Pebble Games with Algebraic Rules

  title={Pebble Games with Algebraic Rules},
  author={Anuj Dawar and Bjarki Holm},
  booktitle={Fundam. Informaticae},
We define a general framework of partition games for formulating two-player pebble games over finite structures. We show that one particular such game, which we call the invertible-map game, yields a family of polynomial-time approximations of graph isomorphism that is strictly stronger than the well-known Weisfeiler-Lehman method. The general framework we introduce includes as special cases the pebble games for finite-variable logics with and without counting. It also includes a matrix… 

Game Comonads & Generalised Quantifiers

A one-sided version of this game is defined which allows us to provide a categorical semantics for a number of logics with generalised quantifiers and a novel notion of tree decomposition that emerges from the construction.

Generalizations of k-Weisfeiler-Leman partitions and related graph invariants

A characterization in terms of an invertible map game (as introduced by Dawar-Holm) on the complex field is proved, which introduces new parameters that allow us to tease apart some subtle variations of the usual Weisfeiler-Leman equivalences.

Descriptive complexity of linear equation systems and applications to propositional proof complexity

  • Martin GroheWied Pakusa
  • Mathematics, Computer Science
    2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
  • 2017
We prove that the solvability of systems of linear equations and related linear algebraic properties are definable in a fragment of fixed-point logic with counting that only allows

On the Relative Power of Linear Algebraic Approximations of Graph Isomorphism

It is shown that the distinguishing power of the monomial calculus is no greater than the invertible map method by simulating the former in a fixed-point logic with solvability operators, and that the distinctions made by this logic can be implemented in the Nullstellensatz calculus.

Limitations of the Invertible-Map Equivalences

The intuition is that two graphs G ≡IM k,Q H cannot be distinguished by iterative refinements of equivalences on k-tuples defined via linear operators on vector spaces over fields of characteristic.

On the relative power of algebraic approximations of graph isomorphism

In positive characteristic it is shown that the invertible map method can simulate the monomial calculus and a potential way to extend this to themonomial calculus is identified.

Symmetric Circuits for Rank Logic

A circuit characterization of fixed-point logic with rank in terms of families of symmetric circuits with rank gates is given, along the lines of that for FPC given by Anderson and Dawar in 2017.


This work shows that the variant of rank logic FPR* with an operator that uniformly expresses the matrix rank over finite fields is more expressive than FPR, and implies that rank logic, in its original definition with a distinct rank operator for every field, fails to capture polynomial time.

Separating Rank Logic from Polynomial Time

  • Moritz Lichter
  • Mathematics, Computer Science
    2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
  • 2021
It is shown that the isomorphism problem for CFI graphs over ${{\mathbb{Z}}_{{2^i}}}$ cannot be defined in rank logic, even if the base graph is totally ordered, but CPT can define this isomorphicism problem.

On the Expressive Power of Linear Algebra on Graphs

This paper considers M A T L A N G, a matrix query language recently introduced, in which some basic linear algebra functionality is supported, and investigates the problem of characterising the equivalence of graphs, represented by their adjacency matrices, for various fragments of M AT L AN G.



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The boundary of definability in $\ensuremath{FP}+\textsf{C}}} is explored with respect to problems from linear algebra and suggestions on how the logic might be extended are looked at.

Logics with Rank Operators

This work introduces extensions of first-order logic (FO) and fixed-point logic (FP) with operators that compute the rank of a definable matrix and shows that FO+rk_p can define deterministic and symmetric transitive closure and captures the complexity class MOD_pL, for all prime values of p.

Logical hierarchies in PTIME

  • L. Hella
  • Computer Science, Mathematics
    [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science
  • 1992
It is proved that for each integer n, there is a property of finite models which is expressible in fixpoint logic, or even in DATALOG, but not in the extension of first-order logic by any set of n-ary quantifiers.

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The subject of this paper is the part of finite model theory intimately related to the classical model theory. In the very beginning of our career in computer science, we attended a few lectures on

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This thesis considers an incongruence between machine computations and logics, and the open question whether there exists a logic which generally captures polynomial-time computations, and introduces a variety of rank logics with the ability to compute the ranks of matrices over (finite) prime fields.

The Graph Isomorphism Problem and approximate categories

  • H. Derksen
  • Mathematics, Computer Science
    J. Symb. Comput.
  • 2013

Describing Graphs: A First-Order Approach to Graph Canonization

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Polynomial time algorithms for modules over finite dimensional algebras

We present polynomial time algorithms for some fundamental tasks from representation theory of finite dimensional algebras. These involve testing (and constructing) isomorphisms of modules as well as

The complexity of relational query languages (Extended Abstract)

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