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}$}$.

We show that many classical decision problems about1-counter ω-languages, context free ω-languages, or infinitary rational relations, are Π½ -complete, hence located at the second level of the analytical hierarchy, and “highly undecidable”.In particular, the universalityproblem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and theunambiguity problem are all Π½ -complete for context-free ω-languages or for infinitary rational relations. Topological and arithmetical properties of1-counter ω-languages, context free ω-languages, or infinitary rational relations, are also highly undecidable.These very surprising results provide the first examples of highly undecidable problems about the behaviour of verysimple finite machines like 1-counter automata or 2-tapeautomata.