Small-Space Analogues of Valiant’s Classes

  • Meena Mahajan
  • B. V. Raghavendra Rao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5699)


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. \({\sf VBP}\in{\it \Sigma}^R \cdot{\sf VL}\). We also show that \({\it \Sigma}^R \cdot {\sf VBP} ={\sf VBP}\), 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.


Arithmetic Computation Polynomial Space Boolean Circuit Output Gate Input Tape 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Meena Mahajan
    • 1
  • B. V. Raghavendra Rao
    • 1
  1. 1.The Institute of Mathematical SciencesChennaiIndia

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