Elements of Information and Numerical Complexity of Polynomial Extremal Problems

Abstract

After the famous works of Church, Godel and Turing (see [Ta 51], [AHU 76]) in 1930-th one can clearly define what the notion of potential solvability of some class of problems means. They established the existense of non-solvable (non-resolvable) problems that arise in logic and in the theory of computation in a natural way. Problem is considered algorithmicaly solvable if it can be solved by appropriate Turing machine. But the partition of all problems in two classes (solvable or nonsolvable) is a very rough classification. Further more detailed separation of solvable problems into subclasses can be done by establishing of upper bounds on computational resources necessary for obtaining their solution. Two main resources used by an algorithm is time (the number of elementary steps of the algorithm) and the memory (working memory area). We can obtain the upper bounds for the necessary resources studying an arbitrary algorithm that solves the given problem. But to get an answer for the question of how good a particular algorithm is we must find the lower bounds for computational resources, the limits that cannot be improved by the “best” algorithm among the potentially possible ones. Obtaining the nontrivial lower bounds is a very hard job in many important cases. But even in the case with no chance for lower and upper bounds to be close enough, one has a possibility to make a comparative classification of the problem by associating it with other problems from a wide enough class. So we have two types of estimates of computational complexity: absolute and relative. As we deal with polynomial functions it is natural for us to consider examples linked with the first order theory of solvability for real numbers.