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 ΓΦ(x)Ω∗ and s...

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Vydáno v:Mathematics (Basel) Ročník 9; číslo 17; s. 2102
Hlavní autoři: Goncharov, Sergey, Nechesov, Andrey
Médium: Journal Article
Jazyk:angličtina
Vydáno: Basel MDPI AG 01.09.2021
Témata:
Fixed points (mathematics)
Gandy’s fixed point theorem
Mathematics
Operators (mathematics)
p-computability
polynomial computability
polynomial operators
Polynomials
semantic programming
Theorems
Δ0p
ISSN:2227-7390, 2227-7390
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Shrnutí: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.
Bibliografie:ObjectType-Article-1
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ISSN:2227-7390
2227-7390
DOI:10.3390/math9172102