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24
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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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...
On the Power of Computational Secret Sharing
, 2003
"... Secret sharing is a very important primitive in cryptography and distributed computing. In this work, we consider computational secret sharing (CSS) which provably allows a smaller share size (and hence greater efficiency) than its informationtheoretic counterparts. Extant CSS schemes result in ..."
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Secret sharing is a very important primitive in cryptography and distributed computing. In this work, we consider computational secret sharing (CSS) which provably allows a smaller share size (and hence greater efficiency) than its informationtheoretic counterparts. Extant CSS schemes result in succinct sharesize and are in a few cases, like threshold access structures, optimal. However, in general, they are not efficient (sharesize not polynomial in the number of players n), since they either assume efficient perfect schemes for the given access structure (as in [10]) or make use of exponential (in n) amount of public information (like in [5]). In this paper, our goal is to explore other classes of access structures that admit of efficient CSS, without making any other assumptions. We construct efficient CSS schemes for every access structure in monotone P . As of now, most of the efficient informationtheoretic schemes known are for access structures in algebraic NC . Monotone P and algebraic NC are not comparable in the sense one does not include other. Thus our work leads to secret sharing schemes for a new class of access structures. In the second part of the paper, we introduce the notion of secret sharing with a semitrusted third party, and prove that in this relaxed model efficient CSS schemes exist for a wider class of access structures, namely monotone NP.
A Criterion for Monotone Circuit Complexity
, 1991
"... In this paper we study the lower bounds problem for monotone circuits. The main goal is to extend and simplify the well known method of approximations proposed by A. Razborov in 1985. The main result is the following combinatorial criterion for the monotone circuit complexity: a monotone Boolean fun ..."
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In this paper we study the lower bounds problem for monotone circuits. The main goal is to extend and simplify the well known method of approximations proposed by A. Razborov in 1985. The main result is the following combinatorial criterion for the monotone circuit complexity: a monotone Boolean function f(X) of n variables X = fx 1 ; : : : ; x n g requires monotone circuits of size exp(\Omega\Gamma t= log t)) if there is a family F ` 2 X such that: (i) each set in F is either a minterm or a maxterm of f; and (ii) D k (F)=D k+1 (F) t for every k = 0; 1; : : : ; t \Gamma 1: Here D k (F) is the kth degree of F , i.e. maximum cardinality of a subfamily H ` F with j " Hj k: 1 Introduction The question of determining how much economy the universal nonmonotone basis f; ; :g provides over the monotone basis f; g has been a long standing open problem in Boolean circuit complexity. In 1985, Razborov [10, 11] achieved a major development in this direction. He worked out the, socalled,...
Monotone circuits for connectivity have depth (log ) 2(1
 SIAM Journal on Computing
, 1998
"... We prove that a monotone circuit of size n d recognizing connectivity must have depth ((log n) 2 = log d). For formulas this implies depth ((log n) 2 = log log n). For ((log n) 2)which is optimal up to a conpolynomialsize circuits the bound becomes stant. Warning: Essentially this paper has been p ..."
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We prove that a monotone circuit of size n d recognizing connectivity must have depth ((log n) 2 = log d). For formulas this implies depth ((log n) 2 = log log n). For ((log n) 2)which is optimal up to a conpolynomialsize circuits the bound becomes stant. Warning: Essentially this paper has been published in SIAM Journal on Computing is hence subject to copyright restrictions. It is for personal use only. 1
Optimal Cryptographic Hardness of Learning Monotone Functions
"... Abstract. A wide range of positive and negative results have been established for learning different classes of Boolean functions from uniformly distributed random examples. However, polynomialtime algorithms have thus far been obtained almost exclusively for various classes of monotone functions, ..."
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Abstract. A wide range of positive and negative results have been established for learning different classes of Boolean functions from uniformly distributed random examples. However, polynomialtime algorithms have thus far been obtained almost exclusively for various classes of monotone functions, while the computational hardness results obtained to date have all been for various classes of general (nonmonotone) functions. Motivated by this disparity between known positive results (for monotone functions) and negative results (for nonmonotone functions), we establish strong computational limitations on the efficient learnability of various classes of monotone functions. We give several such hardness results which are provably almost optimal since they nearly match known positive results. Some of our results show cryptographic hardness of learning polynomialsize monotone circuits to accuracy only slightly greater than 1/2 + 1 / √ n; this accuracy bound is close to optimal by known positive results (Blum et al., FOCS ’98). Other results show that under a plausible cryptographic hardness assumption, a class of constantdepth, subpolynomialsize circuits computing monotone functions is hard to learn; this result is close to optimal in terms of the circuit size parameter by known positive results as well (Servedio, Information and Computation ’04). Our main tool is a complexitytheoretic approach to hardness amplification via noise sensitivity of monotone functions that was pioneered by O’Donnell (JCSS ’04). 1
A Lower Bound for Monotone Arithmetic Circuits Computing 01 Permanent
"... this paper we show that this, in fact, is not the case. We extend the framework in [1] to show that monotone arithmetic circuits require exponential size even for computing the 01 permanent. ..."
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this paper we show that this, in fact, is not the case. We extend the framework in [1] to show that monotone arithmetic circuits require exponential size even for computing the 01 permanent.
Average Circuit Depth and Average Communication Complexity
, 1995
"... We use the techniques of Karchmer and Widgerson [KW90] to derive strong lower bounds on the expected parallel time to compute boolean functions by circuits. By average time, we mean the time needed on a selftimed circuit, a model introduced recently by Jakoby, Reischuk, and Schindelhauer, [JRS94] i ..."
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We use the techniques of Karchmer and Widgerson [KW90] to derive strong lower bounds on the expected parallel time to compute boolean functions by circuits. By average time, we mean the time needed on a selftimed circuit, a model introduced recently by Jakoby, Reischuk, and Schindelhauer, [JRS94] in which gates compute their output as soon as it is determined (possibly by a subset of the inputs to the gate). More precisely, we show that the average time needed to compute a boolean function on a circuit is always greater than or equal to the average number of rounds required in Karchmer and Widgerson's communication game. We also prove a similar lower bound for the monotone case. We then use these techniques to show that, for a large subset of the inputs, the average time needed to compute s \Gamma t connectivity by monotone boolean circuits is\Omega\Gamma/42 2 n). We show, that, unlike the situation for worst case bounds, where the number of rounds characterize circuit depth, in th...
A Simple Lower Bound for Monotone Clique Using a Communication Game
"... We give a simple proof that a monotone circuit for the kclique problem in an nvertex graph p 3 requires depth k,whenk p n=2 2 The proof is based on an equivalence between the depth of a Boolean circuit for a function and the number of rounds required to solve a related communication problem. This ..."
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We give a simple proof that a monotone circuit for the kclique problem in an nvertex graph p 3 requires depth k,whenk p n=2 2 The proof is based on an equivalence between the depth of a Boolean circuit for a function and the number of rounds required to solve a related communication problem. This equivalence was shown by Karchmer and Wigderson. Warning: Essentially this paper has been published in Information Processing Letters and is hence subject to copyright restrictions. It is for personal use only. Key words. computational complexity, theory of computation, circuit complexity, formula
Noncancellative Boolean Circuits: A Generalization of Monotone Boolean Circuits
 Theoretical Computer Science
"... Cancellations are known to be helpful in efficient algebraic computation of polynomials over fields. We define a notion of cancellation in Boolean circuits and define Boolean circuits that do not use cancellation to be noncancellative. Noncancellative Boolean circuits are a natural generalizati ..."
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Cancellations are known to be helpful in efficient algebraic computation of polynomials over fields. We define a notion of cancellation in Boolean circuits and define Boolean circuits that do not use cancellation to be noncancellative. Noncancellative Boolean circuits are a natural generalization of monotone Boolean circuits. We show that in the absence of cancellation, Boolean circuits require superpolynomial size to compute the determinant interpreted over GF(2). This nonmonotone Boolean function is known to be in P . In the spirit of monotone complexity classes, we define complexity classes based on noncancellative Boolean circuits. We show that when the Boolean circuit model is restricted by withholding cancellation, P and popular classes within P are restricted as well, but NP and circuit definable classes above it remain unchanged. Keywords: Boolean circuit, monotonicity, cancellation, determinant, complexity classes. 1 Introduction Using the power of cancellatio...
Lower Bounds for Perfect Matching in Restricted Boolean Circuits
, 1993
"... We consider three restrictions on Boolean circuits: bijectivity, consistency and multilinearity. Our main result is that Boolean circuits require exponential size to compute the bipartite perfect matching function when restricted to be (i) bijective or (ii) consistent and multilinear. As a consequen ..."
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We consider three restrictions on Boolean circuits: bijectivity, consistency and multilinearity. Our main result is that Boolean circuits require exponential size to compute the bipartite perfect matching function when restricted to be (i) bijective or (ii) consistent and multilinear. As a consequence of the lower bound on bijective circuits, we prove an exponential size lower bound for monotone arithmetic circuits that compute the 01 permanent function. We also define a notion of homogeneity for Boolean circuits and show that consistent homogeneous circuits require exponential size to compute the bipartite perfect matching function. Motivated by consistent multilinear circuits, we consider certain restricted (\Phi; ) circuits and obtain an exponential lower bound for computing bipartite perfect matching using such circuits. Finally, we show that the lower bound arguments for the bipartite perfect matching function on all these restricted models can be adapted to prove exponential low...