Enumerations including laconic enumerators

We show that it is possible, for every machine universal for Kolmogorov complexity, to enumerate the lexicographically least description of a length n string in O(n) attempts. In contrast to this positive result for strings, we find that, in any Kolmogorov numbering, no enumerator of nontrivial size can generate a list containing the minimal index of a given partial-computable function. One cannot even achieve a laconic enumerator for nearly-minimal indices of partial-computable functions. 1 Short list approximations for minimal programs No effective algorithm exists which computes shortest descriptions for strings, let alone lexicographically least descriptions. Such an algorithm would contradict the well-known fact that Kolmogorov complexity is not computable [11]. This paper investigates the extent to which one can effectively enumerate a “short” list of candidate indices which includes the lexicographically minimal program for a given string or a function. Definition 1. An enumerator is an algorithm which takes an integer input and, over time, enumerates a list of integers. For an enumerator f , we let f(e) denote the set of all elements which f eventually enumerates on input e. Enumerators with non-trivial list sizes (i.e., of size much smaller than the length of the string x) fail to list-approximate Kolmogorov complexity. ∗Supported in part by NUS grant numbers R146-000-181-112, R252-000-534-112 and C252-000-087-001. †Supported in part by MOE grant number R-252-000-560-112.