Pumping lemma in context-free grammar theory based on complete residuated lattice-valued logic

Residuated lattices are important algebras and have close links with various important algebras. Automata theory based on complete residuated lattice-valued logic, called L-valued automata, has been established by the second author in 2001 and 2002. As a continuation of automata theory based on complete residuated lattice-valued logic, in this paper, we mainly deal with the problem concerning pumping lemma in L-valued context-free languages (L-CFLs). As a generalization of the notion in the theory of formal grammars, the definition of L-valued context-free grammars (L-CFGs) is introduced. We also discuss a special case of L-CFGs, L-right (or left)-linear grammars, and show the equivalence between L-linear grammars and L-regular grammars. This result shows that we generalize the pumping lemma in L-valued regular languages (L-RLs) more recently established by the second author.