Identification of grammars (r. e. indices) for recursively enumerable languages from positive data by algorithmic devices is a well studied problem in learning theory. The present paper considers identification of r. e. languages by machines that have access to membership oracles for noncomputable sets. It is shown that for any set A there exists another set B such that the collections of r. e. languages that can be identified by machines with access to a membership oracle for B is strictly larger than the collections of r. e. languages that can be identified by machines with access to a membership oracle for A. In other words, there is no maximal inference degree for language identification.

Inductive inference considers two types of queries: Queries to a teacher about the function to be learned and queries to a non-recursive oracle. This paper combines these two types --- it considers three basic models of queries to a teacher, namely QEX[Succ], QEX[!] and QEX[+], together with membership queries to some oracle. The results for these three models of queryinference are very similar: If an oracle is already omniscient for query-inference, then it is already omniscient for EX. There is an oracle of trivial EX-degree, which allows nontrivial query-inference. Furthermore, queries to a teacher can not overcome differences between oracles and the query-inference degrees are a proper refinement of the EX-degrees. 1 Introduction One famous example of learning via queries to a teacher is the game Mastermind. The teacher first selects the code -- a quadruple of colours -- that should be learned. Then the learner tries to figure out the code. In each round, the learner makes one gue...

Empirical studies of multitask learning provide some evidence that the performance of a learning system on its intended targets improves by presenting to the learning system related tasks, also called contexts, as additional input. Angluin, Gasarch, and Smith, as well as Kinber, Smith, Velauthapillai, and Wiehagen have provided mathematical justification for this phenomenon in the inductive inference framework. However, their proofs rely heavily on self-referential coding tricks, that is, they directly code the solution of the learning problem into the context. Fulk has shown that for the Ex- and Bc-anomaly hierarchies, such results, which rely on self-referential coding tricks, may not hold robustly. In this work we analyze robust versions of learning aided by context and show that --- in contrast to Fulk's result above --- the robust versions of This work was carried out while J. Case, S. Jain, M. Ott, and F. Stephan were visiting the School of Computer Science and Engineering at ...

A learner noisily infers a function or set, if every correct item is presented infinitely often while in addition some incorrect data ("noise") is presented a finite number of times. It is shown that learning from a noisy informant is equal to finite learning with K-oracle from a usual informant. This result has several variants for learning from text and using different oracles. Furthermore, partial identification of all r.e. sets can cope also with noisy input.

We characterize FIN-, EX- and BC-learning, as well as the corresponding notions of team learning, in terms of isolated branches on uniformly strongly recursive sequences of trees. Further, the more restrictive models of FIN-learning and strong-monotonic BC-learning can be characterized in terms of isolated branches on a single tree. We discuss learning with additional information where the learner receives an index for a strongly recursive tree such that the function to be learned is isolated on this tree. We show that EX-learning with this type of additional information is strictly more powerful than EX-learning. 1 Introduction Inductive inference [1, 2, 4, 6, 10] deals with learning classes of recursive functions in the limit under certain convergence constraints. The most general setting is that of behaviorally correct learning (BC): for each prefix f(0)f(1) : : : f(n) of the recursive function f , the learner guesses a program for f ; the learner succeeds if Universit¨at Heidel...

We investigate the model recBC of learning of r.e. sets, where changes in hypotheses only count when there is an extensional difference. We study the learnability of collections that are uniformly r.e. We prove that, in contrast with the case of uniformly recursive collections, identifiability does not imply recursive BC-identifiability. This answers a question of D. de Jongh. In contrast to the model of recursive identifiability, we prove that the BC-model separates the notions of finite thickness and finite elasticity. 1 Introduction In this paper we consider a model of learning where two hypotheses about the data under consideration are considered equal when they denote the same object, i.e. when they are extensionally the same. This model was first defined for identification of functions in Feldman [6], Barzdin [3]. The first reference for this model in the context of set learning (learning from text) seems to be Osherson and Weinstein [14]. The model, and similar ones, ha...

The present work investigates Gold style algorithmic learning from input-output examples whereby the learner has access to oracles as additional information. Furthermore this access has to be robust, that means that a single learning algorithm has to succeed with every oracle which meets a given specification. The first main result considers oracles of the same Turing degree: Robust learning with any oracle from a given degree does not achieve more than learning without any additional information. The further work considers learning from function oracles which describe the whole class of...

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Inductive inference is a recursion-theoretic theory of learning, first developed by E. M. Gold (1967). This paper surveys developments in probabilistic inductive inference. We mainly focus on finite inference of recursive functions, since this simple paradigm has produced the most interesting (and most complex) results. 1

. Let BC be the model of behaviourally correct function learning as introduced by Barzdins [4] and Case and Smith [8]. We introduce a mind change hierarchy for BC, counting the number of extensional differences in the hypotheses of a learner. We compare the resulting models BCn to models from the literature and discuss confidence, team learning, and finitely defective hypotheses. Among other things, we prove that there is a tradeoff between the number of semantic mind changes and the number of anomalies in the hypotheses. We also discuss consequences for language learning. In particular we show that, in contrast to the case of function learning, the family of classes that are confidently BC-learnable from text is not closed under finite unions. Keywords. Models of grammar induction, inductive inference, behaviourally correct learning. 1 Introduction Gold [10] introduced an abstract model of learning computable functions, where a learner receives increasing amounts of data ...