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The Nature of Computation$
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Cristopher Moore and Stephan Mertens

Print publication date: 2011

Print ISBN-13: 9780199233212

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199233212.001.0001

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The Deep Question: P vs. NP

The Deep Question: P vs. NP

Chapter:
(p.173) Chapter 6 The Deep Question: P vs. NP
Source:
The Nature of Computation
Author(s):

Cristopher Moore

Stephan Mertens

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199233212.003.0006

This chapter considers the consequences of the possibility that P is equal to NP. It examines why it is extremely difficult to prove that P is not equal to NP by focusing on a set of ‘metatheorems’ which show the inadequacy of most of the proof techniques that have been proposed thus far. It also looks at the internal structure of NP and highlights its subclasses that capture various types of mathematical proof. The chapter demonstrates that, unless P is equal to NP, problems float in limbo between P and NP-completeness. Moreover, it describes several situations in which lower bounds can be proven and shows that a problem is outside a certain complexity class, or that one class is strictly more powerful than another. It also proves that polynomial time is less powerful than exponential time. Finally, it explores loopholes that might exist in the P vs. NP question and explains why it will eventually be resolved.

Keywords:   problems, P, NP, metatheorems, mathematical proof, NP-completeness, lower bounds, polynomial time, exponential time

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