ISRN Computational Mathematics

Volume 2012, Article ID 321372, 15 pages

http://dx.doi.org/10.5402/2012/321372

## Physical Portrayal of Computational Complexity

Department of Physics, Institute of Biotechnology and Department of Biosciences, University of Helsinki, 00014 Helsinki, Finland

Received 3 October 2011; Accepted 3 November 2011

Academic Editor: L. Pan

Copyright © 2012 Arto Annila. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process from an initial instance to the final acceptance is motivated because information requires physical representations and because many natural processes complete in nondeterministic polynomial time (*NP*). The irreversible process with three or more degrees of freedom is found intractable when, in terms of physics, flows of energy are inseparable from their driving forces. In computational terms, when solving a problem in the class *NP*, decisions among alternatives will affect subsequently available sets of decisions. Thus the state space of a nondeterministic finite automaton is evolving due to the computation itself, hence it cannot be efficiently contracted using a deterministic finite automaton. Conversely when solving problems in the class *P*, the set of states does not depend on computational history, hence it can be efficiently contracted to the accepting state by a deterministic sequence of dissipative transformations. Thus it is concluded that the state set of class *P* is inherently smaller than the state set of class *NP*. Since the computational time needed to contract a given set is proportional to dissipation, the computational complexity class *P* is a proper (strict) subset of *NP*.