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.

A function f is self-reducible if it can be computed given an oracle for f . In a random-self-reduction 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, average-case complexity, and program checking. A simpler model of randomized self-reducibility 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 random-self-reducible 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 self-reduces if it can be computed efficiently by making queries to f . For a function to be random-self-reducible, the queries must be made at random in such a way that t...

We initiate the study of random-self-reducibility from an information-theoretic point of view. Specifically, we formally define the notion of a random-self-reduction 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 random-self-reducibility and other properties of interest, including self-correctability and NP-hardness. In the case of self-correctability, we show that the information-theoretic definition of random-self-reducibility leads to somewhat different conclusions from those drawn by Feigenbaum, Fortnow, Laplante, and Naik [13], who used the standard definition. In the case of NP-hardness, we use the information-theoretic definition to strengthen the result of Feigenbaum and Fortnow [12], who proved, using the standard definition, that the polyn...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Self-reductions . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . ....

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 read-once branching program of size polynomial in n and with a small (constant) error, 2. any nondeterministic ordered read-k-times 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 read-once branching program of size polynomial in n, and 4. any randomized ordered read-once 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...

Abstract: This paper proves lower bounds of the quantum query complexity of a multiple-block 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 multi-block ordered search needs a large number of nonadaptive oracle queries on a black-box model of quantum computation supplemented by advice. This main theorem can be applied directly to two important notions in structural complexity theory: nonadaptive (truth-table) 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 polynomial-time adaptively probabilistically-autoreducible set which is not polynomial-time nonadaptively quantum-autoreducible even with any help of polynomial advice, and 3) there is a set in ESPACE which is not polynomial-time nonadaptively quantum-autoreducible in polynomial time even in the presence of polynomial advice. For the single-block ordered search problem, our algorithmic argument also shows a large lower bound of the quantum query complexity in the presence of advice.