We consider a panel of experts asked to assign probabilities to events, both logically simple and complex. The events evaluated by different experts are based on overlapping sets of variables but may otherwise be distinct. The union of all the judgments will likely be probabilistic incoherent. We address the problem of revising the probability estimates of the panel so as to produce a coherent set that best represents the group's expertise.

this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld

The algorithm selection problem aims at selecting the best algorithm for a given computational problem instance according to some characteristics of the instance. In this dissertation, we first introduce some results from theoretical investigation of the algorithm selection problem. We show, by Rice's theorem, the nonexistence of an automatic algorithm selection program based only on the description of the input instance and the competing algorithms. We also describe an abstract theoretical framework of instance hardness and algorithm performance based on Kolmogorov complexity to show that algorithm selection for search is also incomputable. Driven by the theoretical results, we propose a machine learning-based inductive approach using experimental algorithmic methods and machine learning techniques to solve the algorithm selection problem. Experimentally, we have

We show the following results regarding complete sets. • NP-complete sets and PSPACE-complete sets are many-one autoreducible. • Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are many-one autoreducible. • EXP-complete sets are many-one mitotic. • NEXP-complete sets are weakly many-one mitotic. • PSPACE-complete sets are weakly Turing-mitotic. • If one-way permutations and quick pseudo-random generators exist, then NP-complete languages are m-mitotic. • If there is a tally language in NP ∩ coNP − P, then, for every ɛ> 0, NP-complete sets are not 2 n(1+ɛ)-immune. These results solve several of the open questions raised by Buhrman and Torenvliet in their 1994 survey paper on the structure of complete sets. 1

If every language in coNP has constant round interactive proof system, then the polynomialtime hierarchy collapses [BHZ87]. On the other hand, the well-known LFKN protocol gives O(n)-round interactive proof systems for all languages in coNP [LFKN92]. We consider the question whether it is possible for coNP to have interactive proof systems with polylogarithmic round complexity. We show that this is unlikely by proving that if a coNP-complete set has a polylogarithmic-round interactive proof system then the exponential-time hierarchy collapses. We also consider exponential versions of the Karp-Lipton theorem and Yap’s theorem.

We describe the properties of various notions of time-bounded Kolmogorov complexity and other connections between Kolmogorov complexity and computational complexity. 1

It is known [BHZ87] that if every language in coNP has a constant-round interactive proof system, then the polynomial hierarchy collapses. On the other hand, Lund et al. [LFKN92] have shown that #SAT, the #P-complete function that outputs the number of satisfying assignments of a Boolean for-mula, can be computed by a linear-round interactive protocol. As a consequence, the coNP-complete set SAT has a proof system with linear rounds of interaction. We show that if every set in coNP has a polylogarithmic-round interactive protocol then the expo-nential hierarchy collapses to the third level. In order to prove this, we obtain an exponential version of Yap’s result [Yap83], and improve upon an exponential version of the Karp-Lipton theorem [KL80], obtained first by Buhrman and Homer [BH92].

This paper introduces a self-training automatic algorithm selection system based on experimental methods and probabilistic learning and reasoning techniques. The system aims to select the most appropriate algorithm according to the characteristics of the input problem instance. The general methodology is described, the system framework is presented, and key research problems are identified.

This thesis studies average-case complexity theory and polynomial-time reducibilities. The issues in average-case complexity arise primarily from Cai and Selman's extension of Levin's denition of average polynomial time. We study polynomial-time reductions between distributional problems. Under strong but reasonable hypotheses we separate ordinary NP-completeness notions.

In recent years, there has been a rising interest in developing online approximation algorithms for data streams. Some of the key challenges are posed by the fact that streaming data can be read only once in a fixed order of arrival and only a limited amount of memory is available for storage. In this paper, we address the problem of approximately counting tree patterns over a stream of labeled trees (e.g., XML documents). We propose a new approximation algorithm called SketchTree that computes a synopsis of the stream in a single pass by processing each tree only once. Using a limited amount of memory, SketchTree provides approximate answers for both ordered and unordered tree pattern counts. Furthermore, we discuss a class of count queries that can be handled by SketchTree and their utility. We provide theoretical analyses to show that our algorithm has provably strong guarantees on the error bounds. Experiments on real datasets demonstrate thatSketchTree can indeed estimate tree pattern counts within 10-15 % relative error with high confidence under various situations.