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23
Uniform ConstantDepth Threshold Circuits for Division and Iterated Multiplication
, 2002
"... this paper. 2.1. Circuit Classes We begin by formally defining the three circuit complexity classes that will concern us here. These are given by combinatorial restrictions on the circuits of the family. We will then define the uniformity restrictions we will use. Finally, we will give the equival ..."
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Cited by 38 (8 self)
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this paper. 2.1. Circuit Classes We begin by formally defining the three circuit complexity classes that will concern us here. These are given by combinatorial restrictions on the circuits of the family. We will then define the uniformity restrictions we will use. Finally, we will give the equivalent formulations of uniform circuit complexity classes in terms of descriptive complexity classes
Proving lower bounds via pseudorandom generators
 FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science, 25th International Conference, Hyderabad, India, December 1518, 2005, Proceedings, volume 3821 of Lecture
, 2005
"... Abstract. In this paper, we formalize two stepwise approaches, based on pseudorandom generators, for proving P � = NP and its arithmetic analog: Permanent requires superpolynomial sized arithmetic circuits. 1 ..."
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Cited by 31 (1 self)
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Abstract. In this paper, we formalize two stepwise approaches, based on pseudorandom generators, for proving P � = NP and its arithmetic analog: Permanent requires superpolynomial sized arithmetic circuits. 1
TimeSpace Lower Bounds for Satisfiability
 JACM
, 2005
"... We establish the first polynomial timespace lower bounds for satisfiability on general models of computation. We show that for any constant c less than the golden ratio there exists a positive constant d such that no deterministic randomaccess Turing machine can solve satisfiability in time n c an ..."
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Cited by 25 (7 self)
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We establish the first polynomial timespace lower bounds for satisfiability on general models of computation. We show that for any constant c less than the golden ratio there exists a positive constant d such that no deterministic randomaccess Turing machine can solve satisfiability in time n c and space n d, where d approaches 1 when c does. On conondeterministic instead of deterministic machines, we prove the same for any constant c less than √ 2. Our lower bounds apply to nondeterministic linear time and almost all natural NPcomplete problems known. In fact, they even apply to the class of languages that can be solved on a nondeterministic machine in linear time and space n 1/c. Our proofs follow the paradigm of indirect diagonalization. We also use that paradigm to prove timespace lower bounds for languages higher up in the polynomialtime hierarchy.
TimeSpace Tradeoffs in the Counting Hierarchy
, 2001
"... We extend the lower bound techniques of [14], to the unboundederror probabilistic model. A key step in the argument is a generalization of Nepomnjasci's theorem from the Boolean setting to the arithmetic setting. This generalization is made possible, due to the recent discovery of logspaceuniform ..."
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Cited by 19 (4 self)
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We extend the lower bound techniques of [14], to the unboundederror probabilistic model. A key step in the argument is a generalization of Nepomnjasci's theorem from the Boolean setting to the arithmetic setting. This generalization is made possible, due to the recent discovery of logspaceuniform TC 0 circuits for iterated multiplication [9]. Here is an
TimeSpace Lower Bounds for the PolynomialTime Hierarchy on Randomized Machines
 SIAM Journal on Computing
, 2006
"... We establish the first polynomialstrength timespace lower bounds for problems in the lineartime hierarchy on randomized machines with twosided error. We show that for any integer ℓ> 1 and constant c < ℓ, there exists a positive constant d such that QSAT ℓ cannot be computed by such machines in ti ..."
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Cited by 15 (5 self)
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We establish the first polynomialstrength timespace lower bounds for problems in the lineartime hierarchy on randomized machines with twosided error. We show that for any integer ℓ> 1 and constant c < ℓ, there exists a positive constant d such that QSAT ℓ cannot be computed by such machines in time n c and space n d, where QSAT ℓ denotes the problem of deciding the validity of a quantified Boolean formula with at most ℓ − 1 quantifier alternations. Moreover, d approaches 1/2 from below as c approaches 1 from above for ℓ = 2, and d approaches 1 from below as c approaches 1 from above for ℓ ≥ 3. In fact, we establish the stronger result that for any constants a ≤ 1 and c < 1+(ℓ −1)a, there exists a positive constant d such that lineartime alternating machines using space n a and ℓ − 1 alternations cannot be simulated by randomized machines with twosided error running in time n c and space n d, where d approaches a/2 from below as c approaches 1 from above for ℓ = 2 and d approaches a from below as c approaches 1 from above for ℓ ≥ 3. Corresponding to ℓ = 1, we prove that there exists a positive constant d such that the set of Boolean tautologies cannot be decided by a randomized machine with onesided error in time n 1.759 and space n d. As a corollary, this gives the same lower bound for satisfiability on deterministic machines, improving on the previously best known such result. 1
Amplifying lower bounds by means of selfreducibility
 In IEEE Conference on Computational Complexity
, 2008
"... We observe that many important computational problems in NC 1 share a simple selfreducibility property. We then show that, for any problem A having this selfreducibility property, A has polynomial size TC 0 circuits if and only if it has TC 0 circuits of size n 1+ɛ for every ɛ>0 (counting the numb ..."
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Cited by 13 (4 self)
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We observe that many important computational problems in NC 1 share a simple selfreducibility property. We then show that, for any problem A having this selfreducibility property, A has polynomial size TC 0 circuits if and only if it has TC 0 circuits of size n 1+ɛ for every ɛ>0 (counting the number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean Formula Evaluation problem (BFE), which is complete for NC 1 and has the selfreducibility property. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC 0 circuits of size n 1+ɛd. If one were able to improve this lower bound to show that there is some constant ɛ>0 such that every TC 0 circuit family recognizing BFE has size n 1+ɛ, then it would follow that TC 0 ̸ = NC 1. We show that proving lower bounds of the form n 1+ɛ is not ruled out by the Natural Proof framework of Razborov and Rudich and hence there is currently no known barrier for separating classes such as ACC 0,TC 0 and NC 1 via existing “natural ” approaches to proving circuit lower bounds. We also show that problems with small uniform constantdepth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known timespace tradeoff lower bounds to show that SAT requires uniform depth d TC 0 and AC 0 [6] circuits of size n 1+c for some constant c depending on d. 1
A Survey of Lower Bounds for Satisfiability and Related Problems
 Foundations and Trends in Theoretical Computer Science
, 2007
"... Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a lineartime, logarithmicspace algorithm for satisfiability was not ruled out. In 1997 Fortnow ..."
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Cited by 12 (1 self)
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Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a lineartime, logarithmicspace algorithm for satisfiability was not ruled out. In 1997 Fortnow, building on earlier work by Kannan, ruled out such an algorithm. Since then there has been a significant amount of progress giving nontrivial lower bounds on the computational complexity of satisfiability. In this article we survey the known lower bounds for the time and space complexity of satisfiability and closely related problems on deterministic, randomized, and quantum models with random access. We discuss the stateoftheart results and present the underlying arguments in a unified framework. 1
TimeSpace Tradeoffs for Counting NP Solutions Modulo Integers
 In Proceedings of the 22nd IEEE Conference on Computational Complexity
, 2007
"... We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km satisf ..."
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Cited by 11 (5 self)
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We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km satisfying assignments, for some integer k. We show for all primes p except for possibly one of them, and for all c < 2cos(π/7) ≈ 1.801, there is a d> 0 such that MODpSat is not solvable in n c time and n d space by general algorithms. That is, there is at most one prime p that does not satisfy the tradeoff. We prove that the same limitation holds for Sat and MOD6Sat, as well as MODmSat for any composite m that is not a prime power. Our main tool is a general method for rapidly simulating deterministic computations with restricted space, by counting the number of solutions to NP predicates modulo integers. The simulation converts an ordinary algorithm into a “canonical ” one that consumes roughly the same amount of time and space, yet canonical algorithms have nice properties suitable for counting.
Improved Bounds on the Weak Pigeonhole Principle and Infinitely Many Primes from Weaker Axioms
, 2001
"... We show that the known boundeddepth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a sizedepth tradeoff upper bound: there are proofs of size n O(d(log(n)) 2=d ) and depth O(d). This solves an open problem ..."
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Cited by 8 (2 self)
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We show that the known boundeddepth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a sizedepth tradeoff upper bound: there are proofs of size n O(d(log(n)) 2=d ) and depth O(d). This solves an open problem of Maciel, Pitassi and Woods (2000). Our technique requires formalizing the ideas underlying Nepomnjascij's Theorem which might be of independent interest. Moreover, our result implies a proof of the unboundedness of primes in I \Delta 0 with a provably weaker `large number assumption' than previously needed.
A quantum timespace lower bound for the counting hierarchy
, 2007
"... We obtain the first nontrivial timespace lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real d and ..."
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Cited by 5 (0 self)
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We obtain the first nontrivial timespace lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real d and every positive real ǫ there exists a real c> 1 such that either: • MajMajSAT does not have a quantum algorithm with bounded twosided error that runs in time n c, or • MajSAT does not have a quantum algorithm with bounded twosided error that runs in time n d and space n 1−ǫ. In particular, MajMajSAT cannot be solved by a quantum algorithm with bounded twosided error running in time n 1+o(1) and space n 1−ǫ for any ǫ> 0. The key technical novelty is a time and spaceefficient simulation of quantum computations with intermediate measurements by probabilistic machines with unbounded error. We also develop a model that is particularly suitable for the study of general quantum computations with simultaneous time and space bounds. However, our arguments hold for any reasonable uniform model of quantum computation. 1