Finite Identification with Positive and with Complete Data
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.
KeywordsFormal learning theory Finite identification Positive data Complete data Indexed family Anti-chains
We thank the two anonymous referees that helped us to clarify a number of issues and improve the paper.
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