Results 1  10
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19
Complexity Classes Defined By Counting Quantifiers
, 1991
"... We study the polynomial time counting hierarchy, a hierarchy of complexity classes related to the notion of counting. We investigate some of their structural properties, settling many open questions dealing with oracle characterizations, closure under boolean operations, and relations with other com ..."
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Cited by 52 (0 self)
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We study the polynomial time counting hierarchy, a hierarchy of complexity classes related to the notion of counting. We investigate some of their structural properties, settling many open questions dealing with oracle characterizations, closure under boolean operations, and relations with other complexity classes. We develop a new combinatorial technique to obtain relativized separations for some of the studied classes, which imply absolute separations for some logarithmic time bounded complexity classes.
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...
The random oracle hypothesis is false
, 1990
"... The Random Oracle Hypothesis, attributed to Bennett and Gill, essentially states that the relationships between complexity classes which holdforalmost all relativized worlds must also hold in the unrelativized case. Although this paper is not the rst to provideacounterexample to the Random Oracle Hy ..."
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Cited by 24 (2 self)
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The Random Oracle Hypothesis, attributed to Bennett and Gill, essentially states that the relationships between complexity classes which holdforalmost all relativized worlds must also hold in the unrelativized case. Although this paper is not the rst to provideacounterexample to the Random Oracle Hypothesis, it does provide a most compelling counterexample by showing that for almost all oracles A, IP A 6=PSPACE A. If the Random Oracle Hypothesis were true, it would contradict Shamir's result that IP = PSPACE. In fact, it is shown that for almost all oracles A, coNP A 6 IP A. These results extend to the multiprover proof systems of BenOr, Goldwasser, Kilian and Wigderson. In addition, this paper shows that the Random Oracle Hypothesis is sensitive to small changes in the de nition. A class IPP, similar to IP, is de ned. Surprisingly, the IPP = PSPACE result holds for all oracle worlds. Warning: Essentially this paper has been published in Information and Computation and is hence subject to copyright restrictions. It is for personal use only. 1
A Downward Collapse Within The Polynomial Hierarchy
, 1998
"... . Downward collapse (also known as upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial hierarchy. In particular, we prove that, for
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Cited by 23 (9 self)
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.<F3.803e+05> Downward collapse (also known as upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial hierarchy. In particular, we prove that, for<F3.319e+05> k ><F3.803e+05> 2, if P<F2.821e+05> #<F2.795e+05> p k<F2.821e+05> [1]<F3.803e+05> = P<F2.821e+05> #<F2.795e+05> p k<F2.821e+05> [2]<F3.803e+05> then #<F2.562e+05> p k<F3.803e+05> = #<F2.562e+05> p k<F3.803e+05> = PH. We extend this to obtain a more general downward collapse result.<F4.005e+05> Key words.<F3.803e+05> computational complexity theory, easyhard arguments, downward collapse, polynomial hierarchy<F4.005e+05> AMS subject classifications.<F3.803e+05> 68Q15, 68Q10, 03D15, 03D10<F4.005e+05> PII.<F3.803e+05> S0097539796306474<F5.353e+05> 1. Introduction.<F4.529e+05> The theory of NPcompleteness does not resolve the issue of whether P and NP are equal. However, it do...
Nondeterministic Polynomial Time versus Nondeterministic Logarithmic Space
 In Proceedings, Twelfth Annual IEEE Conference on Computational Complexity
, 1996
"... We discuss the possibility of using the relatively old technique of diagonalization to separate complexity classes, in particular NL from NP. We show several results in this direction. ffl Any nonconstant level of the polynomialtime hierarchy strictly contains NL. ffl SAT is not simultaneously in ..."
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Cited by 22 (1 self)
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We discuss the possibility of using the relatively old technique of diagonalization to separate complexity classes, in particular NL from NP. We show several results in this direction. ffl Any nonconstant level of the polynomialtime hierarchy strictly contains NL. ffl SAT is not simultaneously in NL and deterministic n log j n time for any j. ffl On the negative side, we present a relativized world where P = NP but any nonconstant level of the polynomialtime hierarchy differs from P. 1 Introduction Separating complexity classes remains the most important and difficult of problems in theoretical computer science. Circuit complexity and other techniques on finite functions have seen some exciting early successes (see the survey of Boppana and Sipser [BS90]) but have yet to achieve their promise of separating complexity classes above logarithmic space. Other techniques based on logic and geometry also have given us separations only on very restricted models. We should turn back to...
ComplexityTheoretic Aspects of Interactive Proof Systems
, 1989
"... In 1985, Goldwasser, Micali and Rackoff formulated interactive proof systems as a tool for developing cryptographic protocols. Indeed, many exciting cryptographic results followed from studying interactive proof systems and the related concept of zeroknowledge. Interactive proof systems also have a ..."
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Cited by 19 (3 self)
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In 1985, Goldwasser, Micali and Rackoff formulated interactive proof systems as a tool for developing cryptographic protocols. Indeed, many exciting cryptographic results followed from studying interactive proof systems and the related concept of zeroknowledge. Interactive proof systems also have an important part in complexity theory merging the well established concepts of probabilistic and nondeterministic computation. This thesis will study the complexity of various models of interactive proof systems. A perfect zeroknowledge interactive protocol convinces a verifier that a string is in a language without revealing any additional knowledge in an information theoretic sense. This thesis will show that for any language that has a perfect zeroknowledge proof system, its complement has a short interactive protocol. This result implies that there are not any perfect zeroknowledge protocols for NPcomplete languages unless the polynomialtime hierarchy collapses. Thus knowledge comp...
Relating Polynomial Time to Constant Depth
 THEORETICAL COMPUTER SCIENCE
, 1998
"... Going back to the seminal paper [FSS84] by Furst, Saxe, and Sipser, analogues between polynomial time classes and constant depth circuit classes have been considered in a number of papers. Oracles separating polynomial time classes have been obtained by diagonalization making essential use of lower ..."
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Cited by 12 (2 self)
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Going back to the seminal paper [FSS84] by Furst, Saxe, and Sipser, analogues between polynomial time classes and constant depth circuit classes have been considered in a number of papers. Oracles separating polynomial time classes have been obtained by diagonalization making essential use of lower bounds for circuit classes. In this note we show how separating oracles can be obtained uniformly from circuit lower bounds without the need of carrying out a particular diagonalization. Our technical tool is the leaf language approach to the definition of complexity classes.
ComplexityRestricted Advice Functions
"... . We consider uniform subclasses of the nonuniform complexity classes defined by Karp and Lipton [23] via the notion of advice functions. These subclasses are obtained by restricting the complexity of computing correct advice. We also investigate the effect of allowing advice functions of limited co ..."
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Cited by 12 (4 self)
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. We consider uniform subclasses of the nonuniform complexity classes defined by Karp and Lipton [23] via the notion of advice functions. These subclasses are obtained by restricting the complexity of computing correct advice. We also investigate the effect of allowing advice functions of limited complexity to depend on the input rather than on the input's length. Among other results, using the notions described above, we give new characterizations of (a) NP NP"SPARSE , (b) NP with a restricted access to an NP oracle and (c) the odd levels of the boolean hierarchy. As a consequence, we show that every set that is nondeterministically truthtable reducible to SAT in the sense of Rich [35] is already deterministically truthtable reducible to SAT. Furthermore, it turns out that the NP reduction classes of bounded versions of this reducibility coincide with the odd levels of the boolean hierarchy. Key words. nonuniform complexity classes, advice classes, optimization functions, restric...
A Lower Bound for Perceptrons and an Oracle Separation of the PP PH Hierarchy
 Journal of Computer and System Sciences
, 1997
"... We show that there are functions computable by linear size boolean circuits of depth k that require superpolynomial size perceptrons of depth k \Gamma 1, for k ! logn=(6loglogn). This result implies the existence of an oracle A such that S p;A k 6` PP S p;A k\Gamma2 and in particular this ..."
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Cited by 8 (0 self)
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We show that there are functions computable by linear size boolean circuits of depth k that require superpolynomial size perceptrons of depth k \Gamma 1, for k ! logn=(6loglogn). This result implies the existence of an oracle A such that S p;A k 6` PP S p;A k\Gamma2 and in particular this oracle separates the levels in the PP PH hierarchy. Using the same ideas, we show a lower bound for another function, which makes it possible to strengthen the oracle separation to D p;A k 6` PP S p;A k\Gamma2 . 1 Introduction There is a strong connection between lower bounds for boolean circuits (consisting of AND, OR, and NOT gates) and relativization results about the polynomial time hierarchy. This fact was first established by Furst, Saxe, and Sipser [5]. Sipser [13] later defined a family of functions that are computable by linear size circuits of depth k, and showed that they require superpolynomial size boolean circuits of depth k \Gamma 1. Yao [14] and Hstad [8, 9] impr...
On Proving Circuit Lower Bounds Against the Polynomialtime Hierarchy: Positive and Negative Results
, 2008
"... We consider the problem of proving circuit lower bounds against the polynomialtime hierarchy. We give both positive and negative results. For the positive side, for any fixed integer k> 0, we give an explicit Σ p 2 language, acceptable by a Σp2machine with running time O(nk2 +k), that requires ci ..."
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Cited by 7 (3 self)
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We consider the problem of proving circuit lower bounds against the polynomialtime hierarchy. We give both positive and negative results. For the positive side, for any fixed integer k> 0, we give an explicit Σ p 2 language, acceptable by a Σp2machine with running time O(nk2 +k), that requires circuit size> nk. This provides a constructive version of an existence theorem of Kannan [Kan82]. Our main theorem is on the negative side. We give evidence that it is infeasible to give relativizable proofs that any single language in the polynomialtime hierarchy requires super polynomial circuit size. Our proof techniques are based on the decision tree version of the Switching Lemma for constant depth circuits and NisanWigderson pseudorandom generator.