Abstract
Taking intuitionistic fuzzy sets as the structures of truth values, we propose the notions of intuitionistic fuzzy context-free grammars (IFCFGs, for short) and pushdown automata with final states (IFPDAs). Then we investigate algebraic characterization of intuitionistic fuzzy recognizable languages including decomposition form and representation theorem. By introducing the generalized subset construction method, we show that IFPDAs are equivalent to their simple form, called intuitionistic fuzzy simple pushdown automata (IF-SPDAs), and then prove that intuitionistic fuzzy recognizable step functions are the same as those accepted by IFPDAs. It follows that intuitionistic fuzzy pushdown automata with empty stack and IFPDAs are equivalent by classical automata theory. Additionally, we introduce the concepts of Chomsky normal form grammar (IFCNF) and Greibach normal form grammar (IFGNF) based on intuitionistic fuzzy sets. The results of our study indicate that intuitionistic fuzzy context-free languages generated by IFCFGs are equivalent to those generated by IFGNFs and IFCNFs, respectively, and they are also equivalent to intuitionistic fuzzy recognizable step functions. Then some operations on the family of intuitionistic fuzzy context-free languages are discussed. Finally, pumping lemma for intuitionistic fuzzy context-free languages is investigated.
Highlights
Intuitionistic fuzzy set (IFS) introduced by Atanassov [1,2,3], which emerges from the simultaneous consideration of the degrees of membership and nonmembership with a degree of hesitancy, has been found to be highly useful in dealing with problems with vagueness and uncertainty
The results of our study indicate that intuitionistic fuzzy context-free languages (IFCFLs) generated by intuitionistic fuzzy context-free grammars (IFCFGs) are equivalent to those generated by intuitionistic fuzzy Greibach normal form (IFGNF) and intuitionistic fuzzy Chomsky normal form (IFCNF), respectively, and they are equivalent to intuitionistic fuzzy recognizable step functions
Using the generalized subset construction method, we show that IFPDAs are equivalent to intuitionistic fuzzy simple pushdown automaton (IFSPDA) and prove that intuitionistic fuzzy step functions are the same as those accepted by IFPDAs
Summary
Intuitionistic fuzzy set (IFS) introduced by Atanassov [1,2,3], which emerges from the simultaneous consideration of the degrees of membership and nonmembership with a degree of hesitancy, has been found to be highly useful in dealing with problems with vagueness and uncertainty. As a continuation of the work in [29,30,31], a fundamental framework of fuzzy pushdown automata theory based on complete residuated lattice-valued logic has been established in recent years by Xing et al [36], and the work generalizes the previous fuzzy automata theory systematically studied by Mordeson and Malik to some extent. Due to pushdown automata being another kind of important computational models [15] and motivated by the importance of grammars, languages and models theory [14], it stands to reason that we ought consider the notions of intuitionistic fuzzy pushdown automata, intuitionistic fuzzy context-free grammars, and fuzzy context-free languages because our discussion in this paper will provide a fundamental framework for studying intuitionistic fuzzy set theory on fuzzy pushdown automata and generators as well.
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