Elsevier

Journal of Complexity

Volume 24, Issue 1, February 2008, Pages 16-38
Journal of Complexity

Characterizing Valiant's algebraic complexity classes

https://doi.org/10.1016/j.jco.2006.09.006Get rights and content
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Abstract

Valiant introduced 20 years ago an algebraic complexity theory to study the complexity of polynomial families. The basic computation model used is the arithmetic circuit, which makes these classes very easy to define and open to combinatorial techniques. In this paper we gather known results and new techniques under a unifying theme, namely the restrictions imposed upon the gates of the circuit, building a hierarchy from formulas to circuits. As a consequence we get simpler proofs for known results such as the equality of the classes VNP and VNPe or the completeness of the Determinant for VQP, and new results such as a characterization of the classes VQP and VP (which we can also apply to the Boolean class LOGCFL) or a full answer to a conjecture in Bürgisser's book [Completeness and reduction in algebraic complexity theory, Algorithms and Computation in Mathematics, vol. 7, Springer, Berlin, 2000]. We also show that for circuits of polynomial depth and unbounded size these models all have the same expressive power and can be used to characterize a uniform version of VNP.

MSC

03D15
68Q15
68Q17

Keywords

Algebraic complexity
Valiant's theory
Polynomials
Straight line programs
Circuits
Permanent
Determinant
Skew circuits
Formulas

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