Affiliations: Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand. [email protected] | École Normale Supérieure, 45 rue d'Ulm, 75005 Paris, France. [email protected]
Note: [] Address for correspondence: Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand
Abstract: We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a decidable and generic (i.e. a set of natural density one) set whose intersection with S is decidable. Every decidable set is almost decidable, but the converse implication is false. We prove the existence of infinitely many universal functions whose halting sets are generic (negligible, i.e. have density zero) and (not) almost decidable. One result—namely, the existence of infinitely many universal functions whose halting sets are generic (negligible) and not almost decidable—solves an open problem in [9]. We conclude with some open problems.
Keywords: Universal function, halting set, density, generic and negligible sets, almost decidable set
