Physical portrayal of computational complexity

TL;DR

This paper explores computational complexity through a physical lens, linking the intractability of NP problems to irreversible physical processes and energy dissipation, suggesting P is a subset of NP.

Contribution

It introduces a physical framework for understanding computational complexity, connecting entropy, energy dissipation, and irreversibility to complexity classes.

Findings

NP problems involve intractable irreversible processes.

P problems can be contracted efficiently due to stationary states.

P is a subset of NP based on physical dissipation principles.

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 many natural processes have been recognized to complete in non-polynomial time (NP). The irreversible process with three or more degrees of freedom is found intractable because, in terms of physics, flows of energy are inseparable from their driving forces. In computational terms, when solving problems in the class NP, decisions will affect subsequently available sets of decisions. The state space of a non-deterministic finite automaton is evolving due to the computation itself hence it cannot be efficiently contracted using a deterministic finite automaton that will arrive at a solution in super-polynomial time. The solution of the NP problem itself is verifiable in polynomial time (P) because the corresponding state is stationary. Likewise the class P 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 class P set of states is inherently smaller than the set of class NP. Since the computational time to contract a given set is proportional to dissipation, the computational complexity class P is a subset of NP.