Resolution with Symmetry Rule applied to Linear Equations

This paper considers the length of resolution proofs when using Krishnamurthy's classic symmetry rules. We show that inconsistent linear equation systems of bounded width over a fixed finite field $\mathbb{F}_p$ with $p$ a prime have, in their standard encoding as CNFs, polynomial length resolutions when using the local symmetry rule (SRC-II). As a consequence it follows that the multipede instances for the graph isomorphism problem encoded as CNF formula have polynomial length resolution proofs. This contrasts exponential lower bounds for individualization-refinement algorithms on these graphs. For the Cai-Furer-Immerman graphs, for which Toran showed exponential lower bounds for resolution proofs (SAT 2013), we also show that already the global symmetry rule (SRC-I) suffices to allow for polynomial length proofs.