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45
Almost Everywhere High Nonuniform Complexity
, 1992
"... . We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuitsize and spacebounded Kolmogorov complexity almost everywhere. (The circuitsize lower bound actually exceeds ..."
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Cited by 169 (34 self)
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. We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuitsize and spacebounded Kolmogorov complexity almost everywhere. (The circuitsize lower bound actually exceeds, and thereby strengthens, the Shannon 2 n n lower bound for almost every problem, with no computability constraint.) In exponential time complexity classes, we prove that the strongest relativizable lower bounds hold almost everywhere for almost all problems. Finally, we show that infinite pseudorandom sequences have high nonuniform complexity almost everywhere. The results are unified by a new, more powerful formulation of the underlying measure theory, based on uniform systems of density functions, and by the introduction of a new nonuniform complexity measure, the selective Kolmogorov complexity. This research was supported in part by NSF Grants CCR8809238 and CCR9157382 and in ...
Some Connections between Bounded Query Classes and NonUniform Complexity
 In Proceedings of the 5th Structure in Complexity Theory Conference
, 1990
"... This paper is dedicated to the memory of Ronald V. Book, 19371997. ..."
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Cited by 70 (23 self)
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This paper is dedicated to the memory of Ronald V. Book, 19371997.
New Collapse Consequences Of NP Having Small Circuits
, 1995
"... . We show that if a selfreducible set has polynomialsize circuits, then it is low for the probabilistic class ZPP(NP). As a consequence we get a deeper collapse of the polynomialtime hierarchy PH to ZPP(NP) under the assumption that NP has polynomialsize circuits. This improves on the wellknown ..."
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Cited by 56 (7 self)
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. We show that if a selfreducible set has polynomialsize circuits, then it is low for the probabilistic class ZPP(NP). As a consequence we get a deeper collapse of the polynomialtime hierarchy PH to ZPP(NP) under the assumption that NP has polynomialsize circuits. This improves on the wellknown result of Karp, Lipton, and Sipser (1980) stating a collapse of PH to its second level \Sigma P 2 under the same assumption. As a further consequence, we derive new collapse consequences under the assumption that complexity classes like UP, FewP, and C=P have polynomialsize circuits. Finally, we investigate the circuitsize complexity of several language classes. In particular, we show that for every fixed polynomial s, there is a set in ZPP(NP) which does not have O(s(n))size circuits. Key words. polynomialsize circuits, advice classes, lowness, randomized computation AMS subject classifications. 03D10, 03D15, 68Q10, 68Q15 1. Introduction. The question of whether intractable sets ca...
Measure, Stochasticity, and the Density of Hard Languages
 SIAM Journal on Computing
, 1994
"... The main theorem of this paper is that, for every real number ff ! 1 (e.g., ff = 0:99), only a measure 0 subset of the languages decidable in exponential time are P n ff \Gammatt reducible to languages that are not exponentially dense. Thus every P n ff \Gammatt hard language for E is exp ..."
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Cited by 43 (13 self)
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The main theorem of this paper is that, for every real number ff ! 1 (e.g., ff = 0:99), only a measure 0 subset of the languages decidable in exponential time are P n ff \Gammatt reducible to languages that are not exponentially dense. Thus every P n ff \Gammatt hard language for E is exponentially dense. This strengthens Watanabe's 1987 result, that every P O(log n)\Gammatt hard language for E is exponentially dense. The combinatorial technique used here, the sequentially most frequent query selection, also gives a new, simpler proof of Watanabe's result. The main theorem also has implications for the structure of NP under strong hypotheses. Ogiwara and Watanabe (1991) have shown that the hypothesis P 6= NP implies that every P btt hard language for NP is nonsparse (i.e., not polynomially sparse). Their technique does not appear to allow significant relaxation of either the query bound or the sparseness criterion. It is shown here that a stronger hypothesis na...
On Pseudorandomness and ResourceBounded Measure
 Theoretical Computer Science
, 1997
"... In this paper we extend a key result of Nisan and Wigderson [17] to the nondeterministic setting: for all ff ? 0 we show that if there is a language in E = DTIME(2 O(n) ) that is hard to approximate by nondeterministic circuits of size 2 ffn , then there is a pseudorandom generator that can be u ..."
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Cited by 42 (3 self)
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In this paper we extend a key result of Nisan and Wigderson [17] to the nondeterministic setting: for all ff ? 0 we show that if there is a language in E = DTIME(2 O(n) ) that is hard to approximate by nondeterministic circuits of size 2 ffn , then there is a pseudorandom generator that can be used to derandomize BP \Delta NP (in symbols, BP \Delta NP = NP). By applying this extension we are able to answer some open questions in [14] regarding the derandomization of the classes BP \Delta \Sigma P k and BP \Delta \Theta P k under plausible measure theoretic assumptions. As a consequence, if \Theta P 2 does not have pmeasure 0, then AM " coAM is low for \Theta P 2 . Thus, in this case, the graph isomorphism problem is low for \Theta P 2 . By using the NisanWigderson design of a pseudorandom generator we unconditionally show the inclusion MA ` ZPP NP and that MA " coMA is low for ZPP NP . 1 Introduction In recent years, following the development of resourcebounded meas...
The Role of Relativization in Complexity Theory
 Bulletin of the European Association for Theoretical Computer Science
, 1994
"... Several recent nonrelativizing results in the area of interactive proofs have caused many people to review the importance of relativization. In this paper we take a look at how complexity theorists use and misuse oracle results. We pay special attention to the new interactive proof systems and progr ..."
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Cited by 40 (9 self)
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Several recent nonrelativizing results in the area of interactive proofs have caused many people to review the importance of relativization. In this paper we take a look at how complexity theorists use and misuse oracle results. We pay special attention to the new interactive proof systems and program checking results and try to understand why they do not relativize. We give some new results that may help us to understand these questions better.
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 bounds. Al ..."
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Cited by 30 (3 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.
Algebrization: A new barrier in complexity theory
 MIT Theory of Computing Colloquium
, 2007
"... Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a ..."
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Cited by 29 (2 self)
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Any proof of P � = NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linearsize circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a lowdegree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known nonrelativizing results based on arithmetization—both inclusions such as IP = PSPACE and MIP = NEXP, and separations such as MAEXP � ⊂ P/poly —do indeed algebrize. Second, we show that almost all of the major open problems—including P versus NP, P versus RP, and NEXP versus P/poly—will require nonalgebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MAprotocol for the Inner Product function with O ( √ n log n) communication (essentially matching a lower bound of Klauck), as well as a communication complexity conjecture whose truth would imply NL � = NP. 1
Circuit Minimization Problem
 In ACM Symposium on Theory of Computing (STOC
, 1999
"... We study the complexity of the circuit minimization problem: given the truth table of a Boolean function f and a parameter s, decide whether f can be realized by a Boolean circuit of size at most s. We argue why this problem is unlikely to be in P (or even in P=poly) by giving a number of surpris ..."
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Cited by 26 (1 self)
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We study the complexity of the circuit minimization problem: given the truth table of a Boolean function f and a parameter s, decide whether f can be realized by a Boolean circuit of size at most s. We argue why this problem is unlikely to be in P (or even in P=poly) by giving a number of surprising consequences of such an assumption. We also argue that proving this problem to be NPcomplete (if it is indeed true) would imply proving strong circuit lower bounds for the class E, which appears beyond the currently known techniques. Keywords: hard Boolean functions, derandomization, natural properties, NPcompleteness. 1 Introduction An nvariable Boolean function f n : f0; 1g n ! f0; 1g can be given by either its truth table of size 2 n , or a Boolean circuit whose size may be significantly smaller than 2 n . It is well known that most Boolean functions on n variables have circuit complexity at least 2 n =n [Sha49], but so far no family of sufficiently hard functions has ...
The Complexity of Planarity Testing
, 2000
"... We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circ ..."
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Cited by 23 (7 self)
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We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degreethree graphs had been shown to be in SL [23, 20].