Structural Complexity I

This first of the two-volume presentation of structural complexity introduces the background necessary for the understanding of complexity theory. It is written for undergraduate students with basic knowledge in Formal Language Theory and contains about 140 exercises and many bibliographical remarks. This second, corrected edition includes an appendix with recent results and an update of the references.

Klappentext

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.

Inhalt
1 Introduction.- 2 Basic Notions About Models of Computation.- 1.1 Introduction.- 1.2 Alphabets, Words, Sets, and Classes.- 1.3 Inclusion Modulo Finite Variants.- 1.4 Boolean Formulas.- 1.5 Models of Computation: Finite Automata.- 1.6 Models of Computation: Taring Machines.- 1.7 Models of Computation: Nondeterministic Turing Machines.- 1.8 Models of Computation: Oracle Turing Machines.- 1.9 Bibliographical Remarks.- 3 Time and Space Bounded Computations.- 2.1 Introduction.- 2.2 Orders of Magnitude.- 2.3 Running Time and Work Space of Turing Machines.- 2.4 Time and Space Constructibility.- 2.5 Bounding Resources: Basic Definitions and Relationships.- 2.6 Bibliographical Remarks.- 4 Central Complexity Classes.- 3.1 Introduction.- 3.2 Definitions, Properties, and Examples.- 3.3 Computing Functions: Invertibility and Honesty.- 3.4 Polynomial Time Many-one Reducibility.- 3.5 Natural NP-complete Sets.- 3.6 Natural PSPACE-complete Sets.- 3.7 Padding Arguments.- 3.8 Space Bounded Reducibility.- 3.9 Exercises.- 3.10 Bibliographical Remarks.- 5 Time Bounded Turing Reducibilities.- 4.1 Introduction.- 4.2 Polynomial Time Turing Reducibility: Relativized Classes.- 4.3 Tally and Sparse Sets in NP.- 4.4 Strong Nondeterministic Polynomial Time Reducibility.- 4.5 Self-Reducibility.- 4.6 Exercises.- 4.7 Bibliographical Remarks.- 6 Nonuniform Complexity.- 5.1 Introduction.- 5.2 Classes Defined by Advice Functions.- 5.3 Boolean Circuit Complexity.- 5.4 Turing Machines and Boolean Circuits.- 5.5 Polynomial Advice.- 5.6 Logarithmic Advice.- 5.7 Self-Producible Circuits.- 5.8 A Lower Bound to the Circuit Size of Boolean Functions.- 5.9 Other Nonuniform Complexity Measures.- 5.10 Exercises.- 5.11 Bibliographical Remarks.- 7 Probabilistic Algorithms.- 6.1 Introduction.- 6.2 TheProbabilistic Computational Model.- 6.3 Polynomial Time Probabilistic Classes.- 6.4 Bounded Error Probability.- 6.5 Nonuniform Properties of BPP.- 6.6 Zero Error Probability.- 6.7 Exercises.- 6.8 Bibliographical Remarks.- 8 Uniform Diagonalization.- 7.1 Introduction.- 7.2 Presentability and Other Properties.- 7.3 The Main Theorem.- 7.4 Applications.- 7.5 Exercises.- 7.6 Bibliographical Remarks.- 9 The Polynomial Time Hierarchy.- 8.1 Introduction.- 8.2 Definition and Properties.- 8.3 Characterization and Consequences.- 8.4 Complete Sets and Presentability.- 8.5 BPP and the Polynomial Time Hierarchy.- 8.6 Exercises.- 8.7 Bibliographical Remarks.- References.- Appendix Complementation via Inductive Counting.- Author Index.- Symbol Index.