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$\mathsf{V}\mathsf{B}\mathsf{P}\in {\mathit{\Sigma }}^R\cdot \mathsf{V}\mathsf{L}$. We also show that ΣR⋅VBP=VBP${\mathit{\Sigma }}^R\cdot \mathsf{V}\mathsf{B}\mathsf{P}=\mathsf{V}\mathsf{B}\mathsf{P}$, 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.