Context-Free Grammar — Part I

The articles [8], [5], [9], [1], [10], [11], [3], [2], [7], [4], and [6] provide the notation and terminology for this paper. We consider tree construction structures as extensions of 1-sorted structure as systems 〈 a carrier, rules 〉, where the carrier is a set and the rules constitute a relation between the carrier and the carrier∗. Let us note that there exists a tree construction structure which is non empty and strict. We consider context-free grammars as extensions of tree construction structure as systems 〈 a carrier, an initial symbol, rules 〉, where the carrier is a set, the initial symbol is an element of the carrier, and the rules constitute a relation between the carrier and the carrier∗. Let us observe that there exists a context-free grammar which is non empty. Let G be a tree construction structure. A symbol of G is an element of G. A string of G is an element of (the carrier of G)∗. Let D be a set and let p, q be elements of D∗. Then p a q is an element of D∗. Let D be a set. One can check that there exists an element of D∗ which is empty. Let D be a set. Then εD is an empty element of D∗. Let D be a non empty set and let d be an element of D. Then 〈d〉 is an element of D∗. Let e be an element of D. Then 〈d,e〉 is an element of D∗. In the sequel G denotes a non empty tree construction structure, s denotes a symbol of G, and n, m denote strings of G. Let us consider G, s and let n be a finite sequence. The predicate s ⇒ n is defined as follows: