  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.
