Computational Complexity and Probability Constructions
Journal of the ACM (JACM)
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The general problem of finding minimal programs realizing given "program descriptions" is considered, where program descriptions may specify arbitrary program properties. The problem of finding minimal programs consistent with finite or infinite input-output lists is a special case (for infinite input-output lists, this is a variant of E.M. Gold's function identification problem; another closely related problem is the grammatical inference problem). Although most program minimization problems are not recursively solvable, they are found to be no more difficult than the problem of deciding whether any given program realizes any given description, or the problem of enumerating programs in order of nondecreasing length (whichever is harder). This result is formulated in terms of k-limiting recursive predicates and functionals, defined by repeated application of Gold's limit operator. A simple consequence is that the program minimization problem is limiting recursively solvable for finite input-output lists and 2- limiting recursively solvable for infinite input-output lists, with weak assumptions about the measure of program size. Gold regarded limiting function identification (more generally," black box" identification) as a model of inductive thought. Intuitively, iterated limiting identification might be regarded as higher-order inductive inference performed collectively by an ever growing community of lower-order inductive inference machines.