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TimeSpace Tradeoffs for Satisfiability
 Journal of Computer and System Sciences
, 1997
"... We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved ..."
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We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for logspace uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomialtime hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...
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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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.
The Descriptive Complexity of the FixedPoints of Bounded Formulas
 Computer Science Logic '2000, 14th Annual Conference of the EACSL, volume 1862 of Lecture Notes in Computer Science
, 2000
"... . We investigate the complexity of the fixedpoints of bounded formulas in the context of finite set theory; that is, in the context of arbitrary classes of finite structures that are equipped with a builtin BIT predicate, or equivalently, with a builtin membership relation between hereditaril ..."
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. We investigate the complexity of the fixedpoints of bounded formulas in the context of finite set theory; that is, in the context of arbitrary classes of finite structures that are equipped with a builtin BIT predicate, or equivalently, with a builtin membership relation between hereditarily finite sets (input relations are allowed). We show that the iteration of a positive bounded formula converges in polylogarithmically many steps in the cardinality of the structure. This extends a previously known much weaker result. We obtain a number of connections with the rudimentary languages and deterministic polynomialtime. Moreover, our results provide a natural characterization of the complexity class consisting of all languages computable by boundeddepth, polynomialsize circuits, and polylogarithmictime uniformity. As a byproduct, we see that this class coincides with LH(P), the logarithmictime hierarchy with an oracle to deterministic polynomialtime. Finally, we dis...
Automated proofs of time lower bounds
, 2007
"... A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6SAT, MajorityofMajoritySAT, and Tautologies, to name a few. The ..."
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A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6SAT, MajorityofMajoritySAT, and Tautologies, to name a few. These lower bound proofs all follow a certain diagonalizationbased proofbycontradiction strategy. A pressing open problem has been to determine how powerful such proofs can possibly be. We propose an automated theoremproving methodology for studying these lower bound problems. In particular, we prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. We describe an implementation of a smallscale theorem prover and discover surprising experimental results. In some settings, our program provides strong evidence that the best known lower bound proofs are already optimal for the current framework, contradicting the consensus intuition; in others, the program guides us to improved lower bounds where none had been known for years.
Learning, Cryptography, and the Average Case
"... This thesis explores problems in computational learning theory from an averagecase perspective. Through this perspective we obtain a variety of new results for learning theory and cryptography. Several major open questions in computational learning theory revolve around the problem of efficiently l ..."
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This thesis explores problems in computational learning theory from an averagecase perspective. Through this perspective we obtain a variety of new results for learning theory and cryptography. Several major open questions in computational learning theory revolve around the problem of efficiently learning polynomialsize DNF formulas, which dates back to Valiant’s introduction of the PAC learning model [Valiant, 1984]. We apply an averagecase analysis to make progress on this problem in two ways. • We prove that Mansour’s conjecture is true for random DNF. In 1994, Y. Mansour conjectured that for every DNF formula on n variables with t terms there exists a polynomial p with tO(log(1/ɛ)) nonzero coefficients such that Ex∈{0,1} n[(p(x)−f(x)) 2] ≤ ɛ. We make the first progress on this conjecture and show that it is true for several natural subclasses of DNF formulas including randomly chosen DNF formulas and readk DNF formulas. Our result yields the first polynomialtime query algorithm for agnostically learning these subclasses of DNF formulas with respect to the uniform distribution on {0, 1} n (for any constant error parameter and constant k). Applying
www.stacsconf.org ALTERNATIONTRADING PROOFS, LINEAR PROGRAMMING, AND
, 2010
"... Abstract. A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6SAT, MajorityofMajoritySAT, and Tautologies, to name ..."
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Abstract. A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6SAT, MajorityofMajoritySAT, and Tautologies, to name a few. The proofs of these lower bounds follow a certain proofbycontradiction strategy that we call alternationtrading. An important open problem is to determine how powerful such proofs can possibly be. We propose a methodology for studying these proofs that makes them amenable to both formal analysis and automated theorem proving. We prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. Implementing a smallscale theorem prover based on this result, we extract new humanreadable time lower bounds for several problems. This framework can also be used to prove concrete limitations on the current techniques. 1.
unknown title
"... interest in STACS has remained at a high level over the past years. The STACS 2010 call for papers led to over 238 submissions from 40 countries. Each paper was assigned to three program committee members. The committee selected 54 papers during a two week electronic meeting held in November. As co ..."
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interest in STACS has remained at a high level over the past years. The STACS 2010 call for papers led to over 238 submissions from 40 countries. Each paper was assigned to three program committee members. The committee selected 54 papers during a two week electronic meeting held in November. As cochairs of the program committee, we would like to sincerely thank its members and the many external referees for their valuable work. In particular, there were intense and interesting discussions. The overall very high quality of the submissions made the selection a difficult task. We would like to express our thanks to the three invited speakers, Mikołaj Bojańczyk, Rolf Niedermeier, and Jacques Stern. Special thanks go to Andrei Voronkov for his EasyChair software (www.easychair.org). Moreover, we would like to warmly thank Wadie Guizani for preparing the conference proceedings and continuous help throughout the conference organization. For the third time, this year’s STACS proceedings are published in electronic form. A printed version was also available at the conference, with ISBN. The electronic proceedings are available through several portals, and in particular through HAL and LIPIcs series. The proceedings of the Symposium, which are published electronically in