Arithmetization: A new method in structural complexity theory

Abstract

We introduce a technique of arithmetization of the process of computation in order to obtain novel characterizations of certain complexity classes viamultivariate polynomials. A variety of concepts and tools of elementary algebra, such as the degree of polynomials and interpolation, becomes thereby available for the study of complexity classes. The theory to be described provides a unified framework from which powerful recent results follow naturally. The central result is a characterization of #P in terms ofarithmetic straight line programs. The consequences include a simplified proof of Toda's Theorem thatPH ⊂P#P; and an infinite class of natural and potentially inequivalent functions, checkable in the sense of Blum et al. Similar characterizations of PSPACE are also given. The arithmetization technique has been introduced independently by Adi Shamir. While this simultaneous discovery was driven by applications to interactive proofs, the present paper demonstrates the applicability of this technique to classical complexity classes.