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42
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 17 (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.
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 16 (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.
Valiant’s Holant Theorem and Matchgate Tensors (Extended Abstract
 In Proceedings of TAMC 2006: Lecture Notes in Computer Science
"... Abstract We propose matchgate tensors as a natural and proper language to develop Valiant's newtheory of Holographic Algorithms. We give a treatment of the central theorem in this theorythe Holant Theoremin terms of matchgate tensors. Some generalizations are presented. 1 Background In a remarka ..."
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Cited by 13 (7 self)
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Abstract We propose matchgate tensors as a natural and proper language to develop Valiant's newtheory of Holographic Algorithms. We give a treatment of the central theorem in this theorythe Holant Theoremin terms of matchgate tensors. Some generalizations are presented. 1 Background In a remarkable paper, Valiant [9] in 2004 has proposed a completely new theory of Holographic Algorithms or Holographic Reductions. In this framework, Valiant has developed a most novel methodology of designing polynomial time (indeed NC2) algorithms, a methodology by which one can design a custom made process capable of carrying out a seemingly exponential computation with exponentially many cancellations so that the computation can actually be done in polynomial time. The simplest analogy is perhaps with Strassen's matrix multiplication algorithm [5]. Here the algorithm computes some extraneous quantities in terms of the submatrices, which do not directly appear in the answer yet only to be canceled later, but the purpose of which is to speedup computation by introducing cancelations. In the several cases such clever algorithms had been found, they tend to work in a linear algebraic setting, in particular the computation of the determinant figures prominently [8, 2, 6]. Valiant's new theory manages to create a process of custom made cancelation which gives polynomial time algorithms for combinatorial problems which do not appear to be linear algebraic. In terms of its broader impact in complexity theory, one can view Valiant's new theory as another algorithmic design paradigm which pushes back the frontier of what is solvable by polynomial time. Admittedly, at this early stage, it is still premature to say what drastic consequence it might have on the landscape of the big questions of complexity theory, such as P vs. NP. But the new theory has already been used by Valiant to devise polynomial time algorithms for a number of problems for which no polynomial time algorithms were known before.
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 10 (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 time both t ..."
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Cited by 10 (0 self)
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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 ...
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...
On The Complexity Of NegationLimited Boolean Networks
 SIAM J. Comput
, 1998
"... . A theorem of Markov precisely determines the number r of NEGATION gates necessary and su#cient to compute a system of boolean functions F . For a system of boolean functions on n variables, r
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.<F3.862e+05> A theorem of Markov precisely determines the number<F3.255e+05> r<F3.862e+05> of NEGATION gates necessary and su#cient to compute a system of boolean functions<F3.255e+05> F<F3.862e+05> . For a system of boolean functions on<F3.255e+05> n<F3.862e+05> variables,<F3.255e+05> r<F3.466e+05> #<F3.255e+05><F3.862e+05><F3.255e+05><F3.862e+05> b(n) =<F3.466e+05><F3.862e+05> #log<F2.754e+05> 2<F3.862e+05><F3.255e+05> (n<F3.862e+05> +<F3.466e+05><F3.862e+05> 1)#. We call a circuit using<F3.255e+05><F3.862e+05><F3.255e+05><F3.862e+05> b(n) NEGATION gates<F3.309e+05> negationlimited<F3.862e+05> . We continue recent investigations into negationlimited circuit complexity, giving both upper and lower bounds. A circuit with inputs<F3.255e+05> x<F2.754e+05> 1<F3.255e+05> , . . . ,<F2.459e+05> xn<F3.862e+05> and outputs<F3.466e+05><F3.255e+05> x<F2.754e+05> 1<F3.255e+05> , . . . ,<F3.466e+05><F3.255e+05><F2.459e+05> xn<F3.862e+05> is called an<F3.309e+05> inverter<F3.862e+05> , ...
Finite Limits and Monotone Computations: The Lower Bounds Criterion
 Proc. of the 12th IEEE Conference on Computational Complexity
, 1997
"... Our main result is a combinatorial lower bounds criterion for monotone circuits over the reals. We allow any unbounded fanin nondecreasing realvalued functions as gates. The only requirement is their "locality ". Unbounded fanin AND and OR gates, as well as any threshold gate T m s (x 1 ; : : : ..."
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Cited by 5 (1 self)
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Our main result is a combinatorial lower bounds criterion for monotone circuits over the reals. We allow any unbounded fanin nondecreasing realvalued functions as gates. The only requirement is their "locality ". Unbounded fanin AND and OR gates, as well as any threshold gate T m s (x 1 ; : : : ; xm ) with small enough threshold value minfs; m \Gamma s + 1g, are simplest examples of local gates. The proof is relatively simple and direct, and combines the bottlenecks counting approach of Haken with the idea of finite limit due to Sipser. Apparently this is the first combinatorial lower bounds criterion for monotone computations. It is symmetric and yields (in a uniform and easy way) exponential lower bounds. 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. The The work was supported by a DFG grant Me 1077/101. Preliminary...
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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Cited by 5 (2 self)
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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,...
Oneway permutations, computational asymmetry and distortion
 Online prepublication DOI: http://dx.doi.org/10.1016/j.jalgebra.2008.05.035 ) (Preprint: ArXiv http://arxiv.org/abs/0704.1569
, 2008
"... Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of oneway transformations. We introduce a computational asymmetry function that measures the amount of onewayness of permutations. We also introduce the wor ..."
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Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of oneway transformations. We introduce a computational asymmetry function that measures the amount of onewayness of permutations. We also introduce the wordlength asymmetry function for groups, which is an algebraic analogue of computational asymmetry. We relate boolean circuits to words in a Thompson monoid, over a fixed generating set, in such a way that circuit size is equal to wordlength. Moreover, boolean circuits have a representation in terms of elements of a Thompson group, in such a way that circuit size is polynomially equivalent to wordlength. We show that circuits built with gates that are not constrained to have fixedlength inputs and outputs, are at most quadratically more compact than circuits built from traditional gates (with fixedlength inputs and outputs). Finally, we show that the computational asymmetry function is closely related to certain distortion functions: The computational asymmetry function is polynomially equivalent to the distortion of the path length in Schreier graphs of certain Thompson groups, compared to the path length in Cayley graphs of certain Thompson monoids. We also show that the results of Razborov and others on monotone circuit complexity lead to exponential lower bounds on certain distortions. 1