Abstract
We study the differences between finite identifiability of recursive languages with positive and with complete data. In finite families the difference lies exactly in the fact that for positive identification the families need to be anti-chains, while in the infinite case it is less simple, being an anti-chain is no longer a sufficient condition. We also study maximal learnable families, identifiable families with no proper extension which can be identified. We show that these often though not always exist with positive identification, but that with complete data there are no maximal learnable families at all. We also investigate a conjecture of ours, namely that each positively identifiable family has either finitely many or uncountably many maximal noneffectively positively identifiable extensions. We verify this conjecture for the restricted case of families of equinumerous finite languages.
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Change history
27 August 2019
By mistake the originally published version of this chapter did not include the acknowledgement text. This has been corrected so that the updated version of the chapter now contains the following acknowledgement: We want to thank Sebastiaan A. Terwijn for his assistance in the methods of the proof of Theorem 11. We thank the two anonymous referees that helped us to clarify a number of issues and improve the paper.
Notes
- 1.
We write \(F_n\) for the finite set with canonical index n.
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Acknowledgment
We want to thank Sebastiaan A. Terwijn for his assistance in the methods of the proof of Theorem 11. We thank the two anonymous referees that helped us to clarify a number of issues and improve the paper.
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de Jongh, D., Vargas-Sandoval, A.L. (2019). Finite Identification with Positive and with Complete Data. In: Silva, A., Staton, S., Sutton, P., Umbach, C. (eds) Language, Logic, and Computation. TbiLLC 2018. Lecture Notes in Computer Science(), vol 11456. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59565-7_3
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DOI: https://doi.org/10.1007/978-3-662-59565-7_3
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