Characterizing Valiant's algebraic complexity classes

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 B

• Type: Journal
• Year: 2008
• Where: JC
• Authors: Guillaume Malod, Natacha Portier
• Keywords: Algebraic Complexity, Algebraic Complexity Theory, Basic Computation Model, JC 2008