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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 41 (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 algorithmic argument for nonadaptive query complexity lower bounds on advised quantum computation
 In Proc. 29th International Symposium on Mathematical Foundations of Computer Science
, 2004
"... Abstract. This paper employs a powerful argument, called an algorithmic argument, to prove lower bounds of the quantum query complexity of a multipleblock ordered search problem in which, given a block number i, we are to find a location of a target keyword in an ordered list of the ith block. Apar ..."
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Cited by 1 (0 self)
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Abstract. This paper employs a powerful argument, called an algorithmic argument, to prove lower bounds of the quantum query complexity of a multipleblock ordered search problem in which, given a block number i, we are to find a location of a target keyword in an ordered list of the ith block. Apart from much studied polynomial and adversary methods for quantum query complexity lower bounds, our argument shows that the multipleblock ordered search needs a large number of nonadaptive oracle queries on a blackbox model of quantum computation that is also supplemented with advice. Our argument is also applied to the notions of computational complexity theory: quantum truthtable reducibility and quantum truthtable autoreducibility.
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...
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...
AMS subject classifications. 81P68, 68Q30, 68Q30 LOWER BOUNDS FOR RANDOMIZED AND QUANTUM QUERY COMPLEXITY USING KOLMOGOROV ARGUMENTS ∗
"... , 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 Ambai ..."
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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 ( p nC0(f), p nC1(f))), where C0, C1 is the certificate complexity, and n is the size of the input. 1. Introduction. 1.1. Overview. In this paper, we study lower bounds for randomized and quantum query complexity. In the query model, the input is accessed using oracle queries, and the query complexity of an algorithm is the number of calls to the oracle. Since it is difficult to obtain lower bounds on time directly, the query model is often used
SIGACT News Complexity Theory Column 40
"... Aduri Pavan very kindly stepped in, after a lastminute guest column cancellation, to write the present article. Warmest thanks to him. Upcoming issues will contain articles by (all topics/titles are of course tentative): M. Li´skiewicz, ..."
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Aduri Pavan very kindly stepped in, after a lastminute guest column cancellation, to write the present article. Warmest thanks to him. Upcoming issues will contain articles by (all topics/titles are of course tentative): M. Li´skiewicz,