Abstract
Prior work of Gavryushkin, Khoussainov, Jain and Stephan investigated what algebraic structures can be realised in worlds given by a positive (= recursively enumerable) equivalence relation which partitions the natural numbers into infinitely many equivalence classes. The present work investigates the infinite one-one numbered recursively enumerable (r.e.) families realised by such relations and asks how the choice of the equivalence relation impacts the learnability properties of these classes when studying learnability in the limit from positive examples, also known as learning from text. For all choices of such positive equivalence relations, for each of the following entries, there are one-one numbered r.e. families which satisfy it: (a) they are behaviourally correctly learnable but not vacillatorily learnable; (b) they are explanatorily learnable but not confidently learnable; (c) they are not behaviourally correctly learnable. Furthermore, there is a positive equivalence relation which enforces that (d) every vacillatorily learnable one-one numbered family of languages closed under this equivalence relation is already explanatorily learnable and cannot be confidently learnable.
D. Belanger (as RF), Z. Gao (as RF) and S. Jain (as Co-PI), F. Stephan (as PI) have been supported by the Singapore Ministry of Education Academic Research Fund grant MOE2016-T2-1-019/R146-000-234-112 and MOE2019-T2-2-121/R146-000-304-112. Furthermore, S. Jain is supported in part by NUS grant C252-000-087-001.
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Notes
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- 2.
However, there are many natural families of languages that are learnable in the limit, such as the class of non-erasing pattern languages (see [1, Example 1]).
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Belanger, D., Gao, Z., Jain, S., Li, W., Stephan, F. (2021). Learnability and Positive Equivalence Relations. In: Leporati, A., MartÃn-Vide, C., Shapira, D., Zandron, C. (eds) Language and Automata Theory and Applications. LATA 2021. Lecture Notes in Computer Science(), vol 12638. Springer, Cham. https://doi.org/10.1007/978-3-030-68195-1_12
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