# Pebble Games with Algebraic Rules

@inproceedings{Dawar2012PebbleGW, title={Pebble Games with Algebraic Rules}, author={Anuj Dawar and Bjarki Holm}, booktitle={Fundam. Informaticae}, year={2012} }

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…

## 21 Citations

### Game Comonads & Generalised Quantifiers

- Computer ScienceCSL
- 2021

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

- MathematicsArXiv
- 2019

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

- Mathematics, Computer Science2017 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

- Mathematics, Computer ScienceMFCS
- 2021

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

- MathematicsJournal of Logic and Computation
- 2022

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

- MathematicsArXiv
- 2021

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

- Mathematics, Computer ScienceCSL
- 2018

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.

### RANK LOGIC IS DEAD, LONG LIVE RANK LOGIC!

- MathematicsThe Journal of Symbolic Logic
- 2019

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

- Mathematics, Computer Science2021 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

- Mathematics, Computer ScienceICDT
- 2019

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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