# The pebbling comonad in Finite Model Theory

@article{Abramsky2017ThePC, title={The pebbling comonad in Finite Model Theory}, author={S. Abramsky and A. Dawar and Pengming Wang}, journal={2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)}, year={2017}, pages={1-12} }

Pebble games are a powerful tool in the study of finite model theory, constraint satisfaction and database theory. Monads and comonads are basic notions of category theory which are widely used in semantics of computation and in modern functional programming. We show that existential k-pebble games have a natural comonadic formulation. Winning strategies for Duplicator in the k-pebble game for structures A and B are equivalent to morphisms from A to B in the coKleisli category for this comonad… Expand

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#### References

SHOWING 1-10 OF 47 REFERENCES

Applications of Categories in Computer Science: Computational comonads and intensional semantics

- Computer Science
- 1992

Abstract We explore some foundational issues in the development of a theory of intensional semantics, in which program denotations may convey information about computation strategy in addition to the… Expand

Logical Hierarchies in PTIME

- Computer Science, Mathematics
- Inf. Comput.
- 1996

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

The sheaf-theoretic structure of non-locality and contextuality

- Physics, Computer Science
- 2011

It is shown that contextuality, and non-locality as a special case, correspond exactly to obstructions to the existence of global sections, and a linear algebraic approach to computing these obstructions is described, which allows a systematic treatment of arguments for non- Locality and contextuality. Expand

Constraint Satisfaction, Bounded Treewidth, and Finite-Variable Logics

- Mathematics, Computer Science
- CP
- 2002

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

Affine Systems of Equations and Counting Infinitary Logic

- Computer Science, Mathematics
- ICALP
- 2007

It is shown that testing the solvability of systems of equations over a finite Abelian group, a tractable CSP that was previously known not to be definable in Datalog, is not Definable in an infinitary logic with counting and hence that it is not definite in least fixed point logic or its extension with counting. Expand

A Game-Theoretic Approach to Constraint Satisfaction

- Computer Science
- AAAI/IAAI
- 2000

Ex existential -pebble games are used to introduce the concept of -locality and show that it constitutes a new tractable case of constraint satisfaction that properly extends the well known case in which establishing strong -consistency implies global consistency. Expand

On preservation under homomorphisms and unions of conjunctive queries

- Mathematics, Computer Science
- PODS '04
- 2004

The homomorphism-preservation theorem holds for several large classes of finite structures of interest in graph theory and database theory, and it is shown that this result holds for all classes of infinite structures of bounded degree, allclasses of finite structure of bounded treewidth, and, more generally, all classesof finite structures whose cores exclude at least one minor. Expand

One Eilenberg Theorem to Rule Them All

- Mathematics, Computer Science
- ArXiv
- 2016

A generic variety theorem that covers e.g. Wilke's and Pin's work on $\infty$-languages, the variety theorem for cost functions of Daviaud, Kuperberg, and Pin, and unifies the two previous categorical approaches of Boja\'nczyk and of Ad\'amek et al. Expand

Notions of Computation and Monads

- Computer Science, Mathematics
- Inf. Comput.
- 1991

Calculi are introduced, based on a categorical semantics for computations, that provide a correct basis for proving equivalence of programs for a wide range of notions of computation. Expand

The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory

- Mathematics, Computer Science
- SIAM J. Comput.
- 1998

This paper isolates a class (of problems specified by) "monotone monadic SNP without inequality" which may exhibit a dichotomy, and explains the placing of all these restrictions by showing, essentially using Ladner's theorem, that classes obtained by using only two of the above three restrictions do not show this dichotomy. Expand