Formalization of the class of problems solvable by a nondeterministic Turing machine

TL;DR

This paper formalizes NP problems using independence systems and shows that NP equals problems solvable without lookahead, implying alternative formulations are needed if polynomial solutions are not found.

Contribution

It introduces a mathematical model for NP problems and proves NP equals the class of problems solvable without lookahead.

Findings

NP class is identical with problems without lookahead

Effective polynomial algorithms are key to solving NP problems

Alternative problem formulations are necessary if polynomial solutions are not found

Abstract

The objective of this article is to formalize the definition of NP problems. We construct a mathematical model of discrete problems as independence systems with weighted elements. We introduce two auxiliary sets that characterize the solution of the problem: the adjoint set, which contains the elements from the original set none of which can be adjoined to the already chosen solution elements; and the residual set, in which every element can be adjoined to previously chosen solution elements. In a problem without lookahead, every adjoint set can be generated by the solution algorithm effectively, in polynomial time. The main result of the study is the assertion that the NP class is identical with the class of problems without lookahead. Hence it follows that if we fail to find an effective (polynomial-time) solution algorithm for a given problem, then we need to look for an alternative formulation of the problem in set of problems without lookahead.