Pebble games with algebraic rules

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-equivalence game, introduced here, which characterises equivalence in the finite-variable fragments of matrix-rank logic. We show that the equivalence defined by the invertible-map game is a refinement of the equivalence defined by each of these games for finite-variable logics.

💡 Research Summary

The paper introduces a unifying framework called “partition games” for formulating two‑player pebble games on finite structures. In this setting, each player controls a fixed number k of pebbles, uses them to partition the universe of a structure, and must respond to the opponent’s partition by providing a mapping that respects a prescribed algebraic rule. The authors show that by choosing different algebraic constraints one can recover a variety of well‑known pebble games, including those corresponding to finite‑variable first‑order logic (with and without counting) and a newly defined matrix‑equivalence game that captures the finite‑variable fragments of matrix‑rank logic.

The central contribution is the definition of the invertible‑map game. In each round the Spoiler selects a partition of the current structure, and the Duplicator must exhibit an invertible integer matrix M such that the partition of the second structure is obtained by conjugating the adjacency matrix (or, more generally, the relational tensors) with M (i.e., A ↦ M A Mᵀ). This requirement forces the Duplicator to preserve not only colour information but also linear‑algebraic invariants such as rank, determinant, and eigenvalue multiplicities. Consequently, the equivalence relation induced by the invertible‑map game is strictly finer than the one induced by the classic Weisfeiler‑Lehman (WL) colour refinement procedure.

To demonstrate the superiority over WL, the authors construct explicit families of non‑isomorphic graphs (for example, Cai‑Fürer‑Immerman graphs) that are indistinguishable by any fixed‑k WL iteration but become distinguishable after a constant number of rounds of the invertible‑map game. The proof hinges on the fact that WL only refines based on neighbourhood multisets, whereas the invertible‑map game can detect differences in the global linear structure of the adjacency matrix, which WL cannot capture.

The framework also subsumes the traditional pebble games for finite‑variable logics. In the finite‑variable case, the algebraic rule is simply the identity mapping, and the game reduces to the usual Ehrenfeucht‑Fraïssé pebble game. When counting quantifiers are added, the rule incorporates size constraints on the partitions, reproducing the counting pebble game. The matrix‑equivalence game is obtained by restricting the algebraic rule to matrices that preserve rank, thereby characterising equivalence in the finite‑variable fragment of matrix‑rank logic.

A key theoretical result is that the equivalence induced by the invertible‑map game refines every equivalence induced by these specialised games. Formally, for any finite‑variable logic L (with or without counting) and any structure pair (A, B), if Duplicator wins the L‑pebble game then she also wins the invertible‑map game with the same number of pebbles. This establishes the invertible‑map game as the most discriminating among the considered algebraic pebble games.

From an algorithmic perspective, the authors argue that each round of the invertible‑map game can be reduced to solving a feasibility problem for integer invertible matrices, which is polynomial‑time solvable using lattice‑basis reduction or integer linear programming techniques. Thus the game yields a hierarchy of polynomial‑time approximations to graph isomorphism: increasing the number of pebbles or the number of rounds yields strictly stronger tests, forming a ladder that sits strictly between WL and the exact isomorphism test.

The paper concludes with a discussion of potential applications. The framework could be adapted to compare database schemas, verify equivalence of circuit representations, or serve as a basis for new graph‑isomorphism heuristics that combine colour refinement with linear‑algebraic checks. By unifying logical and algebraic perspectives on pebble games, the work opens a pathway for designing more powerful, yet still tractable, equivalence tests for complex combinatorial structures.