Deciding Top-Down Determinism of Regular Tree Languages

TL;DR

This paper investigates the computational complexity of deciding whether a regular tree language recognized by a bottom-up automaton can also be recognized by a top-down automaton, establishing quadratic time algorithms for these decisions.

Contribution

It introduces quadratic time algorithms for deciding top-down recognizability of languages recognized by bottom-up automata and for finite unions of DTAs.

Findings

Decidability of top-down recognizability in quadratic time

Decidability for finite unions of DTAs in quadratic time

Recognition limitations for certain finite tree languages

Abstract

It is well known that for a regular tree language it is decidable whether or not it can be recognized by a deterministic top-down tree automaton (DTA). However, the computational complexity of this problem has not been studied. We show that for a given deterministic bottom-up tree automaton it can be decided in quadratic time whether or not its language can be recognized by a DTA. Since there are finite tree languages that cannot be recognized by DTAs, we also consider finite unions of \DTAs and show that also here, definability within deterministic bottom-up tree automata is decidable in quadratic time.