Polynomial Time Learnability of a Sub-class of Linear Languages

We propose some PAC like settings for a learning problem of a sub-class of linear languages, and show its polynomial time learnability in each of our settings. Here, the sub-class of linear languages is newly defined, and it includes the class of regular languages and the class of even linear languages. We show a polynomial time learning algorithm in either of the following settings with a fixed but unknown probability distribution for examples.(1) The first case is when the learner can use randomly drawn examples, membership queries, and a set of representative samples.(2) The second case is when the learner can use randomly drawn examples, membership queries, and both of the size of a grammar which can generate the target language and d. Where d is the probability such that the rarest rule in the target grammar occurs in the derivation of a randomly drawn example. In each case, for the target language Lt, the hypothesis Lhsatisfies thatPr[P(Lh Δ Lt) ≤ ε] ≥ 1 - δ for the error parameter 0 < ε ≤ 1 and the confidential parameter 0 < δ ≤ 1.