Results 1  10
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17
Monotone Complexity
, 1990
"... We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple ..."
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Cited by 2350 (12 self)
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We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic logspace) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = coNL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for stconnectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...
On the power of smalldepth threshold circuits
 Proceedings 31st Annual IEEE Symposium on Foundations of Computer Science
, 1990
"... Abstract. Weinvestigate the power of threshold circuits of small depth. In particular, we give functions that require exponential size unweighted threshold circuits of depth 3 when we restrict the bottom fanin. We also prove that there are monotone functions fk that can be computed in depth k and li ..."
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Cited by 103 (2 self)
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Abstract. Weinvestigate the power of threshold circuits of small depth. In particular, we give functions that require exponential size unweighted threshold circuits of depth 3 when we restrict the bottom fanin. We also prove that there are monotone functions fk that can be computed in depth k and linear size ^ � _circuits but require exponential size to compute by a depth k; 1 monotone weighted threshold circuit. Key words. Circuit complexity, monotone circuits, threshold circuits, lower bounds Subject classi cations. 68Q15, 68Q99 1.
Majority Gates vs. General Weighted Threshold Gates
 Computational Complexity
, 1992
"... . In this paper we study small depth circuits that contain threshold gates (with or without weights) and parity gates. All circuits we consider are of polynomial size. We prove several results which complete the work on characterizing possible inclusions between many classes defined by small depth c ..."
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Cited by 89 (7 self)
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. In this paper we study small depth circuits that contain threshold gates (with or without weights) and parity gates. All circuits we consider are of polynomial size. We prove several results which complete the work on characterizing possible inclusions between many classes defined by small depth circuits. These results are the following: 1. A single threshold gate with weights cannot in general be replaced by a polynomial fanin unweighted threshold gate of parity gates. 2. On the other hand it can be replaced by a depth 2 unweighted threshold circuit of polynomial size. An extension of this construction is used to prove that whatever can be computed by a depth d polynomial size threshold circuit with weights can be computed by a depth d + 1 polynomial size unweighted threshold circuit, where d is an arbitrary fixed integer. 3. A polynomial fanin threshold gate (with weights) of parity gates cannot in general be replaced by a depth 2 unweighted threshold circuit of polynomial size...
Noise sensitivity of Boolean functions and applications to percolation, Inst. Hautes Études
, 1999
"... It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority ..."
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Cited by 72 (16 self)
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It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noisestable. Several necessary and sufficient conditions for noise sensitivity and stability are given. Consider, for example, bond percolation on an n + 1 by n grid. A configuration is a function that assigns to every edge the value 0 or 1. Let ω be a random configuration, selected according to the uniform measure. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges e with ω(e) = 1. By duality, the probability for having a crossing is 1/2. Fix an ǫ ∈ (0,1). For each edge e, let ω ′ (e) = ω(e) with probability 1 − ǫ, and ω ′ (e) = 1 − ω(e)
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 ReadOnce Threshold Formulae and their Randomized Decision Tree Complexity
 Theoretical Computer Science
, 1994
"... TC 0 is the class of functions computable by polynomialsize, constant depth formulae with threshold gates. ReadOnce TC 0 (ROTC 0 ) is the subclass of TC 0 which restricts every variable to occur exactly once in the formula. Our main result is a (tight) linear lower bound on the randomize ..."
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Cited by 17 (0 self)
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TC 0 is the class of functions computable by polynomialsize, constant depth formulae with threshold gates. ReadOnce TC 0 (ROTC 0 ) is the subclass of TC 0 which restricts every variable to occur exactly once in the formula. Our main result is a (tight) linear lower bound on the randomized decision tree complexity of any function in ROTC 0 . This relationship between threshold circuits and decision trees bears significance on both models of computation. Regarding decision trees, this is the first class of functions for which such a strong bound is known. Regarding threshold circuits, it may be considered as a possible first step towards proving TC 0 6= NC 1 ; generalizing our lower bound to all functions in TC 0 would establish this separation. Another structural result we obtain is that a readonce threshold formula uniquely represents the function it computes. 1 Introduction 1.1 Boolean Decision Trees The Boolean decision tree is an extremely simple model for co...
Depth Reduction for Circuits of Unbounded FanIn
, 1991
"... We prove that constant depth circuits of size n over the basis AND, OR, PARITY are no more powerful than circuits of this size with depth four. Similar techniques are used to obtain several other depth reduction theorems; in particular, we show every set in AC can be recognized by a family of ..."
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Cited by 14 (5 self)
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We prove that constant depth circuits of size n over the basis AND, OR, PARITY are no more powerful than circuits of this size with depth four. Similar techniques are used to obtain several other depth reduction theorems; in particular, we show every set in AC can be recognized by a family of depth three . The size bound n is optimal when considering depth reduction over AND, OR, and PARITY. Most of our results hold both for the uniform and the nonuniform case.
Combinatorics of Monotone Computations
 Combinatorica
, 1998
"... Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length 6 d, and (ii) arbitrary realvalued nondecreasing functions on 6 d variables. This r ..."
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Cited by 6 (0 self)
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Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length 6 d, and (ii) arbitrary realvalued nondecreasing functions on 6 d variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, nontrivial lower bounds for circuits computing explicit functions even when d !1. The proof is relatively simple and direct, and combines the bottlenecks counting method of Haken with the idea of finite limit due to Sipser. We demonstrate the criterion by superpolynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial tdesigns. We then derive exponential lower bounds for cliquelike graph functions of Tardos, thus establishing an exponential gap between the monotone real and nonmonotone Boolean circuit complexities. Since we allow real gates, the criterion...