Polynomial Analogue of Gandy’s Fixed Point Theorem

Sergey Goncharov,Andrey Nechesov

doi:10.3390/math9172102

Abstract

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 ΓΦ(x)Ω∗ and states that the smallest fixed point of this operator is a Σ-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 Σ-formula Φ(x¯) 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.

Highlights

In both mathematics and programming, we are increasingly confronted with inductively given constructs
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
Using the induction by complexity r (l ) we show that t( P1 (l )) ≤ 25 · C · r (l ) · |l | p, where the constant C is the maximum for all constants that participate in the splitting function R(l ), in functions γi and in the algorithm for checking the truth of the formula φem (l1, . . . , lnm )

Summary

Introduction

In both mathematics and programming, we are increasingly confronted with inductively given constructs These constructs can be, for example, new types of terms and formulas in logic or programs and new data types in high-level programming languages that are inductively defined using basic tools. All these inductively generated sets can be viewed as the smallest fixed points of a suitable operator. We just talk about the construction of a ∆0 -operator with the smallest fixed point being a p-computable set, which allows us to consider many inductive formulas definable constructions as some polynomially computable set

P-Computability

Word Splitting

Fixed Points of Monotone Locally Finite Operators

A Polynomial Analogue of Gandy’s Theorem p

Corollaries and Applications p

10. Conclusions