Semidefinite programming and arithmetic circuit evaluation

Abstract

A rational number can be naturally presented by an arithmetic computation (AC): a sequence of elementary arithmetic operations starting from a fixed constant, say 1. The asymptotic complexity issues of such a representation are studied e.g. in the framework of the algebraic complexity theory over arbitrary field. Here we study a related problem of the complexity of performing arithmetic operations and computing elementary predicates, e.g. ``='' or ``>'', on rational numbers given by AC. In the first place, we prove that AC can be efficiently simulated by the exact semidefinite programming (SDP). Secondly, we give a BPP-algorithm for the equality predicate. Thirdly, we put ``>''-predicate into the complexity class PSPACE. We conjecture that ``>''-predicate is hard to compute. This conjecture, if true, would clarify the complexity status of the exact SDP - a well known open problem in the field of mathematical programming.