Abstract
Patterns provide a concise, syntactic way of describing a set of strings, but their expressive power comes at a price: a number of fundamental decision problems concerning (erasing) pattern languages, such as the membership problem and inclusion problem, are known to be NP-complete or even undecidable, while the decidability of the equivalence problem is still open; in learning theory, the class of pattern languages is unlearnable in models such as the distribution-free (PAC) framework (if \(\mathcal {P}/poly \ne \mathcal {NP}/poly\)). Much work on the algorithmic learning of pattern languages has thus focussed on interesting subclasses of patterns for which positive learnability results may be achieved. A natural restriction on a pattern is a bound on its variable frequency – the maximum number m such that some variable occurs exactly m times in the pattern. This paper examines the effect of limiting the variable frequency of all patterns belonging to a class \(\varPi \) on the worst-case minimum number of labelled examples needed to uniquely identify any pattern of \(\varPi \) in cooperative teaching-learning models. Two such models, the teaching dimension model as well as the preference-based teaching model, will be considered.
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Notes
- 1.
Roughly speaking, a class of languages is learnable in the limit if there is a learning algorithm such that, given any infinite sequence of all positive examples for any language L in the class, the algorithm outputs a corresponding sequence of guesses for the target language (based on a representation system for the languages in the class) that converges to a fixed representation for L; this model is due to Gold [14].
- 2.
This implies that for every pattern \(\pi \) belonging to any one of these classes, \(L(\pi )\) contains a finite set that distinguishes \(\pi \) from all \(\pi '\) in the class such that \(L(\pi ') \subset L(\pi )\) [4, Theorem 1].
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Acknowledgements
The author was supported (as RF) by the Singapore Ministry of Education Academic Research Fund grant MOE2016-T2-1-019/R146-000-234-112. I sincerely thank Fahimeh Bayeh, Sanjay Jain and Sandra Zilles for proofreading the manuscript; their numerous suggestions for corrections and improvements (such as studying the PBTD of m-quasi-regular patterns over unary alphabets) are gratefully acknowledged. Many thanks are also due to the anonymous referees of this paper for their very helpful comments and suggestions.
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Gao, Z. (2019). The Teaching Complexity of Erasing Pattern Languages with Bounded Variable Frequency. In: Hofman, P., Skrzypczak, M. (eds) Developments in Language Theory. DLT 2019. Lecture Notes in Computer Science(), vol 11647. Springer, Cham. https://doi.org/10.1007/978-3-030-24886-4_11
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