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37
An Exponential Lower Bound for the Size of Monotone Real Circuits
 J. COMP. SYSTEM SCIENCES
, 1995
"... We prove a lower bound, exponential in the eighth root of the input length, on the size of monotone arithmetic circuits that solve an NP problem related to clique detection. The result is more general than the famous lower bound of Razborov and Andreev, because the gates of the circuit are allow ..."
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We prove a lower bound, exponential in the eighth root of the input length, on the size of monotone arithmetic circuits that solve an NP problem related to clique detection. The result is more general than the famous lower bound of Razborov and Andreev, because the gates of the circuit are allowed to compute arbitrary monotone binary realvalued functions (including AND and OR). Our proof is relatively simple and direct, and uses the method of counting bottlenecks. The generalization was proved independently by Pudl'ak using a different method, who also showed that the result can be used to obtain an exponential lower bound on the size of unrestricted cutting plane proofs in the propositional calculus.
Two Lower Bounds for Branching Programs
, 1986
"... . The first result concerns branching programs having width (log n) O(1) . We give an \Omega\Gamma n log n= log log n) lower bound for the size of such branching programs computing almost any symmetric Boolean function and in particular the following explicit function: "the sum of the input vari ..."
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Cited by 19 (1 self)
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. The first result concerns branching programs having width (log n) O(1) . We give an \Omega\Gamma n log n= log log n) lower bound for the size of such branching programs computing almost any symmetric Boolean function and in particular the following explicit function: "the sum of the input variables is a quadratic residue mod p" where p is any given prime between n 1=4 and n 1=3 . This is a strengthening of previous nonlinear lower bounds obtained by Chandra, Furst, Lipton and by Pudl'ak. We mention that by iterating our method the result can be further strengthened to \Omega\Gamma n log n). The second result is a C n lower bound for readonceonly branching programs computing an explicit Boolean function. For n = \Gamma v 2 \Delta , the function computes the parity of the number of triangles in a graph on v vertices. This improves previous exp(c p n) lower bounds for other graph functions by Wegener and Z'ak. The result implies a linear lower bound for the space comp...
Discretely Ordered Modules as a FirstOrder Extension of the Cutting Planes Proof System
"... We define a firstorder extension LK(CP) of the cutting planes proof system CP as the firstorder sequent calculus LK whose atomic formulas are CPinequalities P i a i \Delta x i b (x i 's variables, a i 's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary ..."
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Cited by 14 (0 self)
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We define a firstorder extension LK(CP) of the cutting planes proof system CP as the firstorder sequent calculus LK whose atomic formulas are CPinequalities P i a i \Delta x i b (x i 's variables, a i 's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary a conditional lower bound for LK(CP)proofs. For a subsystem R(CP) of LK(CP), essentially resolution working with clauses formed by CPinequalities, we prove a monotone interpolation theorem obtaining thus an unconditional lower bound (depending on the maximum size of coefficients in proofs and on the maximum number of CPinequalities in clauses). We also give an interpolation theorem for polynomial calculus working with sparse polynomials. The proof relies on a universal interpolation theorem for semantic derivations [16, Thm. 5.1]. LK(CP) can be viewed as a twosorted firstorder theory of Z considered itself as a discretely ordered Zmodule. One sort of variables are module ele...
Symmetric Approximation Arguments for Monotone Lower Bounds without Sunflowers
 Comput. Complexity
, 1997
"... We propose a symmetric version of Razborov's method of approximation to prove lower bounds for monotone circuit complexity. Traditionally, only DNF formulas have been used as approximators, whereas we use both CNF and DNF formulas. As a consequence we no longer need the Sunflower lemma that has been ..."
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We propose a symmetric version of Razborov's method of approximation to prove lower bounds for monotone circuit complexity. Traditionally, only DNF formulas have been used as approximators, whereas we use both CNF and DNF formulas. As a consequence we no longer need the Sunflower lemma that has been essential for the method of approximation. The new approximation...
Characterizing nondeterministic circuit size
 In Proceedings of the 25th STOC
, 1993
"... Consider the following simple communication problem. Fix a universe U and a family Ω of subsets of U. Players I and II receive, respectively, an element a ∈ U and a subset A ∈ Ω. Their task is to find a subset B of U such that A∩B  is even and a ∈ B. With every Boolean function f we associate a co ..."
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Cited by 10 (4 self)
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Consider the following simple communication problem. Fix a universe U and a family Ω of subsets of U. Players I and II receive, respectively, an element a ∈ U and a subset A ∈ Ω. Their task is to find a subset B of U such that A∩B  is even and a ∈ B. With every Boolean function f we associate a collection Ωf of subsets of U = f −1 (0), and prove that its (one round) communication complexity completely determines the size of the smallest nondeterministic circuit for f. We propose a linear algebraic variant to the general approximation method of Razborov, which has exponentially smaller description. We use it to derive four different combinatorial problems (like the one above) that characterize NP. These are tight, in the sense that they can be used to prove superlinear circuit size lower bounds. Combined with Razborov’s method, they present a purely combinatorial framework in which to study the P vs. NP vs. co − NP question.
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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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 ...
On the Bottleneck Counting Argument
 In Twelfth Annual IEEE Conference on Computational Complexity
, 1997
"... Both the bottleneck counting argument [7, 8] and Razborov's approximation method [1, 4, 12] have been used to prove exponential lower bounds for monotone circuits. We show that under the monotone circuit model for every proof by the approximation method, there is a bottleneck counting proof and v ..."
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Cited by 8 (0 self)
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Both the bottleneck counting argument [7, 8] and Razborov's approximation method [1, 4, 12] have been used to prove exponential lower bounds for monotone circuits. We show that under the monotone circuit model for every proof by the approximation method, there is a bottleneck counting proof and vice versa. We also illustrate the elegance of the bottleneck counting technique with a simple selfexplained example: the proof of a (previously known) lower bound for the 3CLIQUEn problem by the bottleneck counting argument. Keywords: Computational complexity; Circuit complexity; Monotone circuit complexity 1 Introduction Razborov's proof of an exponential lower bound on the size of monotone Boolean circuits to detect cliques in a graph [12] [1], represented a breakthrough in the theory of monotone circuit complexity. The proof introduced the method of approximation. The method is roughly as follows. Consider two sets of test inputs, a positive (the output is 1) and a negative one. Gi...
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...
Higher Lower Bounds On Monotone Size
 Proc. of 32nd STOC (2000
, 2000
"... We prove a lower bound of 2\Omega i ( n log n ) 1 3 j on the monotone size of an explicit function in monotoneNP (where n is the number of input variables). This is higher than any previous lower bound on the monotone size of a function. The previous best being a lower bound of about 2 ..."
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Cited by 5 (1 self)
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We prove a lower bound of 2\Omega i ( n log n ) 1 3 j on the monotone size of an explicit function in monotoneNP (where n is the number of input variables). This is higher than any previous lower bound on the monotone size of a function. The previous best being a lower bound of about 2 \Omega\Gamma n 1 4 ) for Andreev's function, proved in [AlBo87]. Our lower bound is proved by the symmetric version of Razborov's method of approximations. However, we present this method in a new and simpler way: Rather than building approximator functions for all the gates in a circuit, we use a gate elimination argument that is based on a Monotone Switching Lemma. The bound applies for a family of functions, each defined by a construction of a small probability space of cwise independent random variables. 1 Introduction 1.1 Background and Previous Work A monotone function is one that can be computed by a monotone circuit i.e., a circuit with only AND and OR gates. The monoton...