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Boundedwidth polynomialsize branching programs recognize exactly those languages
 in NC’, in “Proceedings, 18th ACM STOC
, 1986
"... We show that any language recognized by an NC ’ circuit (fanin 2, depth O(log n)) can be recognized by a width5 polynomialsize branching program. As any boundedwidth polynomialsize branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such prog ..."
Abstract

Cited by 213 (13 self)
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We show that any language recognized by an NC ’ circuit (fanin 2, depth O(log n)) can be recognized by a width5 polynomialsize branching program. As any boundedwidth polynomialsize 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
On Uniformity within NC¹
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1990
"... 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 ..."
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Cited by 127 (19 self)
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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 firstorder formulas [Im87a,Im87b] and a uniformity corresponding to Buss' deterministic logtime reductions [Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for lowlevel 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].
The Complexity of Solving Equations over Finite Groups
 Information and Computation
, 2002
"... 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 ..."
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Cited by 12 (1 self)
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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 NPcomplete for all nonsolvable 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 NPcomplete if G is nonAbelian, and in P otherwise. Finally, we observe some connections between these problems and the theory of nonuniform automata. 1
On Learning Branching Programs and Small Depth Circuits
 Computational Learning Theory: Proc. Third European Conference. Lecture Notes in Articial Intelligence
, 1997
"... 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 ..."
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Cited by 10 (2 self)
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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.