Grammars with Partial Ordering of the Rules

I t is wel] known tha t context-sensitive languages can be generated by means of monotonic grammars (see Ginsburg, Greibach, 1966) i.e. grammars whose rules have the form f --~ g, where I f ] , the length of f, is not greater than I g I . Such a g rammar generates the language, but almost no grammatical structure of the language is given. When such a structure is also investigated the original definition of context-sensitive grammar, i.e. g rammar with the rules of the form x Z y ~ xwy, where I Z t = 1, I w I > 0, must be used, or, better still the explicit context form Z---~ w / x -y is sometimes preferred. This notation is to be understood as " the context-free rule Z --+ w is used, but it can be applied only in the context x y". As context-free rules enable us to introduce a grammatical structure into the language, we are also able in this way to obtain a kind of structure in a context-sensitive language. At this point we may ask what class of languages will be obtained when conditions other than only definite context limit the application of a context-free rule. Two such limitations will be studied in this paper: ( I ) A partial ordering will be introduced into the set of rules and when more rules can be applied to a string the preference must be given to the greater one. ( I I ) Each rule r is assigned the regular language ~( r ) and a rule r can only be applied to a string f if f 6 ~ ( r ) . I t can be seen tha t case I I is a generation of case I. In this paper it is proved tha t even in ease I we obtain non-context-free languages and tha t in case I I we obtain precisely the class of all context-sensitive languages.