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49
The Complexity Of Propositional Proofs
 Bulletin of Symbolic Logic
, 1995
"... This paper of Tseitin is a landmark as the first to give nontrivial lower bounds for propositional proofs; although it predates the first papers on ..."
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Cited by 107 (3 self)
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This paper of Tseitin is a landmark as the first to give nontrivial lower bounds for propositional proofs; although it predates the first papers on
Lower bounds on Hilbert's Nullstellensatz and propositional proofs
 PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
, 1996
"... The socalled weak form of Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Qj(x) = 0, does not have a solution in the algebraic closure if and only if 1 is in the ideal generated by the polynomials (?,(*) • We shall prove a lower bound on the degrees of polyno ..."
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Cited by 60 (19 self)
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The socalled weak form of Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Qj(x) = 0, does not have a solution in the algebraic closure if and only if 1 is in the ideal generated by the polynomials (?,(*) • We shall prove a lower bound on the degrees of polynomials P,(x) such that £, P,(x)Qt(x) = 1. This result has the following application. The modular counting principle states that no finite set whose cardinality is not divisible by q can be partitioned into ^element classes. For each fixed cardinality N, this principle can be expressed as a propositional formula Count^fo,...) with underlying variables xe, where e ranges over <7element subsets of N. Ajtai [4] proved recently that, whenever p,q are two different primes, the propositional formulas Count $ n+I do not have polynomial size, constantdepth Frege proofs from substitution instances of Count/?, where m^O (modp). We give a new proof of this theorem based on the lower bound for Hilbert's Nullstellensatz. Furthermore our technique enables us to extend the independence results for counting principles to composite numbers p and q. This improved lower bound together with new upper bounds yield an exact characterization of when Count, can be proved efficiently from Countp, for all values of p and q.
A Switching Lemma for Small Restrictions and Lower Bounds for kDNF Resolution (Extended Abstract)
 SIAM J. Comput
, 2002
"... We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with kDNFs instead of cla ..."
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Cited by 44 (7 self)
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We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to DNFs with small conjunctions. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with kDNFs instead of clauses. We also obtain an exponential separation between depth d circuits of k + 1.
Circuit Complexity before the Dawn of the New Millennium
, 1997
"... The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bound ..."
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Cited by 34 (3 self)
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The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.
Approximate Lineage for Probabilistic Databases
"... In probabilistic databases, lineage is fundamental to both query processing and understanding the data. Current systems s.a. Trio or Mystiq use a complete approach in which the lineage for a tuple t is a Boolean formula which represents all derivations of t. In large databases lineage formulas can b ..."
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Cited by 34 (9 self)
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In probabilistic databases, lineage is fundamental to both query processing and understanding the data. Current systems s.a. Trio or Mystiq use a complete approach in which the lineage for a tuple t is a Boolean formula which represents all derivations of t. In large databases lineage formulas can become huge: in one public database (the Gene Ontology) we often observed 10MB of lineage (provenance) data for a single tuple. In this paper we propose to use approximate lineage, which is a much smaller formula keeping track of only the most important derivations, which the system can use to process queries and provide explanations. We discuss in detail two specific kinds of approximate lineage: (1) a conservative approximation called sufficient lineage that records the most important derivations for each tuple, and (2) polynomial lineage, which is more aggressive and can provide higher compression ratios, and which is based on Fourier approximations of Boolean expressions. In this paper we define approximate lineage formally, describe algorithms to compute approximate lineage and prove formally their error bounds, and validate our approach experimentally on a real data set. 1.
On learning monotone DNF under product distributions
 In Proceedings of the Fourteenth Annual Conference on Computational Learning Theory
, 2001
"... We show that the class of monotone 2 O( √ log n)term DNF formulae can be PAC learned in polynomial time under the uniform distribution from random examples only. This is an exponential improvement over the best previous polynomialtime algorithms in this model, which could learn monotone o(log 2 n) ..."
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Cited by 32 (11 self)
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We show that the class of monotone 2 O( √ log n)term DNF formulae can be PAC learned in polynomial time under the uniform distribution from random examples only. This is an exponential improvement over the best previous polynomialtime algorithms in this model, which could learn monotone o(log 2 n)term DNF. We also show that various classes of small constantdepth circuits which compute monotone functions are PAC learnable in polynomial time under the uniform distribution. All of our results extend to learning under any constantbounded product distribution.
The complexity of propositional proofs
 BULLETIN OF SYMBOLIC LOGIC
"... Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. Thi ..."
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Cited by 31 (0 self)
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Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.
Pseudorandom Bits for ConstantDepth Circuits with Few Arbitrary Symmetric Gates
 SIAM Journal on Computing
, 2005
"... We exhibit an explicitly computable ‘pseudorandom ’ generator stretching l bits into m(l) = l Ω(log l) bits that look random to constantdepth circuits of size m(l) with log m(l) arbitrary symmetric gates (e.g. PARITY, MAJORITY). This improves on a generator by Luby, Velickovic and Wigderson (ISTCS ..."
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Cited by 29 (12 self)
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We exhibit an explicitly computable ‘pseudorandom ’ generator stretching l bits into m(l) = l Ω(log l) bits that look random to constantdepth circuits of size m(l) with log m(l) arbitrary symmetric gates (e.g. PARITY, MAJORITY). This improves on a generator by Luby, Velickovic and Wigderson (ISTCS ’93) that achieves the same stretch but only fools circuits of depth 2 with one arbitrary symmetric gate at the top. Our generator fools a strictly richer class of circuits than Nisan’s generator for constant depth circuits (Combinatorica ’91) (but Nisan’s generator has a much bigger stretch). In particular, we conclude that every function computable by uniform poly(n)size probabilistic constant depth circuits with O(log n) arbitrary symmetric gates is in TIME 2no(1)�. This seems to be the richest probabilistic circuit class known to admit a subexponential derandomization. Our generator is obtained by constructing an explicit function f: {0, 1} n → {0, 1} that is very hard on average for constantdepth circuits of size nɛ·log n with ɛ log 2 n arbitrary symmetric gates, and plugging it into the NisanWigderson pseudorandom generator construction (FOCS ’88). The proof of the averagecase hardness of this function is a modification of arguments by Razborov and Wigderson (IPL ’93), and Hansen and Miltersen (MFCS ’04), and combines H˚astad’s switching lemma (STOC ’86) with a multiparty communication complexity lower bound by Babai, Nisan and
Static Dictionaries on AC^0 RAMs: Query time Θ(,/log n / log log n) is necessary and sufficient
, 1996
"... In this paper we consider solutions to the static dictionary problem ���� � on RAMs, i.e. random access machines where the only restriction on the finite instruction set is that all computational instructions are ���� � in. Our main result is a tight upper and lower bound ���� � ���©���������������� ..."
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Cited by 21 (6 self)
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In this paper we consider solutions to the static dictionary problem ���� � on RAMs, i.e. random access machines where the only restriction on the finite instruction set is that all computational instructions are ���� � in. Our main result is a tight upper and lower bound ���� � ���©��������������������� of on the time for answering membership queries in a set of � size when reasonable space is used for the data structure storing the set; the upper bound can be obtained using space ������ � �� � ���� �. Several variations of this result are also obtained. Among others, we show a tradeoff between time and circuit depth under the unitcost assumption: any RAM instruction set which permits a linear space, constant query time solution to the static dictionary problem must have an instruction of depth �������©���������������©���� � , where � is the word size of the machine (and ���© � the size of the universe). This matches the depth of multiplication and integer division, used in the perfect hashing scheme by Fredman, Komlós and Szemerédi.
A Satisfiability Algorithm for AC 0
, 2011
"... We consider the problem of efficiently enumerating the satisfying assignments to AC 0 circuits. We give a zeroerror randomized algorithm which takes an AC 0 circuit as input and constructs a set of restrictions which partitions {0, 1} n so that under each restriction the value of the circuit is cons ..."
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We consider the problem of efficiently enumerating the satisfying assignments to AC 0 circuits. We give a zeroerror randomized algorithm which takes an AC 0 circuit as input and constructs a set of restrictions which partitions {0, 1} n so that under each restriction the value of the circuit is constant. Let d denote the depth of the circuit and cn denote the number of gates. This algorithm runs in time C2 n(1−µc,d) where C  is the size of the circuit for µc,d ≥ 1/O[lg c + d lg d] d−1 with probability at least 1 − 2 −n. As a result, we get improved exponential time algorithms for AC 0 circuit satisfiability and for counting solutions. In addition, we get an improved bound on the correlation of AC 0 circuits with parity. As an important component of our analysis, we extend the H˚astad Switching Lemma to handle multiple kcnfs and kdnfs. 1