Some Classifications of Context-Free Languages

The basic definitions and notations of the theory of context-free grammars and languages (briefly grammars and languages) used in this paper are as in Ginsburg (1966). The classification of languages L according to the minimal number of variables in grammars for L was studied in Gruska (1967). In this paper some other classifications of grammars and languages are investigated. They are chosen in such a way as to characterize some aspects of our intuitive notion about complexity (of the description) of grammars and languages and their intrinsic structure. The classifications of languages are indicated by those of grammars. The intrinsic structure of a grammar G is characterized by the number and by the depth of the grammatical levels of G. A grammatical level Go of a grammar G is a maximal set of productions of G the left-side symbols of which are mutually dependent. The basic concepts of grammatical levels and classifications of grammars and languages are given in Sections 2 and 3. Only such classifications K are considered here, wherein for every grammar G (language L) K(G) (K(L)) is an integer. In this paper only nonnegative integers will be considered. A classification K is said to be connected in an alphabet Z if for every integer n there is a language L c Z* such that K(L) = n. Sections 4 to 6 provide the proofs that the classifications according to the number of variables, the number of productions, the number of grammatical levels, the number of non-elementary grammatical levels (that is, the grammatical levels with at least two variables) and the maximal depth of grammatical levels (that is, according to the maximal number of variables in grammatical levels) are connected in any alphabet with