Detection of Generative Ambiguities in Context-Free Mechanical Languages

By a co~tezt:}?ee mechanical language we mean one specified recursivety by a finite set of replacement ruled (a production system) of the form @~} ::= ~,: where each ~p~ is a finite concatenation of characters from a base alphabet (often ealled terminal characters, or vocabulary) and the alphabet of the @,} (often called nonterminal characters, or syntactic types) such tha t no ~b~ is the nullstring. Such a system generates strings of characters f rom the base alphabet called words of the language <cq); the set of generated words, we call the extent of the language @~} (called the support of the language by Sehfitzenberger and Choresky [4]). The bracketing characters "(" and "}" indicate the extent. Each finite program of sequenced applications of these replacement rules beginning with @1} and leading to a word of the language is called a derivation of that word. A product ion system in which some {c~i) never appear in the derivation of a word we will call #enerativdy inadmissible. With the requirement that the recursive generator (i.e. the production system) be generatively admissible, our definition of context-free language agrees in extent with that of Schfitzenberger and Chomsky. We are concerned here with the detection of a type of syntactic ambiguity of words of a context-free language. We call a word generativel~l ambiguous (als0 called structurally ambiguous) with respect to the production system if it has essentially more than one derivation. All context-free languages are decidable. B y this we mean that for each one we can construct a processor called a recognizer which, for any string from the terminal alphabet, can decide in a finite time whether tha t string does or does not belong to the language. I t will be easy to see below that any generatively admissible context-free language possesses an ordinal generator (one which produces all words of the language in sequence) which is approximately monotonic; by this we mean that there is a recursively increasing sequence of natural hum-