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In a previous paper, Duchêne and Rigo introduced the notion of invariance for take-away games on heaps. Roughly speaking, these are games whose rulesets do not depend on the position. Given a sequence S of positive tuples of integers, the question of whether there exists an invariant game having S as set of P -positions is relevant. In particular, it was recently proved by Larsson et al. that if S is a pair of complementary Beatty sequences, then the answer to this question is always positive. In this paper, we show that for a fairly large set of sequences (expressed by infinite words), the answer to this question is decidable. [ABSTRACT FROM AUTHOR]

Abstract: The present paper introduces a new model for teaching randomized learners. Our new model, though based on the classical teaching dimension model, allows to study the influence of the learner’s memory size and of the presence or absence of feedback. Moreover, in the new model the order in which examples are presented may influence the teaching process. The resulting models are related to Markov decision processes, and characterizations of optimal teachers for memoryless learners with feedback and for learners with infinite memory and feedback are shown. Furthermore, in the new model it is possible to investigate new aspects of teaching like teaching from positive data only or teaching with inconsistent teachers. Characterization theorems for teachability from positive data for both ordinary teachers and inconsistent teachers with and without feedback are provided. [Copyright &y& Elsevier]