The traditional universal enumerable measure [40,45,29,16,17,41,14,30] studied in the theory of optimal inductive inference [40,41,24,23] reflects properties of a universal MTM with random input. In what follows we will simply replace the MTM by a more expressive EOM and obtain a nonenumerable measure that (1) dominates the traditional measure, (2) is ``just as computable'' in the limit as the traditional one, (3) is universal in its class. It will turn out though that this approach does not generalize to the case of GTMs. Some new definitions are in order.
Proof. We first show that one can enumerate the CEMs, then construct a universal CEM from the enumeration. Check out the differences to Levin's proofs that there is a universal discrete semimeasure and a universal c.e. semimeasure [45,28], and Li and Vitányi's presentation of the latter [30, p. 273 ff], attributed to J. Tyszkiewicz.
Without loss of generality, consider only EOMs without halt instruction and with fixed input encoding of according to Def. 2.8. Such EOMs are c.e., and correspond to an effective enumeration of all c.e. functions from to . Let denote the -th EOM in the list, and let denote its output after instructions when applied to . The following procedure filters out those that already represent CEMs, and transforms the others into representations of CEMs, such that we obtain a way of generating all and only CEMs.
FOR all DO in dovetail fashion:
START: let and and denote variable functions on . Set , and for all other . Define for undefined . Let denote a string variable.
(1) Lexicographically order and rename all with :
(2) FOR down to 1 DO:
(2.1) Systematically search for the smallest such that AND if ; set .
(3) For all satisfying , set . For all with , set . For all with , set .
If indeed represents a CEM then
each search process in (2.1) will terminate, and
will enumerate the
from below, and
will approximate the true
not necessarily from below though.
Otherwise there will be a nonterminating search
at some point, leaving from the previous loop
as a trivial CEM. Hence we can enumerate all CEMs,
and only those. Now define (compare ):