The bound k(x,y) in limited recursion on notation places a polynomial constraint on the values computed by auxiliary functions.

The availability of such characterizations is often taken to provide additional evidence for the mathematical robustness of classes like \(\textbf{NP}\). 2 generalizes to provide a characterization of the classes which comprise the Polynomial Hierarchy. For instance, the logics \(\Sigma^1_i\) and \(\Pi^1_i\) uniformly capture the complexity classes \(\Sigma^P_i\) and \(\Pi^P_i\) (where \(\mathsf{SO}\exists = \Sigma^1_1\) ). e. full second-order logic) captures \(\textbf{PH}\) itself. On the othe