Results 1  10
of
22
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 ..."
Abstract

Cited by 122 (15 self)
 Add to MetaCart
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. ..."
Abstract

Cited by 32 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 29 (3 self)
 Add to MetaCart
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. ..."
Abstract

Cited by 25 (13 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 17 (7 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
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
 In 6th Annual Conference on Structure in Complexity Theory
, 1991
"... In the boolean decision tree model there is at least a linear gap between the Monte Carlo and the Las Vegas complexity of a function depending on the error probability. We prove for a large class of readonce formulae that this trivial speedup is the best what a Monte Carlo algorithm can achieve. F ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
In the boolean decision tree model there is at least a linear gap between the Monte Carlo and the Las Vegas complexity of a function depending on the error probability. We prove for a large class of readonce formulae that this trivial speedup is the best what a Monte Carlo algorithm can achieve. For every formula F belonging to that class we show that the Monte Carlo complexity of F with two sided error p is (1 \Gamma 2p)R(F ), and with one sided error p is (1 \Gamma p)R(F ), where R(F ) denotes the Las Vegas complexity of F: The result follows from a general lower bound that we derive on the Monte Carlo complexity of these formulae. This bound is analogous to the lower bound due to Saks and Wigderson on their Las Vegas complexity. Key words: boolean decision tree, randomized algorithm, readonce formula, lower bound. 1 Introduction The boolean decision tree is an extremely simple model for computing boolean functions. It computes a function by repeatedly checking input bits until ...
Decision trees with Boolean threshold queries
 Journal of Computer and System Sciences
, 1995
"... We investigate decision trees in which one is allowed to query threshold functions of subsets of variables. We are mainly interested in the case were only queries of AND and OR are allowed. This model is a generalization of the classical decision tree model. Its complexity (depth) is related to the ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
We investigate decision trees in which one is allowed to query threshold functions of subsets of variables. We are mainly interested in the case were only queries of AND and OR are allowed. This model is a generalization of the classical decision tree model. Its complexity (depth) is related to the parallel time that is required to compute Boolean functions in certain CRCW PRAM machines with only one cell of constant size. It is also related to the computation using Ethernet channel. We prove a tight lower bound of `(k log(n=k)) for the required depth of a decision tree for the thresholdk function. As a corollary of the method we also prove a tight lower bound for the 'direct sum' problem of computing simultaneously k copies of threshold2 in this model. Next, the size complexity is considered. A relation to depththree circuits is established and a lower bound is proven. Finally the relation between randomized, nondeterminism and determinism is also investigated, we show separation...