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19
In Search of an Easy Witness: Exponential Time vs. Probabilistic Polynomial Time
, 2002
"... Restricting the search space {0, 1} n to the set of truth tables of “easy ” Boolean functions on log n variables, as well as using some known hardnessrandomness tradeoffs, we establish a number of results relating the complexity of exponentialtime and probabilistic polynomialtime complexity class ..."
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Cited by 54 (7 self)
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Restricting the search space {0, 1} n to the set of truth tables of “easy ” Boolean functions on log n variables, as well as using some known hardnessrandomness tradeoffs, we establish a number of results relating the complexity of exponentialtime and probabilistic polynomialtime complexity classes. In particular, we show that NEXP ⊂ P/poly ⇔ NEXP = MA; this can be interpreted as saying that no derandomization of MA (and, hence, of promiseBPP) is possible unless NEXP contains a hard Boolean function. We also prove several downward closure results for ZPP, RP, BPP, and MA; e.g., we show EXP = BPP ⇔ EE = BPE, where EE is the doubleexponential time class and BPE is the exponentialtime analogue of BPP.
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 35 (4 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.
On the complexity of succinct zerosum games
 IEEE Conference on Computational Complexity
, 2005
"... We study the complexity of solving succinct zerosum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M(i, j) = C(i, j). We complement the known EXPhardness of computing the exact value of a succinct zerosum game by several results on approximating ..."
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Cited by 17 (0 self)
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We study the complexity of solving succinct zerosum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M(i, j) = C(i, j). We complement the known EXPhardness of computing the exact value of a succinct zerosum game by several results on approximating the value. (1) We prove that approximating the value of a succinct zerosum game to within an additive factor is complete for the class promiseS p 2, the. To the best of our knowledge, it is “promise ” version of S p 2 the first natural problem shown complete for this class. (2) We describe a ZPP NP algorithm for constructing approximately optimal strategies, and hence for approximating the value, of a given succinct zerosum game. As a corollary, we obtain, in a uniform fashion, several complexitytheoretic results, e.g., a ZPP NP algorithm for learning circuits for SAT [7] and a recent result by Cai [9] that S p 2 ⊆ ZPPNP. (3) We observe that approximating the value of a succinct zerosum game to within a multiplicative factor is in PSPACE, and that it cannot be in promiseS p 2 unless the polynomialtime hierarchy collapses. Thus, under a reasonable complexitytheoretic assumption, multiplicativefactor approximation of succinct zerosum games is strictly harder
Complete Sets and Structure in Subrecursive Classes
 In Proceedings of Logic Colloquium '96
, 1998
"... In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completene ..."
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Cited by 14 (1 self)
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In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is usually done by imposing time and/or space bounds on the computational power of the device representing the reduction. Early on, Ladner et al. [LLS75] categorized the then known types of reductions and made a comparison between these by constructing sets that are reducible to each other via one type of reduction and not reducible via the other. They however were interested just in the rela...
On helping and interactive proof systems
 International Journal of Foundations of Computer Science
, 1995
"... We investigate the complexity of honest provers in interactive proof systems. This corresponds precisely to the complexity of oracles helping the computation of robust probabilistic oracle machines. We obtain upper bounds for languages in FewEXP and for sparse sets in NP. Further, interactive protoc ..."
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Cited by 10 (4 self)
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We investigate the complexity of honest provers in interactive proof systems. This corresponds precisely to the complexity of oracles helping the computation of robust probabilistic oracle machines. We obtain upper bounds for languages in FewEXP and for sparse sets in NP. Further, interactive protocols with provers that are reducible to sets of low information content are considered. Specifically, if the verifier communicates only with provers in P=poly, then the accepted language is low for \Sigma p 2. In the case that the provers are polynomialtime reducible to logsparse sets or to sets in strongP/log then the protocol can be simulated by the verifier even without the help of provers. As a consequence we obtain new collapse results under the assumption that intractable sets reduce to sets with low information content. 1 Introduction and overview of results Two extensions of the concept of NP (as the class of languages with efficient proofs of
Properties of NPcomplete sets
 In Proceedings of the 19th IEEE Conference on Computational Complexity
, 2004
"... We study several properties of sets that are complete for NP. We prove that if L is an NPcomplete set and S � ⊇ L is a pselective sparse set, then L − S is ≤p mhard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p mhard for NP. Moreo ..."
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We study several properties of sets that are complete for NP. We prove that if L is an NPcomplete set and S � ⊇ L is a pselective sparse set, then L − S is ≤p mhard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p mhard for NP. Moreover, we prove for every L ∈ NP − P, that there exists a sparse S ∈ EXP such that L − S is not ≤ p mhard for NP. Hence, removing sparse information in P from a complete set leaves the set complete, while removing sparse information in EXP from a complete set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes. We use hypotheses about pseudorandom generators and secure oneway permutations to derive consequences for longstanding open questions about whether NPcomplete sets are immune. For example, assuming that pseudorandom generators and secure oneway permutations exist, it follows easily that NPcomplete sets are not pimmune. Assuming only that secure oneway permutations exist, we prove that no NPcomplete set is DTIME(2nɛ)immune. Also, using these hypotheses we show that no NPcomplete set is quasipolynomialclose to P. We introduce a strong but reasonable hypothesis and infer from it that disjoint Turingcomplete sets for NP are not closed under union. Our hypothesis asserts existence of a UPmachine M that accepts 0 ∗ such that for some 0 < ɛ < 1, no 2nɛ timebounded machine can correctly compute infinitely many accepting computations of M. We show that if UP∩coUP contains DTIME(2nɛ)biimmune sets, then this hypothesis is true.
NPHard Sets are Exponentially Dense Unless coNP ⊆ NP/poly
"... We show that hard sets S for NP must have exponential density, for some ɛ> 0 and infinitely many n, unless coNP ⊆ i.e. S=n  ≥ 2nɛ NP/poly and the polynomialtime hierarchy collapses. This result holds for Turing reductions that make n1−ɛ queries. In addition we study the instance complexity of ..."
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We show that hard sets S for NP must have exponential density, for some ɛ> 0 and infinitely many n, unless coNP ⊆ i.e. S=n  ≥ 2nɛ NP/poly and the polynomialtime hierarchy collapses. This result holds for Turing reductions that make n1−ɛ queries. In addition we study the instance complexity of NPhard problems and show that hard sets also have an exponential amount of instances that have instance complexity nδ for some δ> 0. This result also holds for Turing reductions that make n1−ɛ queries. 1
Geometric Complexity Theory V: Equivalence between blackbox derandomization of polynomial identity testing and derandomization of Noether’s Normalization Lemma
"... It is shown that blackbox derandomization of polynomial identity testing (PIT) is essentially equivalent to derandomization of Noether’s Normalization Lemma for explicit algebraic varieties, the problem that lies at the heart of the foundational classification problem of algebraic geometry. Specif ..."
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It is shown that blackbox derandomization of polynomial identity testing (PIT) is essentially equivalent to derandomization of Noether’s Normalization Lemma for explicit algebraic varieties, the problem that lies at the heart of the foundational classification problem of algebraic geometry. Specifically: (1) It is shown that in characteristic zero blackbox derandomization of the symbolic trace identity testing (STIT) brings the problem of derandomizing Noether’s Normalization Lemma for the ring of invariants of the adjoint action of the general linear group on a tuple of matrices from EXPSPACE (where it is currently) to P. Next it is shown that assuming the Generalized Riemann Hypothesis (GRH), instead of the blackbox derandomization hypothesis, brings the problem from EXPSPACE to quasiPH, instead of P. Thus blackbox derandomization
Randomness is Hard
 SIAM Journal on Computing
, 2000
"... We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are: polynomial time CD complexity dened by Sipser, the nondeterministic variant due to Buhrman and Fortnow, and the polynomi ..."
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We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are: polynomial time CD complexity dened by Sipser, the nondeterministic variant due to Buhrman and Fortnow, and the polynomial space bounded Kolmogorov complexity, CS introduced by Hartmanis. For all of these measures we dene the set of random strings R CD t , R CND t , and R CS s as the set of strings x such that CD t (x), CND t (x), and CS s (x) is greater than or equal to the length of x, for s and t polynomials. We show the following: MA NP R CD t , where MA is the class of MerlinArthur games dened by Babai. AM NP R CND t , where AM is the class of ArthurMerlin games. PSPACE NP cR CS s . In the last item cR CS s is the set of pairs <x; y> so that x is random given y. These results show that the set of random strings for various resource bounds is hard for...
Nonuniform Lower Bounds for Exponential Time Classes
 Mathematical Foundations of Computer Science 1995, 20th International Symposium, volume 969 of lncs, pages 159168, Prague, Czech Republic, 1 September 28
, 1993
"... this paper we are interested in absolute results and consider advice classes slightly smaller than P=poly and circuit classes smaller than polynomialsize. And we establish several new lower bounds for exponential time problems with respect to these classes. This is not a new tack to explore. In la ..."
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this paper we are interested in absolute results and consider advice classes slightly smaller than P=poly and circuit classes smaller than polynomialsize. And we establish several new lower bounds for exponential time problems with respect to these classes. This is not a new tack to explore. In last years' Structures Bin Fu [Fu93] considered lower bounds for polynomial time reductions to sparse sets, where limits are placed on the number of queries to the sparse set. The main result of his paper was that there are sets in EXP which are not polynomial time Turing reducible to a sparse set when the reduction is restricted to querying the sparse set no more than n