Algorithmic Learning Theory by Ronald Ortner Hans Ulrich Simon & Sandra Zilles

Author:Ronald Ortner, Hans Ulrich Simon & Sandra Zilles
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

2 Preliminaries

Any unexplained recursion theoretic notation is from [17]. denotes the set of natural numbers . A language is any subset of . We let denote empty set, subset, proper subset, superset and proper superset respectively. denotes the symmetric difference of sets A and B, that is . denotes the complement of L. We let denote the cardinality of a set S. For , let respectively denote maximum and minimum of a set S, where and . We sometimes use sets of rational numbers. In this case, we use to denote the least upper bound of the rational numbers in the set S.

A finite set can be coded as . denotes the finite set A with .

denotes a fixed standard acceptable numbering [17]. denotes the i-th program in the acceptable numbering . Let . Thus, is the language/set enumerated by the i-th grammar in the acceptable programming system . Let be a Blum complexity measure [3] for the programming system. Let

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