic formulations of bounded quantifier rank and variable count (and dually tree widthExpand
Montague Semantics for Lambek Pregroups
TLDR
The main result is a factorisation of RelCoCat models through free cartesian bicategories, and as a corollary the authors get a logspace reduction from the semantics problem to the evaluation of conjunctive queries. Expand
Functorial Question Answering
TLDR
This work studies the relational variant of the categorical compositional distributional models of Coecke et al, where it shows that RelCoCat models factorise through Cartesian bicategories, and defines question answering as an NP-complete problem. Expand
Stone Duality for Relations
TLDR
It is shown how Stone duality can be extended from maps to relations by working order enriched and defining a relation from A to B as both an order-preserving function and as a subobject of A times B. Expand
Whither semantics?
TLDR
How mathematical semantics has evolved is discussed, and some new directions for future work are suggested, on encapsulating model comparison games as comonads in the context of finite model theory. Expand
...
1
2
...

References

SHOWING 1-10 OF 63 REFERENCES
Relating Structure and Power: Comonadic Semantics for Computational Resources - Extended Abstract
TLDR
The 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. Expand
The pebbling comonad in Finite Model Theory
TLDR
It is shown that existential k-pebble games have a natural comonadic formulation and lays the basis for some new and promising connections between two areas within logic in computer science which have largely been disjoint: finite and algorithmic model theory, and semantics and categorical structures of computation. Expand
Comonadic semantics for guarded fragments
  • S. Abramsky, Dan Marsden
  • Computer Science, Mathematics
  • 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
  • 2021
TLDR
A systematic account of how a range of model comparison games which play a central role in finite model theory can be captured in terms of resource-indexed comonads on the category of relational structures, including Ehrenfeucht-Fraïssé, pebbling, and bisimulation games, is extended to quantifier-guarded fragments of first-order logic. Expand
The Freedoms of (Guarded) Bisimulation
  • E. Grädel, Martin Otto
  • Mathematics, Computer Science
  • Johan van Benthem on Logic and Information Dynamics
  • 2014
We survey different notions of bisimulation equivalence that provide flexible and powerful concepts for understanding the expressive power as well as the model-theoretic and algorithmic properties ofExpand
The Quantum Monad on Relational Structures
TLDR
It is shown how quantum strategies for homomorphism games between relational structures can be viewed as Kleisli morphisms for a quantum monad on the (classical) category of relational structures and homomorphisms. Expand
Constraint Satisfaction, Bounded Treewidth, and Finite-Variable Logics
TLDR
This work shows that constraint satisfaction problems on inputs of treewidth less than k are definable using Datalog programs with at most k variables; this provides a new explanation for the tractability of these classes of problems. Expand
Logical Hierarchies in PTIME
  • L. Hella
  • Computer Science, Mathematics
  • Inf. Comput.
  • 1996
TLDR
It is proved that, for each natural numbern, there is a polynomial time computable query which is not definable in any extension of fixpoint logic by any set of generalized quantifiers, which rules out the possibility of characterizing PTIME in terms of definability in fix point logic extended by a finite set of universal quantifiers. Expand
Bisimulation and open maps
  • A. Joyal, M. Nielsen, G. Winskel
  • Mathematics, Computer Science
  • [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science
  • 1993
TLDR
An abstract definition of bisimulation is presented and a promising new model, presheaf on categories of pomsets, into which the usual category of labeled event structures embeds fully and faithfully is presented. Expand
On Elementary Equivalence for Equality-free Logic
TLDR
A model-theoretic study of equality-free logic is worthwhile by itself and it is hoped that the results of this study will contribute to the understanding of the role of equality in mathematical theories and structures. Expand
Finite model theory
TLDR
The text presents the main results of descriptive complexity theory, the connection between axiomatizability of classes of finite structures and their complexity with respect to time and space bounds. Expand
...
1
2
3
4
5
...