Definability of linear equation systems over groups and rings

Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and rings from the viewpoint of logical (inter-)definability. All problems that we consider are decidable in polynomial time, but not expressible in fixed-point logic with counting. They also provide natural candidates for a separation of polynomial time from rank logics, which extend fixed-point logics by operators for determining the rank of definable matrices and which are sufficient for solvability problems over fields. Based on the structure theory of finite rings, we establish logical reductions among various solvability problems. Our results indicate that all solvability problems for linear equation systems that separate fixed-point logic with counting from PTIME can be reduced to solvability over commutative rings. Moreover, we prove closure properties for classes of queries that reduce to solvability over rings, which provides normal forms for logics extended with solvability operators. We conclude by studying the extent to which fixed-point logic with counting can express problems in linear algebra over finite commutative rings, generalising known results on the logical definability of linear-algebraic problems over finite fields

Pebble games with algebraic rules

We define a general framework of $\textit{partition games}$ for formulating two-player pebble games over finite structures. The framework we introduce includes as special cases the pebble games for finite-variable logics with and without counting. It also includes a $\textit{matrix-equivalence game}$, introduced here, which characterises equivalence in the finite-variable fragments of the matrix-rank logic of [Dawar et al. 2009]. We show that one particular such game in our framework, which we call the $\textit{invertible-map game}$, yields a family of polynomial-time approximations of graph isomorphism that is strictly stronger than the well-known Weisfeiler-Leman method. We show that the equivalence defined by this game is a refinement of the equivalence defined by each of the games for finite-variable logics.Research supported by EPSRC grant EP/H026835/1