SOME QUESTIONS OF PHRASE STRUCTURE GRAMMARS, I

where V is a finite set of symbols (LETTERS) called DICTIONARY, and Vt is a (proper) subset of V called TERMINAL DICTIONARY. The elements of V are called TERMINAL LETTERS. The (associative and non-commutative) operation of CONCATENATION (denoted by ""*) is defined on V. From now on the sign of this operation will be omitted (instead of Χ^Ύ we shall write XY). The operation of concatenation yields the set of SEQUENCES (or WORDS) over V. The NULL-SEQUENCE, denoted by E, is also considered (for Ε we have Ε X = Χ Ε = X for every X e V). Σ is the finite set of sequences over V called INTTTAT, SEQUENCES. F is a finite set of pairs (φ, ψ), where φ and ψ are sequences over V, called RULES (of the grammar), φ is called the LEFT-SIDE of the rule and ψ its RIGHT-SIDE. To APPLY a rule f: (φ, ψ) to a sequence χ means to replace in χ the first occurrence of φ withy. If χ does not contain any occurrence of the leftside of a given rule f, we say that f cannot be applied to this x. If; by applying f to χ we obtain x' we shall write f [x] = x' or χ -* χ' and say that x directly generates x'. The sequence of words