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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...
Arithmetic circuits: the chasm at depth four gets wider
"... In their paper on the “chasm at depth four”, Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2 o(m) also admit arithmetic circuits of depth four and size 2 o(m). This theorem shows that for problems such as arithmetic circuit lower ..."
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In their paper on the “chasm at depth four”, Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2 o(m) also admit arithmetic circuits of depth four and size 2 o(m). This theorem shows that for problems such as arithmetic circuit lower bounds or blackbox derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of n×n matrices has circuits of size polynomial inn, then it also has depth 4 circuits of sizen O( √ nlogn) If the original circuit uses only integer constants of polynomial size, then the same is true of the resulting depth four circuit. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials. We also use our techniques to reprove two results on: The existence of nontrivial boolean circuits of constant depth for languages in LOGCFL. Reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree.
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 logspaceuni ..."
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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
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 sat ..."
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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.
Minimizing DNF Formulas and AC^0 Circuits Given a Truth Table
 IN PROCEEDINGS OF THE 21ST ANNUAL IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2006
"... For circuit classes R, the fundamental computational problem MinR asks for the minimum Rsize of a Boolean function presented as a truth table. Prominent examples of this problem include MinDNF, which asks whether a given Boolean function presented as a truth table has a kterm DNF, and MinCircu ..."
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For circuit classes R, the fundamental computational problem MinR asks for the minimum Rsize of a Boolean function presented as a truth table. Prominent examples of this problem include MinDNF, which asks whether a given Boolean function presented as a truth table has a kterm DNF, and MinCircuit (also called MCSP), which asks whether a Boolean function presented as a truth table has a size k Boolean circuit. We present a new reduction proving that MinDNF is NPcomplete. It is significantly simpler than the known reduction of Masek [30], which is from CircuitSAT. We then give a more complex reduction, yielding the result that MinDNF cannot be approximated to within a factor smaller than (logN) γ, for some constant γ> 0, assuming that NP is not contained in quasipolynomial time. The standard greedy algorithm for Set Cover is often used in practice to approximate MinDNF. The question of whether MinDNF can be approximated to within a factor of o(logN) remains open, but we construct an instance of MinDNF on which the solution produced by the greedy algorithm is Ω(logN) larger than optimal. Finally, we turn to the question of approximating circuit size for slightly more general classes of circuits. DNF formulas are depth two circuits of AND and OR gates. Depth d circuits are denoted by AC0 d. We show that it is hard to approximate the size of AC0 d circuits (for large enough d) under cryptographic assumptions.
FixedPolynomial Size Circuit Bounds
"... Abstract—In 1982, Kannan showed that Σ P 2 does not have n ksized circuits for any k. Do smaller classes also admit such circuit lower bounds? Despite several improvements of Kannan’s result, we still cannot prove that P NP does not have linear size circuits. Work of Aaronson and Wigderson provides ..."
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Abstract—In 1982, Kannan showed that Σ P 2 does not have n ksized circuits for any k. Do smaller classes also admit such circuit lower bounds? Despite several improvements of Kannan’s result, we still cannot prove that P NP does not have linear size circuits. Work of Aaronson and Wigderson provides strong evidence – the “algebrization ” barrier – that current techniques have inherent limitations in this respect. We explore questions about fixedpolynomial size circuit lower bounds around and beyond the algebrization barrier. We find several connections, including The following are equivalent: – NP is in SIZE(n k) (has O(n k)size circuit families) for some k – For each c, P NP[nc] k is in SIZE(n) for some k – ONP/1 is in SIZE(n k) for some k, where ONP is the class of languages accepted obliviously by NP machines, with witnesses for “yes ” instances depending only on the input length. For a large number of natural classes C and all k � 1, C is in SIZE(n k) if and only if C/1 ∩P/poly is in SIZE(n k). If there is a d such that MATIME(n) ⊆ NTIME(n d), then P NP does not have O(n k) size circuits for any k> 0. One cannot show n 2size circuit lower bounds for ⊕P without new nonrelativizing techniques. In particular, the proof that PP ̸ ⊆ SIZE(n k) for all k relies on the (relativizing) result that P PP ⊆ MA = ⇒ PP ̸ ⊆ SIZE(n k), and we give an oracle relative to which P ⊕P ⊆ MA and ⊕P ⊆ SIZE(n 2) both hold. I.
AlternationTrading Proofs, Linear Programming, and Lower Bounds
"... 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. The proofs of these lower bounds follow a certain proofbycontradiction strategy, which we call “resourcetrading” or “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. Formalizing the framework, 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 implement a smallscale theorem prover and report surprising results, which allow us to extract new humanreadable time lower bounds for several problems. We also use the framework to prove concrete limitations on the current techniques.
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.
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 hereditarily ..."
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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...