Abstract
Valiant introduced 20 years ago a theory to study the complexity of polynomial families. Using arithmetic circuits as computation model, these classes are easy to define and open to combinatorial techniques. In this paper we gather old and new results under a unifying theme, namely the restrictions imposed upon the gates, 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 class VP or answers to both a conjecture and a problem raised by Bürgisser [1]. We also show that for circuits of polynomial depth and unbounded size these models have the same expressive power and characterize a uniform version of VNP.
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Malod, G., Portier, N. (2006). Characterizing Valiant’s Algebraic Complexity Classes. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_61
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DOI: https://doi.org/10.1007/11821069_61
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37791-7
Online ISBN: 978-3-540-37793-1
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