Cancellation in context-free languages: enrichment by reduction

Jantzen, M., H. Petersen, Cancellation in context-free languages: enrichment by reduction, Theoretical Computer Science 127 (1994) 1499170. The following problem is shown to be decidable: Given a context-free grammar G and a string weX*, does there exist a string ueL(G) such that w is obtained from u by deleting all substrings ui that are elements of the symmetric Dyck set DT? The intersection of any two context-free languages can be obtained from only one context-free language by cancellation either with the smaller semi-Dyck set D\* c 0: or with DT itself. Also, the following is shown here for the first time: if the set EQ:= {xnXn 1 n&N} c D\* is used for this cancellation, then each recursively enumerable set can be obtained from linear context-free languages. Previous work has shown that cancellation of substrings from the semi-Dyck language D>* or from any of the former languages 07, or EQ, allows one to obtain the following from the context-free languages: any terminal Petri net language 114,163, the intersections of any context-free with a terminal Petri net language, and the nested iterated substitution thereof 1151. We generalize this characterization by showing that linear context-free languages suffice for generating terminal Petri net languages. In proving this we obtain a new closure property of the family of Petri net languages which is not shared by the context-free sets.