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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 ..."
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Cited by 209 (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].
Bounds for the Computational Power and Learning Complexity of Analog Neural Nets
 Proc. of the 25th ACM Symp. Theory of Computing
, 1993
"... . It is shown that high order feedforward neural nets of constant depth with piecewise polynomial activation functions and arbitrary real weights can be simulated for boolean inputs and outputs by neural nets of a somewhat larger size and depth with heaviside gates and weights from f\Gamma1; 0; 1g. ..."
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Cited by 60 (12 self)
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. It is shown that high order feedforward neural nets of constant depth with piecewise polynomial activation functions and arbitrary real weights can be simulated for boolean inputs and outputs by neural nets of a somewhat larger size and depth with heaviside gates and weights from f\Gamma1; 0; 1g. This provides the first known upper bound for the computational power of the former type of neural nets. It is also shown that in the case of first order nets with piecewise linear activation functions one can replace arbitrary real weights by rational numbers with polynomially many bits, without changing the boolean function that is computed by the neural net. In order to prove these results we introduce two new methods for reducing nonlinear problems about weights in multilayer neural nets to linear problems for a transformed set of parameters. These transformed parameters can be interpreted as weights in a somewhat larger neural net. As another application of our new proof technique we s...
ON THRESHOLD CIRCUITS AND POLYNOMIAL COMPUTATION
"... A Threshold Circuit consists of an acyclic digraph of unbounded fanin, where each node computes a threshold function or its negation. This paper investigates the computational power of Threshold Circuits. A surprising relationship is uncovered between Threshold Circuits and another class of unbound ..."
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Cited by 52 (1 self)
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A Threshold Circuit consists of an acyclic digraph of unbounded fanin, where each node computes a threshold function or its negation. This paper investigates the computational power of Threshold Circuits. A surprising relationship is uncovered between Threshold Circuits and another class of unbounded fanin circuits which are denoted Finite Field ZP (n) Circuits, where each node computes either multiple sums or products of integers modulo a prime P (n). In particular, it is proved that all functions computed by Threshold Circuits of size S(n) n and depth D(n) can also be computed by ZP (n) Circuits of size O(S(n) log S(n)+nP (n) log P (n)) and depth O(D(n)). Furthermore, it is shown that all functions computed by ZP (n) Circuits of size S(n) and depth D(n) can be computed by Threshold Circuits of size O ( 1 2 (S(n) log P (n)) 1+) and depth O ( 1 5 D(n)). These are the main results of this paper. There are many useful and quite surprising consequences of this result. For example, integer reciprocal can be computed in size n O(1) and depth O(1). More generally, any analytic function with a convergent rational polynomial power series (such as sine, cosine, exponentiation, square root, and logarithm) can be computed within accuracy 2,nc, for any constant c, by Threshold Circuits of
Simulating Threshold Circuits by Majority Circuits
 SIAM Journal on Computing
, 1994
"... We prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomialsize depth 2 majority circuit. In general we show that a depth d threshold circuit can be simulated uniformly by a majority circuit of depth d + 1. Goldmann, Hastad, and Razborov showed in [10 ..."
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Cited by 35 (0 self)
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We prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomialsize depth 2 majority circuit. In general we show that a depth d threshold circuit can be simulated uniformly by a majority circuit of depth d + 1. Goldmann, Hastad, and Razborov showed in [10] that a nonuniform simulation exists. Our construction answers two open questions posed in [10]: we give an explicit construction whereas [10] uses a randomized existence argument, and we show that such a simulation is possible even if the depth d grows with the number of variables n (the simulation in [10] gives polynomialsize circuits only when d is constant). 1 A preliminary version of this paper appeared in Proc. 25th ACM STOC (1993), pp. 551560. 2 Laboratory for Computer Science, MIT, Cambridge MA 02139, Email: migo@theory.lcs.mit.edu. This author 's work was done at Royal Institute of Technology in Stockholm, and while visiting the University of Bonn 3 Department of Com...
Circuit Complexity before the Dawn of the New Millennium
, 1997
"... The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Al ..."
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Cited by 30 (3 self)
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The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.
The Permanent Requires Large Uniform Threshold Circuits
, 1999
"... We show that the permanent cannot be computed by uniform constantdepth threshold circuits of size T (n) for any function T such that for all k, T (k) (n) = o(2 n ). More generally, we show that any problem that is hard for the complexity class C=P requires circuits of this size (on the unif ..."
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Cited by 27 (8 self)
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We show that the permanent cannot be computed by uniform constantdepth threshold circuits of size T (n) for any function T such that for all k, T (k) (n) = o(2 n ). More generally, we show that any problem that is hard for the complexity class C=P requires circuits of this size (on the uniform constantdepth threshold circuit model). In particular, this lower bound applies to any problem that is hard for the complexity classes PP or #P.
SizeDepth Tradeoffs for Threshold Circuits (Extended Abstract)
 SIAM J. Comput
, 1997
"... Russell Impagliazzo , Ramamohan Paturi , Michael E. Saks y Abstract The following sizedepth tradeoff for threshold circuits is obtained: any threshold circuit of depth d that computes the parity function on n variables must have at least n 1+c` \Gammad edges, where constants c ? 0 and ..."
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Cited by 21 (1 self)
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Russell Impagliazzo , Ramamohan Paturi , Michael E. Saks y Abstract The following sizedepth tradeoff for threshold circuits is obtained: any threshold circuit of depth d that computes the parity function on n variables must have at least n 1+c` \Gammad edges, where constants c ? 0 and ` 3 are constants independent of n and d. Previously known constructions show that up to the choice of c and ` this bound is best possible. In particular, the lower bound implies an affirmative answer to the conjecture of Paturi and Saks that a bounded depth threshold circuit that computes parity requires a superlinear number of edges. This is the first super linear lower bound for an explicit function that holds for any fixed depth, and the first that applies to threshold circuits with unrestricted weights. The tradeoff is obtained as a consequence of a general restriction theorem for threshold circuits with a small number of edges: For any threshold circuit with n inputs, de...
Counting Hierarchies: Polynomial Time And Constant Depth Circuits
, 1990
"... In the spring of 1989, Seinosuke Toda of the University of ElectroCommunications in Tokyo, Japan, proved that the polynomial hierarchy is contained in P PP [To89]. In this Structural Complexity Column, we will briefly review Toda's result, and explore how it relates to other topics of interest i ..."
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Cited by 19 (4 self)
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In the spring of 1989, Seinosuke Toda of the University of ElectroCommunications in Tokyo, Japan, proved that the polynomial hierarchy is contained in P PP [To89]. In this Structural Complexity Column, we will briefly review Toda's result, and explore how it relates to other topics of interest in computer science. In particular, we will introduce the reader to The Counting Hierarchy: a hierarchy of complexity classes contained in PSPACE and containing the Polynomial Hierarchy. Threshold Circuits: circuits constructed of MAJORITY gates; this notion of circuit is being studied not only by complexity theoreticians, but also by researchers in an active subfield of AI studying "neural networks". Along the way, we'll review the important notion of an operator on a complexity class. 1. The Counting Hierarchy, and Operators on Complexity Classes The counting hierarchy was defined in [Wa86] and independently by Parberry and Schnitger in [PS88]. (The motivation for [Wa86] was the desir...
On TC^0, AC^0, and Arithmetic Circuits
 Journal of Computer and System Sciences
, 2000
"... 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 constantdepth polynomialsize ..."
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Cited by 18 (6 self)
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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 constantdepth polynomialsize arithmetic circuits of unbounded fanin 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.