This is the first volume of a systematic two-volume presentation of the various areas of research on structural complexity. The theory of algorithmic complexity, a part of the mathematical theory of computation, can be approached from several points of view, one of which is the structural one. This volume is written for undergraduate students who have taken a first course in Formal Language Theory. It presents the basic concepts of structural complexity, thus providing the background necessary for the understanding of complexity theory. The second corrected edition has been extended by an appendix with recent results on nondeterministic space classes and updated with regard to the bibliographical remarks and the references.

Structural Complexity I

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

Almost everywhere high nonuniform complexity

The strong exponential hierarchy collapses

Generalized Kolmogorov Complexity and the Structure of Feasible Computations (Preliminary Report)

Computation times of NP sets of different densities

In this paper we study the computational complexity of sets of different densities in NP. We show that the deterministic computation time for sets in NP can depend on their density if and only if there is a collapse or partial collapse of the corresponding higher nondeterministic and deterministic time bonded complexity classes. We show also that for NP sets of different densities there exist complete sets of the corresponding density under polynomial time Turing reductions. Finally, we show that these results can be interpreted as results about the complexity of theorem proving and proof presentation in axiomatized mathematical systems. This interpretation relates fundamental questions about the complexity of our intellectual tools to basic structural problems about P, NP, CoNP, and PSPACE, discussed in this paper.

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Yaacov Yesha

Papers

Computation times of NP sets of different densities

Computation Times of NP Sets of Different Densities