A Note on Structure and Looking Back Applied to the Complexity of Computable Functions

Abstract A strong connection is established between the structural and the looking back techniques for manipulating the relative complexity of computable functions and exploring the nature of subrecursive reducibilities. Looking back serves as a basis for a simple and general structural result which can be used to derive many fundamental properties of subrecursive degrees and complexity classes. For example, as has been shown by L. H. Landweber, R. J. Lipton, and E. L. Robertson (Theor. Comput. Sci., in press), there is a minimal pair of polynomial time degrees below any nonzero computable degree. In addition, the structural method is used to settle a problem concerning the enumeration properties of classes of computable functions. N P-P cannot be effectively presented by domain (i.e. by r.e. indices). However, it can be effectively presented by Δ2 indices.