Identification with Probability One of Stochastic Deterministic Linear Languages

Learning context-free grammars is generally considered a very hard task. This is even more the case when learning has to be done from positive examples only. In this context one possibility is to learn stochastic context-free grammars, by making the implicit assumption that the distribution of the examples is given by such an object. Nevertheless this is still a hard task for which no algorithm is known. We use recent results to introduce a proper subclass of linear grammars, called deterministic linear grammars, for which we prove that a small canonical form can be found. This has been a successful condition for a learning algorithm to be possible. We propose an algorithm for this class of grammars and we prove that our algorithm works in polynomial time, and structurally converges to the target in the paradigm of identification in the limit with probability 1. Although this does not ensure that only a polynomial size sample is necessary for learning to be possible, we argue that the criterion means that no added (hidden) bias is present.