In this report, we will describe the problem of 'hard to describe' classes return sequences, that is, such that the prediction behavior of such sequences is an algorithmically undecidable task. Integer classes will be considered sequences whose generating functions are composed compositions of polynomials (with integer coefficients) and some functions f. The main question under consideration is what should be this f, so that the resulting class is hard to describe. Such functions f we called border. Evidence will be presented that several wide families of functions are boundary. All this evidence stem from one 'central' theorem, for proof of which Minsky machine simulations are used sequences, i.e. it is constructively proved that from the system functions 'polynomials plus f' (for each of the considered f) it is possible to 'build' a universal computing device.

Makeev Sergey Danilovich,  Lomonosov Moscow State University