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

Polynomial time machines having restricted access to an NP oracle are investigated. Restricted access means that the number of queries to the oracle is restricted or the way in which the queries are made is restricted (e.g., queries made during truth-table reductions). Very different kinds of such restrictions result in the same or comparable complexity classes. In particular, the class $P^{\text{NP}}[O(\log n)]$ can be characterized in very different ways. Furthermore, the Boolean hierarchy is generalized in such a way that it is possible to characterize $P^{\text{NP}}$ and $P^{\text{NP}}[O(\log n)]$ in terms of the generalization.

Bounded query classes

1. Mathematical Preliminaries. 2. Elements of Computability Theory. 4. The Class P. 5. The Glass NP. 6. The Complexity of Optiimzation Problems. 7. Beyond NP. 8. Space-Complexity Classes. 9. Probabiillistic. 10. Algorithms and Complexity Classes. 11. Interactivite Proof. 12. Systems. 13. Models of Parallel Computer. 14. Parallel Algorithms.

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Introduction to the theory of complexity

The strong exponential hierarchy collapses

We introduce the notion of combinatorial vote, where a group of agents (or voters) is supposed to express preferences and come to a common decision concerning a set of non-independent variables to assign. We study two key issues pertaining to combinatorial vote, namely preference representation and the automated choice of an optimal decision. For each of these issues, we briefly review the state of the art, we try to define the main problems to be solved and identify their computational complexity.

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Logical Preference Representation and Combinatorial Vote

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

Perfectionnement de la hierarchie de Hausdorff generee par NP. Les nouvelles classes permettent une exacte classification de la complexite de certains problemes de denombrement. Les classes sont caracterisees en termes de machines de Turing polynomiales non deterministes

Counting classes with finite acceptance types

The complexity classification of problems defined by restricting NP-complets problems to those instances having unique solutions requires still finer hierarchies within BC(NP) (the Boolean closure of NP) than that introduced in [Wec 85] (see also [WeWa 85], [GuWe 85], [CaHe 85] and [KoSc 85]) which will be called the Hausdorff hierarchy generated by NP. In this paper an extremely fine hierarchy within BC(NP) is proposed. The classes of this hierarchy are characterized by nondeter-ministic polynomial time Turing machines with suitably modified acceptance notions (Section 2). Complete sets for the classes of the hierarchy are presented in Section 5. The hierarchy is studied under relativizations (Sections 3 and 5). Section 4 yields more insight in the structure of the hierarchy.

Thomas Gundermann

Papers

The Boolean hierarchy I: structural properties

The Boolean hierarchy II: applications

Counting classes with finite acceptance types

Nondeterministic turing machines with modified acceptance