Polynomial Analogue of Gandy's Fixed Point Theorem

The paper suggests a general method for proving the fact whether a certain set is p-computable or not. The method is based on a polynomial analogue of the classical Gandy's fixed point theorem. Classical Gandy's theorem deals with the extension of a predicate through a special operator Gamma(Omega)(phi(x))* and states that the smallest fixed point of this operator is a Sigma-set. Our work uses a new type of operator which extends predicates so that the smallest fixed point remains a p-computable set. Moreover, if in the classical Gandy's fixed point theorem, the special Sigma-formula Phi((x) over bar) is used in the construction of the operator, then a new operator uses special generating families of formulas instead of a single formula. This work opens up broad prospects for the application of the polynomial analogue of Gandy's 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.

Automated summary

This paper proposes a general method for proving whether a set is p-computable, based on a polynomial analogue of Gandy's theorem. The method introduces a new type of operator that extends predicates to maintain the smallest fixed point as a p-computable set, promising broad applications in constructing new data types and programs of polynomial computational complexity.