Lattice-valued tree pushdown automata: Pumping lemma and closure properties

Abstract

In this paper, we introduce the concept of lattice-valued tree pushdown automata that extends the concepts of lattice-valued pushdown automata, lattice-valued tree automata and fuzzy tree pushdown automata. Also, we consider fuzzy tree languages recognized by lattice-valued (deterministic) tree pushdown automata (called lattice-valued context-free tree languages). Then, we prove the pumping lemma that can be used in proving that certain sets of tree languages are not recognizable. Also, we study closure properties of lattice-valued context-free tree languages with respect to some operations. We show that lattice-valued context-free tree languages recognized by lattice-valued deterministic tree pushdown automata are closed under union, concatenation, star, linear and alphabetic tree homomorphism and inverse tree homomorphism. By applying the pumping lemma on an example, we show that lattice-valued context-free tree languages are not closed under intersection. However, we prove that the intersection of lattice-valued context-free tree language and lattice-valued regular tree language is recognizable by a lattice-valued tree pushdown automaton. Also, we show that lattice-valued non-deterministic tree pushdown automata are only closed under union. However, if L satisfies distributivity, then non-deterministic tree pushdown automata are closed under all of these operations.