FUNCTION OPERATORS SPANNINGTHE ARITHMETICAL AND THE POLYNOMIALHIERARCHY

A modified version of the classical μ-operator as well asthe first value operator and the operator of inverting unary functions,applied in combination with the composition of functions and startingfrom the primitive recursive functions, generate all arithmetically representablefunctions. Moreover, the nesting levels of these operatorsare closely related to the stratification of the arithmetical hierarchy.The same is shown for some further function operators known fromcomputability and complexity theory. The close relationships betweennesting levels of operators and the stratification of the hierarchy alsohold for suitable restrictions of the operators with respect to the polynomialhierarchy if one starts with the polynomial-time computablefunctions. It follows that questions around P vs. NP and NP vs. coNPcan equivalently be expressed by closure properties of function classesunder these operators. The polytime version of the first value operatorcan be used to establish hierarchies between certain consecutive levelswithin the polynomial hierarchy of functions, which are related togeneralizations of the Boolean hierarchies over the classes Σp