Small space analogues of Valiant\u27s classes and the limitations of skew formula

Abstract

In the uniform circuit model of computation, the width of a boolean circuit exactly characterises the ``space\u27\u27 complexity of the computed function. Looking for a similar relationship in Valiant\u27s 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, to define algebraic variants of non-deterministic space-bounded classes, we introduce the notion of ``read-once\u27\u27 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, ie $mbox{VBP}inSigma^R cdotmbox{VL}$. We also show that $Sigma^R cdot mbox{VBP} =mbox{VBP}$, ie VBPs are stable under read-once exponential sums. Further, we show that read-once exponential sums over 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. We also study the power of skew formulas and show that exponential sums of a skew formula cannot represent the determinant polynomial