Toward a Theory of Enumerations

Abstract

An attempt is made to show that there is much work in pure recursion theory which implicitly treats computational complexity of algorithmic devices which enumerate sets. The emphasis is on obtaining results which are independent of the particular model one uses for the enumeration technique and which can be obtained easily from known results and known proofs in pure recursion theory. First, it is shown that it is usually impossible to define operators on sets by examining the structure of the enumerating devices unless the same operator can be defined merely by examining the behavior of the devices. However, an example is given of an operator which can be defined by examining the structure but which cannot be obtained merely by examining the behavior. Next, an example is given of a set which cannot be enumerated quickly because there is no way of quickly obtaining large parts of it (perhaps with extraneous elements). By way of contrast, sets are constructed whose elements can be obtained rapidly in conjunction with the enumeration of a second set, but which themselves cannot be enumerated rapidly because there is no easy way to eliminate the members of the second set. Finally, it is shown how some of the elementary parts of the Hartmanis-Stearns theory can be obtained in a general setting.