Small-Space Analogues of Valiant’s Classes

Abstract

In the uniform circuit model of computation, the width of a boolean circuit exactly characterises the space complexity of the computed function. Looking for a similar relationship in Valiant's algebraic model of computation, we propose width of an arithmetic circuit as a possible measure of space. We introduce the class VL as an algebraic variant of deterministic log-space L. In the uniform setting, we show that our definition coincides with that of VPSPACE at polynomial width. Further, todefinealgebraicvariants ofnon-deterministic space-bounded classes, we introduce the notion of certificates for arithmetic circuits. We show that polynomial-size algebraic branching programs can be expressed as a read-once exponential sum over polynomials in VL, i.e. VBP ∈ ΣR ċ VL. We also show that ΣR ċ VBP = VBP, i.e. VBPs are stable under read-once exponential sums. Further, we show that read-once exponential sumsover a restricted class of constant-width arithmetic circuits are within VQP, and this is the largest known such subclass of poly-log-width circuits with this property.