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Lower bounds in a parallel model without bit operations
 TO APPEAR IN THE SIAM JOURNAL ON COMPUTING
, 1997
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Potential of the Approximation Method
 Proc. of the 37th IEEE Symp. on the Foundations of Computer Science
, 1996
"... Developing some techniques for the approximation method, we establish precise versions of the following statements concerning lower bounds for circuits that detect cliques of size s in a graph with m vertices: For 5 s m=4, a monotone circuit computing CLIQUE(m; s) contains at least (1=2)1:8 min( ..."
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Developing some techniques for the approximation method, we establish precise versions of the following statements concerning lower bounds for circuits that detect cliques of size s in a graph with m vertices: For 5 s m=4, a monotone circuit computing CLIQUE(m; s) contains at least (1=2)1:8 min( p s01=2;m=(4s)) gates: If a nonmonotone circuit computes CLIQUE using a "small" amount of negation, then the circuit contains an exponential number of gates. The former is proved very simply using so called bottleneck counting argument within the framework of approximation, whereas the latter is verified introducing a notion of restricting negation and generalizing the sunflower contraction. 1. Introduction Since Razborov, based on the approximation method, succeeded to obtain a superpolynomial lower bound on the size of monotone circuits computing the clique function, much effort has been devoted to explore the method and derive good lower bounds[K, NM, R1, R2, RR]. Employing the appr...
On Ultrafilters and NP
"... We further develop the Fusion Method by exploring its similarities with the Ultraproduct Construction in Model Theory. We use this analogy to reprove a result of Sipser regarding countable circuits, in a simpler way. In the finite case this analogy allows us to give a new characterization of coNP ..."
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We further develop the Fusion Method by exploring its similarities with the Ultraproduct Construction in Model Theory. We use this analogy to reprove a result of Sipser regarding countable circuits, in a simpler way. In the finite case this analogy allows us to give a new characterization of coNP in terms of the CLIQUE function. This gives a natural interpretation to the NPcompleteness of the CLIQUE function. Introduction In this paper we further develop the analogy between Razborov's Generalized Approximation Method and the Ultraproduct Construction in Model Theory [3]. This analogy, in all its variations, has been named the Fusion Method [10]. The Fusion Method has many different incarnations (see [10] for a survey). Razborov originally used it to characterize the classes P and NL [6, 7]. He also used it to give a superlinear lower bound for the complexity of the Majority function on SwitchingandRectifying Networks [7]. Karchmer [3] shows that the monotone size lower bounds for...
On superlinear lower bounds in complexity theory
 In Proc. 10th Annual IEEE Conference on Structure in Complexity Theory
, 1995
"... This paper first surveys the neartotal lack of superlinear lower bounds in complexity theory, for “natural” computational problems with respect to many models of computation. We note that the dividing line between models where such bounds are known and those where none are known comes when the mode ..."
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This paper first surveys the neartotal lack of superlinear lower bounds in complexity theory, for “natural” computational problems with respect to many models of computation. We note that the dividing line between models where such bounds are known and those where none are known comes when the model allows nonlocal communication with memory at unit cost. We study a model that imposes a “fair cost ” for nonlocal communication, and obtain modest superlinear lower bounds for some problems via a Kolmogorovcomplexity argument. Then we look to the larger picture of what it will take to prove really striking lower bounds, and pull from ours and others’ work a concept of information vicinity that may offer new tools and modes of analysis to a young field that rather lacks them.
Finite Limits and Lower Bounds for Circuits Size
, 1994
"... The lower bounds problem in circuit complexity theory may be looked as the problem about the possibility to diagonalize over finite sets of computations. Our goal here is to show that Sipser's notion of "finite limit" is the right diagonal in different situations. ..."
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The lower bounds problem in circuit complexity theory may be looked as the problem about the possibility to diagonalize over finite sets of computations. Our goal here is to show that Sipser's notion of "finite limit" is the right diagonal in different situations.
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"... The lectures are devoted to boolean circuit lower bounds. We consider circuits with gates ∧; ∨; ¬. Suppose L ∈ {0;1} ∗ is a language. Let Ln = L ∩ {0;1} n. We say that L is computed by a family of circuits C1;C2;::: if on an input x = (x1;:::;xn), Cn(x) is 1 when x ∈ Ln and is 0 when x = ∈ Ln. For ..."
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The lectures are devoted to boolean circuit lower bounds. We consider circuits with gates ∧; ∨; ¬. Suppose L ∈ {0;1} ∗ is a language. Let Ln = L ∩ {0;1} n. We say that L is computed by a family of circuits C1;C2;::: if on an input x = (x1;:::;xn), Cn(x) is 1 when x ∈ Ln and is 0 when x = ∈ Ln. For a circuit C, we de ne size(C) to be the number of edges in the graph representing C, and depth(C) to be the length of the longest path from an input to the output. We say that L ∈ P/poly if there exists a family C1;C2;::: computing L such that size(Cn) = n O(1). It is easy to see that if a Turing machine computes L in time T(n), then there exists a family of circuits C1;C2;::: computing L so that size(Cn) ≤ (T(n)) 2. If a Turing machine is given a circuit C and an input x, then it can compute C(x) in time size(C). However, there exist languages that are computable by families of circuits but are not computable by Turing machines. The simplest example is L = Halting problem, Cn(x) = 1 i L(n) = 1 for x  = n. But in some sense, this is not an interesting example. Theorem 1 (Karp and Lipton). If NP ⊆ P/poly, then PH collapses: PH = 2 = 2. Exercise 1. Prove the Karp{Lipton Theorem. What we would really like to prove is that NP ⊆ P/poly implies P = NP but this is currently
Local Rainbow Colorings
"... Given a graph H, we denote by C(n, H) the minimum number k such that the following holds. There are n colorings of E(Kn) with kcolors, each associated with one of the vertices of Kn, such that for every copy T of H in Kn, at least one of the colorings that are associated with V (T) assigns distinct ..."
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Given a graph H, we denote by C(n, H) the minimum number k such that the following holds. There are n colorings of E(Kn) with kcolors, each associated with one of the vertices of Kn, such that for every copy T of H in Kn, at least one of the colorings that are associated with V (T) assigns distinct colors to all the edges of E(T). We characterize the set of all graphs H for which C(n, H) is bounded by some absolute constant c(H), prove a general upper bound and obtain lower and upper bounds for several graphs of special interest. A special case of our results partially answers an extremal question of Karchmer and Wigderson motivated by the investigation of the computational power of span programs.
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"... The lectures are devoted to boolean circuit lower bounds. We consider circuits with gates ∧, ∨, ¬. Suppose L ∈ {0, 1} ∗ is a language. Let Ln = L ∩ {0, 1} n. We say that L is computed by a family of circuits C1, C2,... if on an input x = (x1,..., xn), Cn(x) is 1 when x ∈ Ln and is 0 when x / ∈ Ln. ..."
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The lectures are devoted to boolean circuit lower bounds. We consider circuits with gates ∧, ∨, ¬. Suppose L ∈ {0, 1} ∗ is a language. Let Ln = L ∩ {0, 1} n. We say that L is computed by a family of circuits C1, C2,... if on an input x = (x1,..., xn), Cn(x) is 1 when x ∈ Ln and is 0 when x / ∈ Ln. For a circuit C, we define size(C) to be the number of edges in the graph representing C, and depth(C) to be the length of the longest path from an input to the output. We say that L ∈ P /poly if there exists a family C1, C2,... computing L such that size(Cn) = n O(1). It is easy to see that if a Turing machine computes L in time T (n), then there exists a family of circuits C1, C2,... computing L so that size(Cn) ≤ (T (n)) 2. If a Turing machine is given a circuit C and an input x, then it can compute C(x) in time size(C). However, there exist languages that are computable by families of circuits but are not computable by Turing machines. The simplest example is L = Halting problem, Cn(x) = 1 iff L(n) = 1 for x  = n. But in some sense, this is not an interesting example.