Fuzzy Deterministic Top-down Tree Automata

TL;DR

This paper introduces fuzzy deterministic top-down tree automata over lattices, establishing their properties, decidability results, and their relation to fuzzy path languages, expanding the theoretical understanding of fuzzy tree automata.

Contribution

It defines and studies fuzzy DT-tree automata over lattices, proving a Pumping Lemma, closure properties, and characterizations, and explores their decidability and relation to fuzzy path languages.

Findings

DT-recognizable fuzzy tree languages form a proper subfamily of all regular fuzzy tree languages.

A Pumping Lemma for DT-recognizable languages is established.

Decidability results for DT-recognizability and path closure are proved.

Abstract

In this paper we introduce and study fuzzy deterministic top-down (DT) tree automata over a lattice L. The L-fuzzy tree languages recognized by these automata are said to be DT-recognizable, and they form a proper subfamily $DRec_L$ of the family of $Rec_L$ of all regular L-fuzzy tree languages. We prove a Pumping Lemma for $DRec_L$ from which several decidability results follow. The closure properties of $DRec_L$ under various operations are established. We also characterize DT-recognizability in terms of L-fuzzy path languages, and prove that the path closure of any regular L-fuzzy tree language is DT-recognizable, and that it is decidable whether a regular L-fuzzy tree language is DT-recognizable. In most of the paper, L is just any nontrivial bounded lattice, but sometimes it is assumed to be distributive or even a bounded chain.