Small-Space Analogues of Valiant’s Classes

Abstract

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, 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 ∈ Σ R · VL. We also show that Σ R · 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 arithmetic circuits are within VQP, and this is the largest known such subclass of poly-log-width circuits with this property.

Counting Classes and the Fine Structure between NC 1 and L

“…Also, as in [Ogi98], we use the polynomial technique developed earlier ( [BRS95,FR96]) to show closure properties of the complexity class PP. A new ingredient we require is the notion of read-once certified circuits and exponential sums from [MR09].…”

“…In the GapBWBP setting, things are a bit more complicated since we have only O(1) storage. We get around this by using the notion of exponential sums over read-once certified circuits, introduced in [MR09]. The GapBWBP family computing the desired h is described in Section 4.4, completing the proof of Theorem 20.…”

“…We use the equivalent characterization of GapNC 1 as arithmetic bounded-width branching programs GapBWBP, and establish the normal form here. Another difficulty is computing an exponential sum; we use the notion of read-once certified circuits and read-once exponential sums, introduced in [MR09], to to carry the proof through.…”

“…< i m such that variable y j appears only between layer i j−1 and i j . By specialising the arguments in [MR09] to the counting classes, we can compute exponential sums over the variables in Y efficiently.…”

“…Proposition 7 (adapted from Theorem 3(2), [MR09]). Let f (X, Y ) be a function computed by an a-BP B of size s and width w, read-once certified in Y .…”