Polynomial analogue of Gandhi's fixed point theorem

Abstract

The problem to be solved in this paper is to construct a general method of proving whether a certain set is p-computable or not. The method is based on a polynomial analogue of the classical Gandhi's fixed point theorem. The classical Gandhi theorem uses the extension of the predicate with the help of the special operator $\Gamma^{\Omega^*}_{\Phi(x)}$ whose smallest fixed point is the $\Sigma$-set. The work uses a new type of operator - $\Delta_0^p$-operator $\Gamma_{F_{P_1^{+}},...,F_{P_n^{+}}}^{\mathfrak{M}}$, which extends predicates so that the smallest fixed point remains a p-computable set. Moreover, if in the classical Gandhi's fixed point theorem the special $\Sigma$-formula $\Phi(\overline {x})$ is used in the construction of the operator, then in the new operator, instead of a single formula, special generating families of formulas $F_ {P_1 ^ {+}},...,F_{P_n^{+}}$. This work opens up broad prospects for the application of the polynomial analogue of the Gandhi theorem in the construction of new types of terms and formulas, in the construction of new data types and programs of polynomial computational complexity in Turing complete languages.