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

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.Comment: 10 pages, 2 figure

About set-theoretic properties of one-way functions

We investigate the problem of cryptanalysis as a problem belonging to the class NP. A class of problems UF is defined for which the time constructing any feasible solution is polynomial. The properties of the problems of NP, which may be one-way functions, are established.Comment: 5 page

A class of problems of NP to be worth to search an efficient solving algorithm

We examine possibility to design an efficient solving algorithm for problems of the class \np. It is introduced a classification of \np problems by the property that a partial solution of size $k$ can be extended into a partial solution of size $k+1$ in polynomial time. It is defined an unique class problems to be worth to search an efficient solving algorithm. The problems, which are outside of this class, are inherently exponential. We show that the Hamiltonian cycle problem is inherently exponential.Comment: 9 pages, 1 figure