The set of minimal indices of a G#del numbering ' is deøned as MIN' = fe : (8i ! e)[' i 6= 'e ]g. It has been known since 1972 that MIN' jT ; 00 , but beyond this MIN' has remained mostly uninvestigated. This thesis collects the scarce results on MIN' from the literature and adds some new observations including that MIN' is autoreducible, but neither regressive nor (1; 2)- computable. We also study several variants of MIN' that have been deøned in the literature like size-minimal indices, shortest descriptions, and minimal indices of decision tables. Some challenging open problems are left for the adventurous reader. 1 Introduction How long is the shortest program that solves your problem? There are at least two ways to interpret this question depending on the type of problem involved. If the program's task is to output one speciøc object, we are looking for a shortest description of that object. This interpretation is closely related to Kolmogorov complexity. Although we have sev...

Abstract. In this paper we consider learnability in some special numberings, such as Friedberg numberings, which contain all the recursively enumerable languages, but have simpler grammar equivalence problem compared to acceptable numberings. We show that every explanatorily learnable class can be learnt in some Friedberg numbering. However, such a result does not hold for behaviourally correct learning or finite learning. One can also show that some Friedberg numberings are so restrictive that all classes which can be explanatorily learnt in such Friedberg numberings have only finitely many infinite languages. We also study similar questions for several properties of learners such as consistency, conservativeness, prudence, iterativeness and non U-shaped learning. Besides Friedberg numberings, we also consider the above problems for programming systems with K-recursive grammar equivalence problem. 1

If we study the boolean algebra generated by the recursively enumerable sets we are naturally led to the difference hierarchy of Ershov [6,7,8]. If a set A can be represented as (A1 — A2) U... U (An_x — An) for r.e. sets An c An_x £... £ Ax we say that A is «-r.e. In particular A is r.e. if A is 1-r.e. A set A is called d.r.e. if it is 2-r.e.

. In any learnability setting, hypotheses are conjectured from some hypothesis space. Studied herein are the effects on learnability of the presence or absence of certain control structures in the hypothesis space. First presented are control structure characterizations of some rather specific but illustrative learnability results. Then presented are the main theorems. Each of these characterizes the invariance of a learning class over hypothesis space V (and a little more about V ) as: V has suitable instances of all denotational control structures. 1 Introduction In any learnability setting, hypotheses are conjectured from some hypothesis space, for example, in [OSW86] from general purpose programming systems, in [ZL95, Wie78] from subrecursive systems, and in [Qui92] from very simple classes of classificatory decision trees. 3 Much is known theoretically about the restrictions on learning power resulting from restricted hypothesis spaces [ZL95]. In the present paper we begin to...

ing from concrete machine models the question translates into minimal indices with respect to a numbering of the computable, partial functions. The first part of the paper tells the history of this problem collecting the known results. The second part offers some new observations, and the last part concludes with a list of open problems. We will only consider Godel numberings. A Godel numbering is an effective numbering ' of all computable partial functions such that for every effective numbering / a '-index can be computed from a /-index. We will also use Kolmogorov numberings. A Godel numbering is a Kolmogorov numbering, if there is a linearly bounded computable function that transforms /-indices into '-indices. It is well known that Kolmogorov numberings exist. Definition 1.1 Let ' be a Godel numbering. Define MIN' := fe : (8i ! e)[' i 6= ' e ]g; the set of minimal indices of '. What would happen if instead of Godel numberings arbitrary numberings of the computable, partial fun...

Abstract. This work extends studies of Angluin, Lange and Zeugmann on the dependence of learning on the hypotheses space chosen for the class. In subsequent investigations, uniformly recursively enumerable hypotheses spaces have been considered. In the present work, the following four types of learning are distinguished: class-comprising (where the learner can choose a uniformly recursively enumerable superclass as hypotheses space), class-preserving (where the learner has to choose a uniformly recursively enumerable hypotheses space of the same class), prescribed (where there must be a learner for every uniformly recursively enumerable hypotheses space of the same class) and uniform (like prescribed, but the learner has to be synthesized effectively from an index of the hypothesis space). While for explanatory learning, these four types of learnability coincide, some or all are different for other learning criteria. For example, for conservative learning, all four types are different. Several results are obtained for vacillatory and behaviourally correct learning; three of the four types can be separated, however the relation between prescribed and uniform learning remains open. It is also shown that every (not necessarily uniformly recursively enumerable) behaviourally correct learnable class has a prudent learner, that is, a learner using a hypotheses space such that it learns every set in the hypotheses space. Moreover the prudent learner can be effectively built from any learner for the class. 1

Abstract. This paper extends previous studies on learnability in non-acceptable numberings by considering the question: for which criteria which numberings are optimal, that is, for which numberings it holds that one can learn every learnable class using the given numbering as hypothesis space. Furthermore an effective version of optimality is studied as well. It is shown that the effectively optimal numberings for finite learning are just the acceptable numberings. In contrast to this, there are non-acceptable numberings which are optimal for finite learning and effectively optimal for explanatory, vacillatory and behaviourally correct learning. The numberings effectively optimal for explanatory learning are the K-acceptable numberings. A similar characterization is obtained for the numberings which are effectively optimal for vacillatory learning. Furthermore, it is studied which numberings are optimal for one and not for another criterion: among the criteria of finite, explanatory, vacillatory and behaviourally correct learning all separations can be obtained; however every numbering which is optimal for explanatory learning is also optimal for consistent learning. 1

An effectively closed set, or Π 0 1 class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of Π 0 1 classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of Π 0 1 classes and for the subclasses of decidable and of homogeneous Π 0 1 classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends. 1

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Abstract. In this paper we survey some results in inductive inference showing how learnability of a class of languages may depend on hypothesis space chosen. We also discuss results which consider how learnability is effected if one requires learning with respect to every suitable hypothesis space. Additionally, optimal hypothesis spaces, using which every learnable class is learnable, is considered. 1