this paper. 2.1. Circuit Classes We begin by formally defining the three circuit complexity classes that will concern us here. These are given by combinatorial restrictions on the circuits of the family. We will then define the uniformity restrictions we will use. Finally, we will give the equivalent formulations of uniform circuit complexity classes in terms of descriptive complexity classes

We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The Blum-Shub-Smale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model ” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer N, decide whether N> 0. • In the Blum-Shub-Smale model, polynomial time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. • The Generic Task of Numerical Computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.

Given a description of a probabilistic automaton (one-head probabilistic finite automaton or probabilistic Turing machine) and an input string x of length n, we ask how much space does a deterministic Turing machine need in order to decide the acceptance of the input string by that automaton? The question is interesting even in the case of one-head one-way probabilistic finite automata (PFA). We call (rational) stochastic languages (S ? rat ) the class of languages recognized by PFA's whose transition probabilities and cutpoints (i.e. recognition thresholds) are rational numbers. The class S ? rat contains context-sensitive languages that are not context free, but on the other hand there are context-free languages not included in S ? rat . Our main results are as follows: ffl The (proper) inclusion of S ? rat in Dspace(log n), which is optimal (i.e. S ? rat 6ae Dspace(o(log n))). The previous upper bounds were Dspace(n) [Dieu 1972], [Wang 1992] and Dspace(log n log log n)...

Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC [CMTV96], we study the class of functions . One way to define #AC is as the class of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. In contrast to the preceding # Part of this research was done while visiting the University of Ulm under an Alexander von Humboldt Fellowship.

Integer division has been known to lie in P-uniform TC 0 since the mid-1980's, and recently this was improved to L- uniform TC 0 . At the time that the results in this paper were proved and submitted for conference presentation, it was unknown whether division lay in DLOGTIME-uniform TC 0 (also known as FOM). We obtain tight bounds on the uniformity required for division, by showing that division is complete for the complexity class FOM + POW obtained by augmenting FOM with a predicate for powering modulo small primes. We also show that, under a well-known number-theoretic conjecture (that there are many "smooth" primes), POW (and hence division) lies in FOM. Building on this work, Hesse has shown recently that division is in FOM [17].

We define the counting classes #NC 1 , GapNC 1 , PNC 1 and C=NC 1 . We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC 1 ` #L and that C=NC 1 ` L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separation of ACC 0 from MOD-PH as well as that of TC 0 from the counting hierarchy. Moreover we obtain that dlogtimeuniformity and logspace-uniformity for AC 0 coincide if and only if the polynomial time hierarchy equals PSPACE .