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

Abstract

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 polylogarithmically many iterations of the fixed-point operators. This enables us to separate the descriptive complexity of solving linear equations from full fixed-point logic with counting by logical means. As an application of these results, we separate an extension of first-order logic with a rank operator from fixed-point logic with counting, solving an open problem due to Holm [21]. We then draw a connection from this work in descriptive complexity theory to graph isomorphism testing and propositional proof complexity. Answering an open question from [7], we separate the strength of certain algebraic graph-isomorphism tests. This result can also be phrased as a separation of the algebraic propositional proof systems “Nullstellensatz” and “monomial PC”.