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27
Complexity Measures and Decision Tree Complexity: A Survey
 Theoretical Computer Science
, 2000
"... We discuss several complexity measures for Boolean functions: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial. We survey the relations and biggest gaps known between these measures, and show how they give bounds for the decision tr ..."
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Cited by 123 (14 self)
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We discuss several complexity measures for Boolean functions: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial. We survey the relations and biggest gaps known between these measures, and show how they give bounds for the decision tree complexity of Boolean functions on deterministic, randomized, and quantum computers. 1 Introduction Computational Complexity is the subfield of Theoretical Computer Science that aims to understand "how much" computation is necessary and sufficient to perform certain computational tasks. For example, given a computational problem it tries to establish tight upper and lower bounds on the length of the computation (or on other resources, like space). Unfortunately, for many, practically relevant, computational problems no tight bounds are known. An illustrative example is the well known P versus NP problem: for all NPcomplete problems the current upper and lower bounds lie exponentially ...
Limits on the Provable Consequences of Oneway Functions
, 1989
"... This technical point will prevent the reader from suspecting any measuretheoretic fallacy. ..."
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Cited by 32 (1 self)
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This technical point will prevent the reader from suspecting any measuretheoretic fallacy.
The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
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Cited by 30 (3 self)
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This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
Separability and Oneway Functions
, 2000
"... We settle all relativized questions of the relationships between the following ve propositions: P = NP P = UP P = NP \ coNP All disjoint pairs of NP sets are Pseparable. ..."
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Cited by 26 (12 self)
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We settle all relativized questions of the relationships between the following ve propositions: P = NP P = UP P = NP \ coNP All disjoint pairs of NP sets are Pseparable.
Characterization of nondeterministic quantum query and quantum communication complexity
 In Proceedings of the 15th Annual IEEE Conference on Computational Complexity
, 2000
"... It is known that the classical and quantum query complexities of a total Boolean function f are polynomially related to the degree of its representing polynomial, but the optimal exponents in these relations are unknown. We show that the nondeterministic quantum query complexity of f is linearly re ..."
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Cited by 18 (8 self)
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It is known that the classical and quantum query complexities of a total Boolean function f are polynomially related to the degree of its representing polynomial, but the optimal exponents in these relations are unknown. We show that the nondeterministic quantum query complexity of f is linearly related to the degree of a “nondeterministic ” polynomial for f. We also prove a quantumclassical gap of 1 vs. n for nondeterministic query complexity for a total f. In the case of quantum communication complexity there is a (partly undetermined) relation between the complexity of f and the logarithm of the rank of its communication matrix. We show that the nondeterministic quantum communication complexity of f is linearly related to the logarithm of the rank of a nondeterministic version of the communication matrix. We also exhibit an exponential quantumclassical gap for nondeterministic communication complexity.
On The Relativized Power of Additional Accepting Paths
, 1989
"... The class UP is the class of languages in NP that are accepted by machines with at most one accepting path on each input. We define U k(n) P as the class of languages in NP that are accepted by machines with at most k(n) accepting paths on each input of length n. Obviously P ` UP ` U k(n) P ` U k(n ..."
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Cited by 14 (2 self)
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The class UP is the class of languages in NP that are accepted by machines with at most one accepting path on each input. We define U k(n) P as the class of languages in NP that are accepted by machines with at most k(n) accepting paths on each input of length n. Obviously P ` UP ` U k(n) P ` U k(n)+1 P ` FewP ` NP for every polynomial k(n) 2, where FewP is the class of languages in NP that are accepted by machines with a polynomialbounded number of accepting paths on each input. It is an open question whether any of the containments is proper. A proper containment would imply P 6= NP. We consider the relativized class U k(n) P A , and we construct an oracle A such that P A ae UP A ae U k(n) P A ae U k(n)+1 P A ae FewP A ae NP A for every polynomial k(n) 2. Relative to a random oracle A, we prove that P A ae UP A ae U k P A ae U k+1 P A ae FewP A for every constant k 2, and we prove similar separations when k(n) is not a constant. Previously, Hartmanis and H...
Efficiently Approximable RealValued Functions
 Electronic Colloquium on Computational Complexity
, 2000
"... We consider a class, denoted APP, of realvalued functions f : f0; 1g n ! [0; 1] such that f can be approximated, to within any ffl ? 0, by a probabilistic Turing machine running in time poly(n; 1=ffl). We argue that APP can be viewed as a generalization of BPP, and show that APP contains a nat ..."
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Cited by 12 (2 self)
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We consider a class, denoted APP, of realvalued functions f : f0; 1g n ! [0; 1] such that f can be approximated, to within any ffl ? 0, by a probabilistic Turing machine running in time poly(n; 1=ffl). We argue that APP can be viewed as a generalization of BPP, and show that APP contains a natural complete problem: computing the acceptance probability of a given Boolean circuit; in contrast, no complete problems are known for BPP. We observe that all known complexitytheoretic assumptions under which BPP is easy (i.e., can be efficiently derandomized) imply that APP is easy; on the other hand, we show that BPP may be easy while APP is not, by constructing an appropriate oracle. 1 Introduction The complexity class BPP is traditionally considered a class of languages that can be efficiently decided with the help of randomness. While it does contain some natural problems, the "semantic" nature of its definition (on every input, a BPP machine must have either at least 3=4 or at...
On the Monte Carlo Boolean decision tree complexity of readonce formulae
 Random Structures Algorithms
, 1995
"... ..."
Search problems in the decision tree model
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 1995
"... We study the relative power of determinism, randomness and nondeterminism for search problems in the Boolean decision tree model. We show that the gaps between the nondeterministic, the randomized and the deterministic complexities can be arbitrary large for search problems. We also mention an inter ..."
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Cited by 11 (0 self)
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We study the relative power of determinism, randomness and nondeterminism for search problems in the Boolean decision tree model. We show that the gaps between the nondeterministic, the randomized and the deterministic complexities can be arbitrary large for search problems. We also mention an interesting connection of this model to the complexity of resolution proofs.