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Enumerations of the Kolmogorov Function
"... We consider the hardness of enumerating k possible values for the Kolmogorov complexity function C(x) so that one of them is correct. We show several results including Any computable enumerator for C(x) must enumerate n) possibilities. If a kenumerator (fixed k) for C is reducible to an r.e. ..."
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We consider the hardness of enumerating k possible values for the Kolmogorov complexity function C(x) so that one of them is correct. We show several results including Any computable enumerator for C(x) must enumerate n) possibilities. If a kenumerator (fixed k) for C is reducible to an r.e. set A then A is Turingequivalent to the halting problem. Every nonrecursive set is not weaktruthtable reducible to some 2enumerator for C(x) or any other recursivelybounded function. The timebounded enumeration question gives a new characterization of the class symP dened by Russell and Sundaram. Enumerating O(log n) values of the spacebounded Kolmogorov function remains hard for PSPACE.
The complexity of joint computation
, 2012
"... Joint computation is the ubiquitous scenario in which a computer is presented with not one, but many computational tasks to perform. A fundamental question arises: when can we cleverly combine computations, to perform them with greater efficiency or reliability than by tackling them separately? This ..."
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Joint computation is the ubiquitous scenario in which a computer is presented with not one, but many computational tasks to perform. A fundamental question arises: when can we cleverly combine computations, to perform them with greater efficiency or reliability than by tackling them separately? This thesis investigates the power and, especially, the limits of efficient joint computation, in several computational models: query algorithms, circuits, and Turing machines. We significantly improve and extend past results on limits to efficient joint computation for multiple independent tasks; identify barriers to progress towards better circuit lower bounds for multipleoutput operators; and begin an original line of inquiry into the complexity of joint computation. In more detail, we make contributions in the following areas: Improved direct product theorems for randomized query complexity: The "direct product problem" seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following direct product theorem (DPT) for query complexity: if every Tquery algorithm has success proba
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"... In computer science, often computational problems can be transformed into communication problems. The most illustrative example is probably the evaluation of a function via a Boolean formula. The correspondence to an equivalent communication ..."
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In computer science, often computational problems can be transformed into communication problems. The most illustrative example is probably the evaluation of a function via a Boolean formula. The correspondence to an equivalent communication
Choosing, Agreeing, and Eliminating in Communication Complexity
"... We consider several questions inspired by the directsum problem in (twoparty) communication complexity. In all questions, there are k fixed Boolean functions f1,..., fk and Alice and Bob have k inputs x1,..., xk and y1,..., yk, respectively. In the eliminate problem, Alice and Bob should output a ..."
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We consider several questions inspired by the directsum problem in (twoparty) communication complexity. In all questions, there are k fixed Boolean functions f1,..., fk and Alice and Bob have k inputs x1,..., xk and y1,..., yk, respectively. In the eliminate problem, Alice and Bob should output a vector σ1,..., σk such that fi(xi) ̸ = σi for at least one i (i.e., their goal is to eliminate one of the 2 k output vectors); in choose, Alice and Bob should return (i, fi(xi, yi)) and in agree they should return fi(xi, yi), for some i. The question, in each of the three cases, is whether one can do better than solving one (say, the first) instance. We study these three problems and prove various positive and negative results.
Communication Complexity and Data Compression
"... A Result Of Ahlswede and Cai for the 2party communication complexity of set intersection is generalized to a multiparty model. There are relations to several areas as to the directsum conjecture and amortized complexity in computational complexity or interactive communication in information theor ..."
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A Result Of Ahlswede and Cai for the 2party communication complexity of set intersection is generalized to a multiparty model. There are relations to several areas as to the directsum conjecture and amortized complexity in computational complexity or interactive communication in information theory as well as to wireless sensor networks and even quantum communication. The aim of the paper is mostly to survey these different applications and to draw the attention of researchers in one area to the results and applications in other areas.