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Function operators spanning the arithmetical and the polynomial hierarchy

Published online by Cambridge University Press:  05 October 2010

Armin Hemmerling
Armin Hemmerling*
Affiliation:
Ernst-Moritz-Arndt–Universität Greifswald, Institut für Mathematik und Informatik. Walther-Rathenau-Str. 47, 17487 Greifswald, Germany; hemmerli@uni-greifswald.de
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Abstract

A modified version of the classical µ-operator as well as the first value operator and the operator of inverting unary functions, applied in combination with the composition of functions and starting from the primitive recursive functions, generate all arithmetically representable functions. Moreover, the nesting levels of these operators are closely related to the stratification of the arithmetical hierarchy. The same is shown for some further function operators known from computability and complexity theory. The close relationships between nesting levels of operators and the stratification of the hierarchy also hold for suitable restrictions of the operators with respect to the polynomial hierarchy if one starts with the polynomial-time computable functions. It follows that questions around P vs. NP and NP vs. coNP can equivalently be expressed by closure properties of function classes under these operators. The polytime version of the first value operator can be used to establish hierarchies between certain consecutive levels within the polynomial hierarchy of functions, which are related to generalizations of the Boolean hierarchies over the classes $\mbox{$\Sigma^p_{k}$}$.

Keywords

Arithmetical hierarchypolynomial hierarchyBoolean hierarchyP versus NPNP versus coNPfirst value operatorminimalizationinversion of functions
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 44 , Issue 3 , July 2010 , pp. 379 - 418
Copyright
© EDP Sciences, 2010

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