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Graph Nonisomorphism Has Subexponential Size Proofs Unless The PolynomialTime Hierarchy Collapses
 SIAM Journal on Computing
, 1998
"... We establish hardness versus randomness tradeoffs for a broad class of randomized procedures. In particular, we create efficient nondeterministic simulations of bounded round ArthurMerlin games using a language in exponential time that cannot be decided by polynomial size oracle circuits with acce ..."
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Cited by 108 (6 self)
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We establish hardness versus randomness tradeoffs for a broad class of randomized procedures. In particular, we create efficient nondeterministic simulations of bounded round ArthurMerlin games using a language in exponential time that cannot be decided by polynomial size oracle circuits with access to satisfiability. We show that every language with a bounded round ArthurMerlin game has subexponential size membership proofs for infinitely many input lengths unless exponential time coincides with the third level of the polynomialtime hierarchy (and hence the polynomialtime hierarchy collapses). This provides the first strong evidence that graph nonisomorphism has subexponential size proofs. We set up a general framework for derandomization which encompasses more than the traditional model of randomized computation. For a randomized procedure to fit within this framework, we only require that for any fixed input the complexity of checking whether the procedure succeeds on a given ...
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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Cited by 29 (1 self)
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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...
On Membership Comparable Sets
 Journal of Computer and System Sciences
, 1999
"... A set A is k(n) membership comparable if there is a polynomial time computable function that, given k(n) instances of A of length at most n, excludes one of the 2 k(n) possibilities for the memberships of the given strings in A. We show that if SAT is O(log n) membership comparable, then Unique ..."
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Cited by 15 (1 self)
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A set A is k(n) membership comparable if there is a polynomial time computable function that, given k(n) instances of A of length at most n, excludes one of the 2 k(n) possibilities for the memberships of the given strings in A. We show that if SAT is O(log n) membership comparable, then UniqueSAT 2 P. This extends the work of Ogihara; Beigel, Kummer, and Stephan; and Agrawal and Arvind [Ogi94, BKS94, AA94], and answers in the affirmative an open question suggested by Buhrman, Fortnow, and Torenvliet [BFT97]. Our proof also shows that if SAT is o(n) membership comparable, then UniqueSAT can be solved in deterministic time 2 o(n) . Our main technical tool is an algorithm of Ar et al. [ALRS92] to reconstruct polynomials from noisy data through the use of bivariate polynomial factorization.
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
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
A time lower bound for satisfiability
 In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP
, 2004
"... Abstract. We show that a deterministic Turing machine with one ddimensional work tape and random access to the input cannot solve satisfiability in time na for a < p(d + 2)/(d + 1). For conondeterministic machines, we obtain a similar lower bound for any a such that a3 < 1 + a/(d + 1). The same boun ..."
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Cited by 6 (1 self)
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Abstract. We show that a deterministic Turing machine with one ddimensional work tape and random access to the input cannot solve satisfiability in time na for a < p(d + 2)/(d + 1). For conondeterministic machines, we obtain a similar lower bound for any a such that a3 < 1 + a/(d + 1). The same bounds apply to almost all natural NPcomplete problems known. 1 Introduction Proving time lower bounds for natural problems remains the most difficultchallenge in computational complexity. We know exponential lower bounds on severely restricted models of computation (e.g., for parity on constant depthcircuits) and polynomial lower bounds on somewhat restricted models (e.g., for palindromes on single tape Turing machines) but no nontrivial lower bounds ongeneral randomaccess machines. In this paper, we exploit the recent timespace lower bounds for satisfiability on general randomaccess machines to establishnew lower bounds of the second type, namely a time lower bound for satisfiability on Turing machines with one multidimensional work tape and random accessto the input.
Derandomizing ArthurMerlin Games
, 1998
"... We establish hardness versus randomness tradeooes for ArthurMerlin games. We create eOEcient nondeterministic simulations of bounded round ArthurMerlin games, using a language in exponential time which small circuits cannot decide given access to an oracle for satisøability. Our results yield sub ..."
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Cited by 2 (1 self)
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We establish hardness versus randomness tradeooes for ArthurMerlin games. We create eOEcient nondeterministic simulations of bounded round ArthurMerlin games, using a language in exponential time which small circuits cannot decide given access to an oracle for satisøability. Our results yield subexponential size proofs for graph nonisomorphism at inønitely many lengths unless the polynomialtime hierarchy collapses. The same holds for any other language with a bounded round interactive proof system. We also apply our techniques to related complexity classes. 1 Introduction Randomness helps computer scientists considerably in designing algorithms. Several computational problems seem easier to solve when we can AEip coins. From a complexity theoretic point of view though, the power of random bits remains widely open: Can they reduce the time or space we need to compute solutions? As of now, we do not know the answer to such fundamental questions about randomness. However, a major ins...
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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Cited by 2 (0 self)
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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.
TimeSpace Lower Bounds for Satisfiability
 In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP
, 2001
"... We survey the recent lower bounds on the running time of generalpurpose randomaccess machines that solve satisfiability in a small amount of work space, and related lower bounds for satisfiability in nonuniform models. ..."
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Cited by 1 (1 self)
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We survey the recent lower bounds on the running time of generalpurpose randomaccess machines that solve satisfiability in a small amount of work space, and related lower bounds for satisfiability in nonuniform models.