Some extensions are considered of Gold's influential model of language learning by machine from positive data. Studied are criteria of successful learning featuring convergence in the limit to vacillation between several alternative correct grammars. The main theorem of this paper is that there are classes of languages that can be learned if convergence in the limit to up to (n+1) exactly correct grammars is allowed but which cannot be learned if convergence in the limit is to no more than n grammars, where the no more than n grammars can each make finitely many mistakes. This contrasts sharply with results of Barzdin and Podnieks and, later, Case and Smith, for learnability from both positive and negative data. A subset principle from a 1980 paper of Angluin is extended to the vacillatory and other criteria of this paper. This principle, provides a necessary condition for circumventing overgeneralization in learning from positive data. It is applied to prove another theorem to the eff...

. Overregularization seen in child language learning, re verb tense constructs, involves abandoning correct behaviors for incorrect ones and later reverting to correct behaviors. Quite a number of other child development phenomena also follow this U-shaped form of learning, unlearning, and relearning. A decisive learner doesn't do this and, in general, never abandons an hypothesis H for an inequivalent one where it later conjectures an hypothesis equivalent to H. The present paper shows that decisiveness is a real restriction on Gold's model of iteratively (or in the limit) learning of grammars for languages from positive data. This suggests that natural U-shaped learning curves may not be a mere accident of evolutionary genetic algorithms, but may be necessary for learning. The result also solves an open problem. Second-time decisive learners conjecture each of their hypotheses for a language at most twice. By contrast, they are shown not to restrict Gold's model of lea...

this paper we assume, without loss of generality, that for all oe ` ø , [M(oe) 6=?] ) [M(ø) 6=?].

A team of learning machines is essentially a multiset of learning machines.

We extend Angluin's (1980) theorem to characterize identifiability of indexed families of r.e. languages, as opposed to indexed families of recursive languages. We also prove some variants characterizing conservativity and two other similar restrictions, paralleling Zeugmann, Lange, and Kapur's (1992, 1995) results for indexed families of recursive languages. 1 Introduction A significant portion of the work of recent years in the field of inductive inference of formal languages, as initiated by Gold 1967, stems from Angluin's (1980b) theorem, which characterizes when an indexed family of recursive languages is identifiable in the limit from positive data in the sense of Gold. Up until around 1980, a prevalent view had been that inductive inference from positive data is too weak to be of much theoretical interest. This misconception was due to the negative result in Gold's original paper, which says that any class of languages that contains every finite language and at least one infini...

An index for an r.e. class of languages (by definition) is a procedure which generates a sequence of grammars defining the class. An index for an indexed family of languages (by definition) is a procedure which generates a sequence of decision procedures defining the family. Studied is the metaproblem of synthesizing from indices for r.e. classes and for indexed families of languages various kinds of language-learners for the corresponding classes or families indexed. Many positive results, as well as some negative results, are presented regarding the existence of such synthesizers. The negative results essentially provide lower bounds for the positive results. The proofs of some of the positive results yield, as pleasant corollaries, subset-principle or tell-tale style characterizations for the learnability of the corresponding classes or families indexed. For example, the indexed families of recursive languages that can be behaviorally correctly identified from positive data are surprisingly characterized by Angluin’s (1980b) Condition 2 (the subset principle for circumventing overgeneralization). 1

The present paper considers the effects of introducing inaccuracies in a learner’s environ-ment in Gold’s learning model of identification in the limit. Three kinds of inaccuracies are considered: presence of spurious data is modeled as learning from a noisy environment, miss-ing data is modeled as learning from incomplete environment, and the presence of a mixture of both spurious and missing data is modeled as learning from imperfect environment. Two learning domains are considered, namely, identification of programs from graphs of computable functions and identification of grammars from positive data about recursively enumerable languages. Many hierarchies and tradeoffs resulting from the interplay between the number of errors allowed in the final hypotheses, the number of inaccuracies in the data, the types of inaccuracies, and the type of success criteria are derived. An interesting result is that in the context of function learning, incomplete data is strictly worse for learning than noisy data. 1

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

This paper deals with two problems: 1) what makes languages to be learnable in the limit by natural strategies of varying hardness; 2) what makes classes of languages to be the hardest ones to learn. To quantify hardness of learning, we use intrinsic complexity based on reductions between learning problems. Two types of reductions are considered: weak reductions mapping texts (representations of languages) to texts, and strong reductions mapping languages to languages. For both types of reductions, characterizations of complete (hardest) classes in terms of their algorithmic and topological potentials have been obtained. To characterize the strong complete degree, we discovered a new and natural complete class capable of "coding" any learning problem using density of the set of rational numbers. We have also discovered and characterized rich hierarchies of degrees of complexity based on "core" natural learning problems. The classes in these hierarchies contain "mul...

In their pioneering work, Mukouchi and Arikawa modeled a learning situation in which the learner is expected to refute texts which are not representative of L, the class of languages being identified. Lange and Watson extended this model to consider justified refutation in which the learner is expected to refute texts only if it contains a finite sample unrepresentative of the class L. Both the above studies were in the context of indexed families of recursive languages. We extend this study in two directions. Firstly, we consider general classes of recursively enumerable languages. Secondly, we allow the machine to either identify or refute the unrepresentative texts (respectively, texts containing finite unrepresentative samples). We observe some surprising differences between our results and the results obtained for learning indexed families by Lange and Watson. 1