In this paper, we obtain some refinement of representation theorems for context-free languages by using Dyck languages, insertion systems, strictly locally testable languages, and morphisms. For instance, we improved the Chomsky-Schützenberger representation theorem and show that each context-free language L can be represented in the form L=h(D ∪ R), where D is a Dyck language, R is a strictly 3-testable language, and h is a morphism. A similar representation for context-free languages can be obtained, using insertion systems of weight (3,0) and strictly 4-testable languages.

We introduce a subclass of context-free languages, called pure context-free (PCF) languages, which is generated by context-free grammars with only one type of symbol (i. e. , terminals and nonterminals are not distinguished), and consider the problem of identifying paralleled even monogenic pure context-free (pem-PCF) languages, PCF languages with restricted and enhanced features, from positive data only. In this paper we show that the ploblem of identifying the class of pem-PCF languages is reduced to the ploblem of identifying the class of monogenic PCF (mono-PCF), by decomposing each string of pem-PCF languages. Then, with its result, we show that the class of pem-PCF languages is polynomial time identifiable in the limit from positive data. Further, we refer to properties of its identification algorithm.

A subclass of context-free languages, called pure context-free languages, which is generated by context-free grammar with only one type of symbol (i.e., terminals and nonterminals are not distinguished), is introduced and the problem of identifying from positive data a restricted class of monogenic pure context-free languages (mono-PCF languages, in short) is investigated. The class of mono-PCF languages is incomparable to the class of regular languages. In this paper we show that the class of mono-PCF languages is polynomial time identifiable from positive data. That is, there is an algorithm that, given a mono-PCF language L, identifies from positive data, a grammar generating L, called a monogenic pure context-free grammar (mono-PCF grammar, in short) satisfying the property that the time for updating a conjecture is bounded by O(N3), where N is the sum of lengths of all positive data provided. This is in contrast with another result in this paper that the class of PCF languages is not identifiable in the limit from positive data.