Polynomial-time identification of very simple grammars from positive data

This paper concerns a subclass of simple deterministic grammars, called very simple grammars, and studies the problem of identifying the subclass in the limit from positive data. The class of very simple languages forms a proper subclass of simple deterministic languages and is incomparable to the class of regular languages. This class of languages is also known as the class of left Szilard languages of context-free grammars. After providing some properties of very simple languages, we show that the class of very simple grammars is polynomial-time identifiable in the limit from positive data in the following sense. That is, we show that there effectively exists an algorithm that, given a target very simple grammar G_* over alphabet Σ, identifies a very simple grammar G equivalent to G_* in the limit from positive data, satisfying the property that the time for updating a conjecture is bounded by O(m), and the total number of prediction errors made by the algorithm is bounded by O(n), where n is the size of G_*, m = Max{N^|Σ|+1,|Σ|³} and N is the total length of all positive data provided.