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43
Expander Graphs and their Applications
, 2003
"... Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . ..."
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Cited by 189 (5 self)
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Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 Derandomizing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Magical Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 A Super Concentrator with O(n) edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 Derandomizing Random Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Resolution is Not Automatizable Unless W[P] is Tractable
 In 42nd Annual IEEE Symposium on Foundations of Computer Science
, 2001
"... We show that neither Resolution nor treelike Resolution is automatizable unless the class W[P] from the hierarchy of parameterized problems is xedparameter tractable by randomized algorithms with onesided error. ..."
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Cited by 50 (0 self)
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We show that neither Resolution nor treelike Resolution is automatizable unless the class W[P] from the hierarchy of parameterized problems is xedparameter tractable by randomized algorithms with onesided error.
A Combinatorial Characterization of Resolution Width
 In 18th IEEE Conference on Computational Complexity
, 2002
"... We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result i ..."
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Cited by 34 (4 self)
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We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum space of refuting a 3CNF formula is always bounded from below by the minimum width of refuting it (minus 3). This solves a wellknown open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the Dense Linear Order Principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the sizewidth relationship is tight.
Tautologies From PseudoRandom Generators
, 2001
"... We consider tautologies formed from a pseudorandom number generator, dened in Krajcek [12] and in Alekhnovich et.al. [2]. We explain a strategy of proving their hardness for EF via a conjecture about bounded arithmetic formulated in Krajcek [12]. Further we give a purely nitary statement, in a ..."
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Cited by 17 (0 self)
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We consider tautologies formed from a pseudorandom number generator, dened in Krajcek [12] and in Alekhnovich et.al. [2]. We explain a strategy of proving their hardness for EF via a conjecture about bounded arithmetic formulated in Krajcek [12]. Further we give a purely nitary statement, in a form of a hardness condition posed on a function, equivalent to the conjecture. This is accompanied by a brief explanation, aimed at nonlogicians, of the relation between propositional proof complexity and bounded arithmetic. It is a fundamental problem of mathematical logic to decide if tautologies can be inferred in propositional calculus in substantially fewer steps than it takes to check all possible truth assignments. This is closely related to the famous P/NP problem of Cook [3]. By propositional calculus I mean any textbook system based on a nite number of inference rules and axiom schemes that is sound and complete. The qualication substantially less means that the nu...
Elusive functions and lower bounds for arithmetic circuits
 Electronic Colloquium in Computational Complexity
, 2007
"... A basic fact in linear algebra is that the image of the curve f(x) = (x1, x2, x3,..., xm), say over C, is not contained in any m − 1 dimensional affine subspace of Cm. In other words, the image of f is not contained in the image of any polynomialmapping1 Γ: Cm−1 → Cm of degree 1 (that is, an affin ..."
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Cited by 13 (2 self)
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A basic fact in linear algebra is that the image of the curve f(x) = (x1, x2, x3,..., xm), say over C, is not contained in any m − 1 dimensional affine subspace of Cm. In other words, the image of f is not contained in the image of any polynomialmapping1 Γ: Cm−1 → Cm of degree 1 (that is, an affine mapping). Can one give an explicit example for a polynomial curve f: C → Cm, such that, the image of f is not contained in the image of any polynomialmapping Γ: Cm−1 → Cm of degree 2? In this paper, we show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. For example, any explicit f as above (with the right notion of explicitness2), of degree up to 2mo(1) , implies superpolynomial lower bounds for computing the permanent over C. More generally, we say that a polynomialmapping f: Fn → Fm is (s, r)elusive, if for every polynomialmapping Γ: Fs → Fm of degree r, Image(f) � ⊂ Image(Γ).
Goldreich’s oneway function candidate and myopic backtracking algorithms
 In Proceedings of the 6th Theory of Cryptography Conference (TCC
, 2009
"... Abstract. Goldreich (ECCC 2000) proposed a candidate oneway function construction which is parameterized by the choice of a small predicate (over d = O(1) variables) and of a bipartite expanding graph of rightdegree d. The function is computed by labeling the n vertices on the left with the bits o ..."
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Cited by 13 (0 self)
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Abstract. Goldreich (ECCC 2000) proposed a candidate oneway function construction which is parameterized by the choice of a small predicate (over d = O(1) variables) and of a bipartite expanding graph of rightdegree d. The function is computed by labeling the n vertices on the left with the bits of the input, labeling each of the n vertices on the right with the value of the predicate applied to the neighbors, and outputting the nbit string of labels of the vertices on the right. Inverting Goldreich’s oneway function is equivalent to finding solutions to a certain constraint satisfaction problem (which easily reduces to SAT) having a “planted solution, ” and so the use of SAT solvers constitutes a natural class of attacks. We perform an experimental analysis using MiniSat, which is one of the best publicly available algorithms for SAT. Our experiment shows that the running time required to invert the function grows exponentially with the length of the input, and that such an attack becomes infeasible
PEBBLE GAMES, PROOF COMPLEXITY AND TIMESPACE TRADEOFFS
, 2010
"... Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolutionbased proof systems when compari ..."
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Cited by 8 (5 self)
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Pebble games were extensively studied in the 1970s and 1980s in a number of different contexts. The last decade has seen a revival of interest in pebble games coming from the field of proof complexity. Pebbling has proven to be a useful tool for studying resolutionbased proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing sizespace tradeoffs. This is a survey of research in proof complexity drawing on results and tools from pebbling, with a focus on proof space lower bounds and tradeoffs between proof size and proof space.
Constantdepth frege systems with counting axioms polynomially simulate nullstellensatz refutations. August 05 2003
 Comment
, 1998
"... Abstract. We show that constantdepth Frege systems with counting axioms modulo m polynomially simulate Nullstellensatz refutations modulo m. Central to this is a new definition of reducibility from propositional formulas to systems of polynomials. Using our definition of reducibility, most previous ..."
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Cited by 4 (0 self)
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Abstract. We show that constantdepth Frege systems with counting axioms modulo m polynomially simulate Nullstellensatz refutations modulo m. Central to this is a new definition of reducibility from propositional formulas to systems of polynomials. Using our definition of reducibility, most previously studied propositional formulas reduce to their polynomial translations. When combined with a previous result of the authors, this establishes the first size separation between Nullstellensatz and polynomial calculus refutations. We also obtain new upper bounds on refutation sizes for certain CNFs in constantdepth Frege with counting axioms systems. 1
A dichotomy for local smallbias generators
 Electronic Colloquium on Computational Complexity, 2011. [AIK06] [AIK08] Benny Applebaum, Yuval Ishai, and Eyal Kushilevitz. Cryptography in NC 0
"... We consider pseudorandom generators in which each output bit depends on a constant number of input bits. Such generators have appealingly simple structure: they can be described by a sparse inputoutput dependency graph G and a small predicate P that is applied at each output. Following the works of ..."
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Cited by 3 (0 self)
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We consider pseudorandom generators in which each output bit depends on a constant number of input bits. Such generators have appealingly simple structure: they can be described by a sparse inputoutput dependency graph G and a small predicate P that is applied at each output. Following the works of Cryan and Miltersen (MFCS’01) and by Mossel et al (STOC’03), we ask: which graphs and predicates yield “smallbias ” generators (that fool linear distinguishers)? We identify an explicit class of degenerate predicates and prove the following. For most graphs, all nondegenerate predicates yield smallbias generators, f: {0, 1} n → {0, 1} m, with output length m = n 1+ɛ for some constant ɛ> 0. Conversely, we show that for most graphs, degenerate predicates are not secure against linear distinguishers, even when the output length is linear m = n + Ω(n). Taken together, these results expose a dichotomy: every predicate is either very hard or very easy, in the sense that it either yields a smallbias generator for almost all graphs or fails to do so for almost all graphs. As a secondary contribution, we attempt to support the view that smallbias is a good measure of pseudorandomness for local functions with large stretch. We do so by demonstrating that resilience to linear distinguishers implies resilience to a larger class of attacks.
Optimality of sizedegree tradeoffs for polynomial calculus
 ACM Transactions on Computational Logic
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