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Lower bounds for randomized and quantum query complexity using Kolmogorov arguments
 in Proc. of the 19th IEEE Conference on Computational Complexity
, 2004
"... Abstract. We prove a very general lower bound technique for quantum and randomized query complexity that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to query complexity. Our technique generalizes the weighted and unweighted methods of Ambainis ..."
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Cited by 40 (5 self)
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Abstract. We prove a very general lower bound technique for quantum and randomized query complexity that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to query complexity. Our technique generalizes the weighted and unweighted methods of Ambainis and the spectral method of Barnum, Saks, and Szegedy. As an immediate consequence of our main theorem, it can be shown that adversary methods can only prove lower bounds for Boolean functions f in O(min ( √ nC0(f), √ nC1(f))), where C0,C1 is the certificate complexity and n is the size of the input.
Stronger Separations for RandomSelfReducibility, Rounds, and Advice
 In IEEE Conference on Computational Complexity
, 1999
"... A function f is selfreducible if it can be computed given an oracle for f . In a randomselfreduction the queries must be made in such a way that the distribution of the ith query is independent of the input that gave rise to it. Randomself reductions have many applications, including countless c ..."
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Cited by 4 (2 self)
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A function f is selfreducible if it can be computed given an oracle for f . In a randomselfreduction the queries must be made in such a way that the distribution of the ith query is independent of the input that gave rise to it. Randomself reductions have many applications, including countless cryptographic protocols, probabilistically checkable proofs, averagecase complexity, and program checking. A simpler model of randomized selfreducibility is coherence, in which the only condition on the queries is that the input itself may not be among the queries. We show that there is a function which is randomselfreducible with 2 rounds of queries, but which is not even coherent, even if polynomial advice is allowed, when the queries must be made in a single round. 1 Introduction Informally, we say that a function f selfreduces if it can be computed efficiently by making queries to f . For a function to be randomselfreducible, the queries must be made at random in such a way that t...
COLLAPSING AND SEPARATING COMPLETENESS NOTIONS UNDER AVERAGECASE AND WORSTCASE HYPOTHESES
, 2010
"... This paper presents the following results on sets that are complete for NP. (i) If there is a problem in NP that requires 2 nΩ(1) time at almost all lengths, then every manyone NPcomplete set is complete under lengthincreasing reductions that are computed by polynomialsize circuits. (ii) If ther ..."
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Cited by 2 (2 self)
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This paper presents the following results on sets that are complete for NP. (i) If there is a problem in NP that requires 2 nΩ(1) time at almost all lengths, then every manyone NPcomplete set is complete under lengthincreasing reductions that are computed by polynomialsize circuits. (ii) If there is a problem in coNP that cannot be solved by polynomialsize nondeterministic circuits, then every manyone complete set is complete under lengthincreasing reductions that are computed by polynomialsize circuits. (iii) If there exist a oneway permutation that is secure against subexponentialsize circuits and there is a hard tally language in NP∩coNP, then there is a Turing complete language for NP that is not manyone complete. Our first two results use worstcase hardness hypotheses whereas earlier work that showed similar results relied on averagecase or almosteverywhere hardness assumptions. The use of averagecase and worstcase hypotheses in the last result is unique as previous results obtaining the same consequence relied on almosteverywhere hardness results.
An InformationTheoretic Treatment of RandomSelfReducibility
 Proc. of the 14'th Symposium on Theoretical Aspects of Computer Science
, 1997
"... We initiate the study of randomselfreducibility from an informationtheoretic point of view. Specifically, we formally define the notion of a randomselfreduction that, with respect to a given ensemble of distributions, leaks a limited number bits, i.e., produces target instances y1 ; : : : ; yk ..."
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Cited by 1 (1 self)
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We initiate the study of randomselfreducibility from an informationtheoretic point of view. Specifically, we formally define the notion of a randomselfreduction that, with respect to a given ensemble of distributions, leaks a limited number bits, i.e., produces target instances y1 ; : : : ; yk in such a manner that each y i has limited mutual information with the input x. We argue that this notion is useful in studying the relationships between randomselfreducibility and other properties of interest, including selfcorrectability and NPhardness. In the case of selfcorrectability, we show that the informationtheoretic definition of randomselfreducibility leads to somewhat different conclusions from those drawn by Feigenbaum, Fortnow, Laplante, and Naik [13], who used the standard definition. In the case of NPhardness, we use the informationtheoretic definition to strengthen the result of Feigenbaum and Fortnow [12], who proved, using the standard definition, that the polyn...
Kolmogorov Techniques in Computational Complexity Theory
, 1997
"... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 iHow Hard is this Problem?j . . . . . . . . . . . . . . . . . . . . 1 1.1.1 An introduction to computational complexity . . . . . . . 3 ..."
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Cited by 1 (1 self)
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 iHow Hard is this Problem?j . . . . . . . . . . . . . . . . . . . . 1 1.1.1 An introduction to computational complexity . . . . . . . 3 1.1.2 An introduction to Kolmogorov complexity . . . . . . . . 6 1.2 Overview of the contributions and previous work . . . . . . . . . 9 1.2.1 Selfreductions . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Extractors . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Indistinguishability . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Collaboration and publication status . . . . . . . . . . . . 11 2. PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Kolmogorov complexity . . . . . . ....
Structure and Complexity
, 1996
"... S On the Power of Randomized Branching Programs Farid Ablayev Kazan University (joint work with Marek Karpinski, Universitat Bonn) We define a notion of randomized branching programs in a natural way similar to the notion of randomized circuits. We present two explicit boolean functions f n : f0; 1 ..."
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S On the Power of Randomized Branching Programs Farid Ablayev Kazan University (joint work with Marek Karpinski, Universitat Bonn) We define a notion of randomized branching programs in a natural way similar to the notion of randomized circuits. We present two explicit boolean functions f n : f0; 1g 4n ! f0; 1g and g n : f0; 1g n ! f0; 1g such that: 1. f n can be computed by a randomized ordered readonce branching program of size polynomial in n and with a small (constant) error, 2. any nondeterministic ordered readktimes branching program that computes f n needs exponential size (the size is\Omega\Gamma/35 (n=(2k \Gamma 1)))), 3. g n can be computed by a nondeterministic readonce branching program of size polynomial in n, and 4. any randomized ordered readonce branching program that computes g n with a constant error ffl has size no less than exp(c(ffl)n= log n). Recent Progress on the Isomorphism Conjecture Eric Allender Rutgers University http://www.cs.rutgers.edu/~all...
An Algorithmic Argument for Query Complexity Lower Bounds of Advised Quantum Computation ∗
, 2003
"... Abstract: This paper proves lower bounds of the quantum query complexity of a multipleblock ordered search problem, which is a natural generalization of the ordered search problems. Apart from much studied polynomial and adversary methods for quantum query complexity lower bounds, our proof employs ..."
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Abstract: This paper proves lower bounds of the quantum query complexity of a multipleblock ordered search problem, which is a natural generalization of the ordered search problems. Apart from much studied polynomial and adversary methods for quantum query complexity lower bounds, our proof employs an argument that (i) commences with the faulty assumption that a quantum algorithm of low query complexity exists, (ii) select any incompressible input, and (iii) constructs another algorithm that compresses the input, which leads to a contradiction. Using this “algorithmic” argument, we show that the multiblock ordered search needs a large number of nonadaptive oracle queries on a blackbox model of quantum computation supplemented by advice. This main theorem can be applied directly to two important notions in structural complexity theory: nonadaptive (truthtable) reducibility and autoreducibility. In particular, we prove: 1) there is an oracle A relative to which there is a set in P A which is not quantumly nonadaptively reducible to A in polynomial time even with polynomial advice, 2) there is a polynomialtime adaptively probabilisticallyautoreducible set which is not polynomialtime nonadaptively quantumautoreducible even with any help of polynomial advice, and 3) there is a set in ESPACE which is not polynomialtime nonadaptively quantumautoreducible in polynomial time even in the presence of polynomial advice. For the singleblock ordered search problem, our algorithmic argument also shows a large lower bound of the quantum query complexity in the presence of advice.
Submitted to the Symposium on Theoretical Aspects of Computer Science www.stacsconf.org COLLAPSING AND SEPARATING COMPLETENESS NOTIONS UNDER AVERAGECASE AND WORSTCASE HYPOTHESES
"... Abstract. This paper presents the following results on sets that are complete for NP. (i) If there is a problem in NP that requires 2 nΩ(1) time at almost all lengths, then every manyone NPcomplete set is complete under lengthincreasing reductions that are computed by polynomialsize circuits. (ii ..."
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Abstract. This paper presents the following results on sets that are complete for NP. (i) If there is a problem in NP that requires 2 nΩ(1) time at almost all lengths, then every manyone NPcomplete set is complete under lengthincreasing reductions that are computed by polynomialsize circuits. (ii) If there is a problem in coNP that cannot be solved by polynomialsize nondeterministic circuits, then every manyone complete set is complete under lengthincreasing reductions that are computed by polynomialsize circuits. (iii) If there exist a oneway permutation that is secure against subexponentialsize circuits and there is a hard tally language in NP ∩ coNP, then there is a Turing complete language for NP that is not manyone complete. Our first two results use worstcase hardness hypotheses whereas earlier work that showed similar results relied on averagecase or almosteverywhere hardness assumptions. The use of averagecase and worstcase hypotheses in the last result is unique as previous results obtaining the same consequence relied on almosteverywhere hardness results. 1.