We show that any language recognized by an NC ’ circuit (fan-in 2, depth O(log n)) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC’. Further, following

In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by first-order formulas [Im87a,Im87b] and a uniformity corresponding to Buss' deterministic log-time reductions [Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for low-level circuit complexity classes. We show that recent results on the structure of NC¹ [Ba89] still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Th'erien, and Thomas [STT88]. A preliminary version of this work appeared as [BIS88].

This paper studies the learnability of branching programs and small depth circuits with modular and threshold gates in both the exact and PAC learning models with and without membership queries. Some of the results extend earlier works in [GG95, ERR95, BTW95]. The main results are as follows. For branching programs we show the following.

We study the computational complexity of solving systems of equations over a finite group. An equation over a group G is an expression of the form w1 w2 wk = id where each w i is either a variable, an inverted variable, or group constant and id is the identity element of G. A solution to such an equation is an assignment of the variables (to values in G) which realizes the equality. A system of equations is a collection of such equations; a solution is then an assignment which simultaneously realizes each equation. We show that the problem of determining if a (single) equation has a solution is NP-complete for all non-solvable groups G. For nilpotent groups, this same problem is shown to be in P. The analogous problem for systems of such equations is shown to be NP-complete if G is non-Abelian, and in P otherwise. Finally, we observe some connections between these problems and the theory of non-uniform automata. 1

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