The Equivalence Problem for LL- and LR-Regular Grammars

Questions whether or not two grammars belonging to a family of grammars generate the same language have extensively been studied in the literature. These problems are called equivalence problems and if there exists an algorithm which for each pair of grammars of this family gives an answer to this question then the equivalence problem for this family of grammars is said to be decidable. Otherwise the problem is said to be undecidable. For example, the equivalence problem for the family of regular grammars is decidable. On the other hand, the equivalence problem for the family of context-free grammars is known to be undecidable. The equivalence problem ~s open for various classes of grammars which generate deterministic languages. For simple deterministic and LL(k) grammars the problem has been solved. In this paper we study the equivalence problem for the class of LLregular grammars and languages. The class of LL-regular grammars is obtained from the class of LL(k) grammars by allowing regular look-ahead instead of finite look-ahead, cf. Jarzabek and Krawczyk 18], Nijholt II0,11,12] and Poplawski ]16] for results on LL-regular grammars and languages. The class of LL(k) grammars is properly included in the class of LL-regular gra~m~ars and the class of LL(k) languages is properly included in the class of LL-regular languages. Contrary to the other families of languages which have been studied from the point of view of the equivalence problem, the class of LL-regular languages contains languages which are not deterministic. It will be shown that the equivalence problem for LL-regular grammars is decidable. Apart from extending the known result for LL(k) grammar equivalence to LLregular grammar equivalence, we obtain an alternative proof of the decidability of