Results 11  20
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61
Why is Boolean Complexity Theory Difficult?
, 1992
"... this paper we shall assume that S is a commutative ring with identity. Then each instruction f i can be identified with the polynomial that is computed at f i , if\Omega and \Phi are interpreted as the ring operations in the polynomial ring S[x 1 ; \Delta \Delta \Delta ; x n ]. Among natural multiva ..."
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Cited by 34 (0 self)
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this paper we shall assume that S is a commutative ring with identity. Then each instruction f i can be identified with the polynomial that is computed at f i , if\Omega and \Phi are interpreted as the ring operations in the polynomial ring S[x 1 ; \Delta \Delta \Delta ; x n ]. Among natural multivariate polynomials whose complexity in this model is of interest are Hamiltonian circuits (HC), the permanent (PERM) and the determinant (DET). These are defined over a matrix X of indeterminates fx 11 ; \Delta \Delta \Delta ; x nn g where x ij
JAMRESISTANT COMMUNICATION WITHOUT SHARED SECRETS THROUGH THE USE OF CONCURRENT CODES
, 2007
"... We consider the problem of establishing jamresistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this problem, the BBC algorithm, and give several instantiations of it. We develop an ..."
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Cited by 19 (9 self)
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We consider the problem of establishing jamresistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this problem, the BBC algorithm, and give several instantiations of it. We develop and analyze this algorithm within the framework of a new type of code, concurrent codes, which are those superimposed codes that allow efficient decoding. Finally, we propose the Universal Concurrent Code algorithm, and prove that it covers all possible concurrent codes, and give connections between its theory and that of monotone Boolean functions.
Monotone Simulations of Nonmonotone Proofs
, 2001
"... We show that an LK proof of size m of a monotone sequent (a sequent that contains only formulas in the basis ; ) can be turned into a proof containing only monotone formulas of size O(log m) and with the number of proof lines polynomial in m. Also we show that some interesting special cases, n ..."
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Cited by 18 (2 self)
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We show that an LK proof of size m of a monotone sequent (a sequent that contains only formulas in the basis ; ) can be turned into a proof containing only monotone formulas of size O(log m) and with the number of proof lines polynomial in m. Also we show that some interesting special cases, namely the functional and the onto versions of PHP and a version of the Matching Principle, have polynomial size monotone proofs. We prove that LK is polynomially bounded if and only if its monotone fragment is.
Jam resistant communications without shared secrets
 in Proceedings of the 3 rd International Conference on Information Warfare and Security
, 2008
"... Distribution A, Approved for public release, distribution unlimited Abstract. We consider the problem of establishing jamresistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this prob ..."
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Cited by 11 (1 self)
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Distribution A, Approved for public release, distribution unlimited Abstract. We consider the problem of establishing jamresistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this problem, the BBC algorithm, and give several instantiations of it. We develop and analyze this algorithm within the framework of a new type of code, concurrent codes, which are those superimposed codes that allow efficient decoding. Finally, we propose the Universal Concurrent Code algorithm, and prove that it covers all possible concurrent codes, and give connections between its theory and that of monotone Boolean functions.
The Fusion Method for Lower Bounds in Circuit Complexity
 Keszthely (Hungary
, 1993
"... This paper coins the term "The Fusion Method" to a recent approach for proving circuit lower bounds. It describes the method, and surveys its achievements, potential and challenges. 1 Introduction In a recent paper, Karchmer [6] suggested an elegant way in which one can view at the same t ..."
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This paper coins the term "The Fusion Method" to a recent approach for proving circuit lower bounds. It describes the method, and surveys its achievements, potential and challenges. 1 Introduction In a recent paper, Karchmer [6] suggested an elegant way in which one can view at the same time both the "approximation method" of Razborov [13] and the "topological approach" of Sipser [15] for proving circuit lower bounds. In Karchmer's setting the lower bound prover shows that a given circuit C is too small for computing a given function f by contradiction, in the following way. She tries to combine (or 'fuse', as we propose calling it) correct accepting computations of inputs in f \Gamma1 (1) by C into an incorrect accepting computation of an input in f \Gamma1 (0). It turns out that this "Fusion Method" reduces the dynamic computation of f by C into a static combinatorial cover problem, which provides the lower bound. Moreover, different restrictions on how we can fuse computations ...
Averagecase complexity of detecting cliques
, 2010
"... The computational problem of testing whether a graph contains a complete subgraph of size k is among the most fundamental problems studied in theoretical computer science. This thesis is concerned with proving lower bounds for kClique, as this problem is known. Our results show that, in certain mod ..."
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Cited by 8 (0 self)
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The computational problem of testing whether a graph contains a complete subgraph of size k is among the most fundamental problems studied in theoretical computer science. This thesis is concerned with proving lower bounds for kClique, as this problem is known. Our results show that, in certain models of computation, solving kClique in the average case requires Ω(n k/4) resources (moreover, k/4 is tight). Here the models of computation are boundeddepth Boolean circuits and unboundeddepth monotone circuits, the complexity measure is the number of gates, and the input distributions are random graphs with an appropriate density of edges. Such random graphs (the wellstudied ErdősRényi graphs) are widely believed to be a source of computationally hard instances for clique problems, a hypothesis first articulated by Karp in 1976. This thesis gives the first unconditional lower bounds supporting this hypothesis. Significantly, our result for boundeddepth Boolean circuits breaks out of the traditional
The effect of nullchains on the complexity of contact schemes
 Proc. of FCT, LNCS 380
, 1989
"... The contact scheme complexity of Boolean functions has been studied for a long time but its main problem remains unsolved: we have no example of a simple function (say in NP) that requires ~(n 3) contact scheme size. The reason is, perhaps, that although the contact scheme model is elegantly simple, ..."
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Cited by 7 (2 self)
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The contact scheme complexity of Boolean functions has been studied for a long time but its main problem remains unsolved: we have no example of a simple function (say in NP) that requires ~(n 3) contact scheme size. The reason is, perhaps, that although the contact scheme model is elegantly simple, our understanding of the way it computes is vague. On the other hand, it is known (see, e.g. [2,3]) that the main tool to reduce the size of schemes is to use "nullchains", i.e. chains with zero conduct iv i ty. (These chains enable one to merge non isomorphic subschemes). So, in order to better understand the power of this tool, it is desirable to have lower bound arguments for schemes with various restr ict ions on nullchains. In this report such an arguments are descr ibed for schemes without nul lchains (Theorems 12), for schemes with restr icted
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...