The book is outstanding and admirable in many respects. ... is necessary reading for all kinds of readers from undergraduate students to top authorities in the field. Journal of Symbolic Logic Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and their applications of Kolmogorov complexity. The book presents a thorough treatment of the subject with a wide range of illustrative applications. Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing. It will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics. The book is self-contained in that it contains the basic requirements from mathematics and computer science. Included are also numerous problem sets, comments, source references, and hints to solutions of problems. New topics in this edition include Omega numbers, KolmogorovLoveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information and others.

An Introduction to Kolmogorov Complexity and Its Applications

In this paper, two interesting complexity classes, PP and $ \oplus {\text{P}}$, are compared with PH, the polynomial-time hierarchy. It is shown that every set in PH is polynomial-time Turing reducible to a set in PP, and PH is included in ${\text{BP}} \cdot \oplus {\text{P}}$. As a consequence of the results, it follows that ${\text{PP}} \subseteq {\text{PH}}$ (or $\oplus {\text{P}} \subseteq {\text{PH}}$) implies a collapse of PH. A stronger result is also shown: every set in PP(PH) is polynomial-time Turing reducible to a set in PP.

PP is as hard as the polynomial-time hierarchy

Publisher Summary This chapter discusses the concepts needed for defining the complexity classes. A complexity class is a set of problems of related resource-based complexity. A typical complexity class has a definition of the form—the set of problems that can be solved by an abstract machine M using O(f(n)) of resource R , where n is the size of the input. The simpler complexity classes are defined by various factors. The type of computational problem in which the most commonly used problems are decision problems. However, complexity classes can be defined based on function problems, counting problems, optimization problems, promise problems, etc. The most common model of computation is the deterministic Turing machine, but many complexity classes are based on nondeterministic Turing machines, etc.

A catalog of complexity classes

The rapidly growing field of computational social choice, at the intersection of computer science and economics, deals with the computational aspects of collective decision making. This handbook, written by thirty-six prominent members of the computational social choice community, covers the field comprehensively. Chapters devoted to each of the field's major themes offer detailed introductions. Topics include voting theory (such as the computational complexity of winner determination and manipulation in elections), fair allocation (such as algorithms for dividing divisible and indivisible goods), coalition formation (such as matching and hedonic games), and many more. Graduate students, researchers, and professionals in computer science, economics, mathematics, political science, and philosophy will benefit from this accessible and self-contained book.

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Handbook of Computational Social Choice

This is the second volume of a systematic two-volume presentation of the various areas of research in the field of structural complexity. The mathematical theory of computation has developed into a broad and rich discipline within which the theory of algorithmic complexity can be approached from several points of view. This volume is addressed to graduate students and researchers and assumes knowledge of the topics treated in the first volume but is otherwise nearly self-contained. Topics covered include vector machines, parallel computation, alternation, uniform circuit complexity, isomorphism, biimmunity and complexity cores, relativization and positive relativization, the low and high hierarchies, Kolmogorov complexity and probability classes. Numerous exercises and references are given.

Structural Complexity II

In this paper, we study the complexity of sets formed by boolean operations (union, intersection, and complement) on NP sets. These are the sets accepted by trees of hardware with NP predicates as leaves, and together these form the boolean hierarchy.We present many results about the structure of the boolean hierarchy: separation and immunity results, natural complete languages, and structural asymmetries between complementary classes.We show that in some relativized worlds the boolean hierarchy is infinite, and that for every k there is a relativized world in which the boolean hierarchy extends exactly k levels. We prove natural languages, variations of VERTEX COVER, complete for the various levels of the boolean hierarchy. We show the following structural asymmetry: though no set in the boolean hierarchy is ${\text{D}}^{\text{P}} $-immune, there is a relativized world in which the boolean hierarchy contains ${\text{coD}}^{\text{P}} $-immune sets.Thus, this paper explores the structural properties of the...

The Boolean hierarchy I: structural properties

The Boolean Hierarchy I: Structural Properties [J. Cai et al., SIAM J. Comput ., 17 (1988), pp. 1232–252] explores the structure of the boolean hierarchy, the closure of NP with respect to boolean ...

The Boolean hierarchy II: applications

By endowing usual nondeterministic Turing machines with new modes of acceptance we introduce new machines whose computational power is bounded by that of alternating Turing machines making only one alternation. The polynomial time classes of these machines are exactly the levels of the Boolean closure of NP which can be defined in a natural way. For all these classes natural problems can be found which are proved to be ≤ m P -complete in these classes.

On the Boolean closure of NP

Consideration is given to polynomial-time machines. Among these classes are EP and PP. The authors prove P/sup EP(log)/ 25 PP, investigate the Boolean closure BC(EP) of EP, and give a relativization principle which allows them to completely separate BC(EP) in a suitable relativized world and to give simple proofs for known relativization results. Further results concerning the relationships of such classes in unrelativized and relativized worlds are given. >

Gerd Wechsung

Papers

The Boolean hierarchy I: structural properties

The Boolean hierarchy II: applications

On the Boolean closure of NP

A survey on counting classes

Embedding ladders and caterpillars into the hypercube