Randomness and Completeness in Computational Complexity

By: Dieter van Melkebeek

Write A Review

Paperback | 13 December 2000

At a Glance

Paperback
220 Pages

Dimensions(cm)
22.86 x 15.88 x 1.27

Paperback

$84.99

or 4 interest-free payments of $21.25 with

Ships in 5 to 7 business days

This book contains a revised version of the dissertation the author wrote at the Department of Computer Science of the University of Chicago. The thesis was submitted to the Faculty of Physical Sciences in conformity with the requirements for the PhD degree in June 1999. It was honored with the 1999 ACM Doctoral Dissertation Award in May 2000. Summary Computational complexity is the study of the inherent di culty of compu- tional problems and the power of the tools we may use to solve them. It aims to describe how many resources we need to compute the solution as a function of the problem size. Typical resources include time on sequential and parallel architectures and memory space. As we want to abstract away from details of input representation and speci cs of the computer model, we end up with classes of problems that we can solve within certain robust resource bounds such as polynomial time, parallel logarithmic time, and logarithmic space. Research in complexity theory boils down to determining the relationships between these classes { inclusions and separations. In this dissertation, we focus on the role of randomness and look at various properties of hard problems in order to obtain separations. We also investigate the power of nondeterminism and alternation, as well as space versus time issues. Randomness provides a resource that seems to help in various situations.

Shipping

	Standard Shipping	Express Shipping
Metro postcodes:	$9.99	$14.95
Regional postcodes:	$9.99	$14.95
Rural postcodes:	$9.99	$14.95

Orders over $0.00 qualify for free shipping.

How to return your order

At Booktopia, we offer hassle-free returns in accordance with our returns policy. If you wish to return an item, please get in touch with Booktopia Customer Care.

Additional postage charges may be applicable.

Defective items

If there is a problem with any of the items received for your order then the Booktopia Customer Care team is ready to assist you.

For more info please visit our Help Centre.

You Can Find This Book In

Non-Fiction Computing & I.T.Computer Science Mathematical Theory of Computation Mathematics Discrete Mathematics Computer Architecture & Logic Design Computer Programming & Software Development Algorithms & Data Structures

Introduction	p. 1
Issues in Computational Complexity	p. 1
The Power of Randomness	p. 1
The Power of Guessing	p. 5
The Power of Memory	p. 6
Contributions in This Dissertation	p. 8
Simulations	p. 8
Separations	p. 9
Preliminaries	p. 13
Computational Problems	p. 13
Models of Computation, Resource Bounds, and Complexity Classes	p. 18
Turing Machines	p. 19
Uniform Families of Boolean Circuits	p. 21
Nondeterministic Turing Machines	p. 24
Alternating Turing Machines	p. 27
Randomness	p. 31
Randomized Computations	p. 32
Randomized Proof Checking	p. 35
Pseudo-Random Generators	p. 37
Reductions and Completeness	p. 42
Relativization	p. 42
Reductions	p. 44
Complete Problems	p. 46
Resource-Bounded Measure	p. 47
Motivation	p. 47
Martingales	p. 48
Properties	p. 50
Derandomizing Arthur-Merlin Games	p. 53
Introduction	p. 53
Notation	p. 56
Derandomizing Arthur-Merlin Games	p. 57
Proof of Lemma 3.3.1	p. 60
Proof of Theorem 3.3.1	p. 61
Proof of Theorem 3.3.2	p. 63
A General Framework for Derandomization	p. 64
More Applications	p. 66
Valiant-Vazirani	p. 66
Learning Circuits	p. 69
Rigid Matrices	p. 70
Universal Traversal Sequences	p. 72
Conclusion and Open Questions	p. 76
Sparseness of Complete Languages	p. 77
Introduction	p. 77
The Sparse Hard Language Problem for NP	p. 77
The Sparse Hard Language Problem for P	p. 78
Overview of this Chapter	p. 79
Deterministic Reductions	p. 81
Previous Work	p. 81
Main Result	p. 85
Generic Theorem for P	p. 92
Extension to Classes Other Than P	p. 95
Randomized Reductions	p. 98
Previous Work	p. 98
Main Result	p. 99
Randomized Generic Theorem for P	p. 106
Extension to Classes Other Than P	p. 109
Conclusion and Open Questions	p. 112
Autoreducibility of Complete Languages	p. 113
Introduction	p. 113
Definitions	p. 115
Nonautoreducibility Results	p. 115
Adaptive Autoreductions	p. 116
Nonadaptive Autoreductions	p. 121
Autoreducibility Results	p. 122
Adaptive Autoreductions	p. 123
Nonadaptive Autoreductions	p. 130
Randomized and Nonuniform Autoreductions	p. 134
Separation Results	p. 137
Conclusion and Open Questions	p. 139
The Size of Randomized Polynomial Time	p. 141
Introduction	p. 141
The Zero-One Law for BPP	p. 142
Generalization	p. 143
Conclusion and Open Questions	p. 144
The Frequency of Complete Languages	p. 145
Introduction	p. 145
Complete Languages under Nonadaptive Reductions with n^¿ Queries and a Small Span Theorem	p. 146
Complete Languages for EXP under Adaptive Reductions with n^c Queries	p. 152
Complete Languages for EXP under Nonadaptive Reductions	p. 154
Conclusion and Open Questions	p. 159
The Frequency of Autoreducible Languages	p. 161
Introduction	p. 161
Betting Games	p. 163
From Betting Games to Martingales	p. 165
Sampling Results	p. 168
Autoreducible Languages	p. 172
Adaptively Autoreducible Languages	p. 172
Nonadaptively Autoreducible Languages	p. 174
Covering Autoreducible Languages by Martingales	p. 175
Conclusion and Open Questions	p. 181
References	p. 183
Notation Index	p. 191
Subject Index	p. 195
Table of Contents provided by Publisher. All Rights Reserved.

Randomness and Completeness in Computational Complexity

At a Glance

Paperback

Shipping

How to return your order

Defective items

You Can Find This Book In

More in Computer Science

Don't Burn Anyone at the Stake Today

(and other lessons from history about living through an information crisis)

The Laws of Thought

The Quest for a Mathematical Theory of the Mind

Nexus

A Brief History of Information Networks from the Stone Age to AI

Storytelling with Data

A Data Visualization Guide for Business Professionals, 10th Anniversary Edition

Microsoft 365 Excel All-in-One For Dummies

Excel for Dummies

The AI-Driven Leader

Harnessing AI to Make Faster, Smarter Decisions

Python All-in-One For Dummies

3rd Edition

Microsoft 365 Excel For Dummies

For Dummies (Computer/Tech)

Hope in the Library

How Libraries Can Help Shape Our Future with Artificial Intelligence

Decoding Despair

How AI is Reshaping Psychiatry

A Practitioner's Guide to Advancing Behavioral Health Care with Artificial Intelligence

The AI Cybersecurity Handbook

The AI Con

How To Fight Big Tech's Hype and Create the Future We Want

AI for Business

A Guide to AI Adoption

Microsoft Power BI For Dummies

For Dummies (Computer/Tech)

Agentic AI For Dummies

For Dummies (Computer/Tech)

AI and Data Science in Healthcare 5.0

Medical Robots and Devices: New Developments and Advances

Wearable Exoskeleton Systems

Design, Control and Applications

Risk-First Software Development, Second Edition

Deliver Better Systems in a Post-Agile, AI World

Python Workout, Second Edition

200 ten-minute exercises

The Challenge of the Future

What Should We Keep from Yesterday as We Rush into Tomorrow?

REVOLUTIONIZING DRUG DEVELOPMENT

HARNESSING AI AND COMPUTATIONAL BIOLOGY

Python Crash Course

3rd Edition - A Hands-On, Project-Based Introduction to Programming

The ChatGPT Revolution

2nd Edition - Get Curious, Get Productive and Get Creative with AI

This product is categorised by