Small-Space Analogues of Valiant's Classes

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, to define algebraic variants of non-deterministic space-bounded classes, we introduce the notion of “read-once” 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 · VL. We also show that ΣR · VBP = VBP, i.e. VBPs are stable under read-once exponential sums. Further, we show that read-once exponential sums over a restricted class of constant-width arithme...

• Type: Conference
• Year: 2009
• Where: FCT
• Authors: Meena Mahajan, B. V. Raghavendra Rao
• Keywords: Algebraic Variants, Arithmetic Circuits, FCT 2009, Read-once Exponential Sums, Theoretical Computer Science