Uniformity and bounded arithmetic below P

Abstract

We study some of the complexity classes below P and, in particular, we concentrate on $AC\sp0\subseteq NC\sp1\subseteq L=$ LogSpace. We also study the nondeterministic classes $NAC\sp{i}$ and $NNC\sp{i},$ for $i\ge0,$ which are the counterparts to the more familiar class NP. In the final part of this work we characterize the so-called Steven's Class $SC=\bigcup\sb{i\ge1}Sc\sp{i}.$

We start by proving that certain basic arithmetic operations such as Count, Multiplication, Multiple Addition, Sorting, etc. can be carried out in Uniform-$NC\sp1$ and that similar results hold in the class Uniform-$AC\sp0$ when dealing with sufficiently "small" numbers. The proofs are carried out by using algebraic characterizations of the previous classes as developed, for example, in (C14) and (CT2).

We continue with the classes $NAC\sp0\subseteq NNC\sp1\subseteq\cdots\subseteq NP$ introduced in (Ta2) and prove that, in fact, all of them coincide and therefore are equal to NPolyTime.

Finally, we move further up and consider the complexity class SC as defined in (Co2). We introduce the notion of Extended k-Bounded Recursion on Notation $(E\sb{k}BRN)$ and prove that the class $SC\sp{k}$ equals the closure of the set of basic functions INITIAL (see for example, (CT2)), under composition, CRN and $E\sb{k}BRN.$