Undecidable properties of deterministic top-down tree transducers

Abstract

Decidability questions concerning ranges of deterministic top-down tree transducers are considered. It is shown that the following nine problems are undecidable, for ranges L 1 and L 2 of arbitrary two deterministic, nondeleting and finite copying top-down tree transducers: Is L 1 ∩ L 2 empty (infinite, recognizable)? Is the complement of L 1 empty (infinite, recognizable)? Is L 1 recognizable? Is L 1 = L 2 ( L 1 ⊆ L 2)? A deterministic top-down tree transducer is a special terminating and confluent term rewriting system. Hence, its range is the set of irreducible elements derivable from a recognizable tree language, namely from its domain. The questions corresponding to the above nine ones are considered and shown to be undecidable for terminating and confluent term rewriting systems as well. For example, the result corresponding to the undecidability of “Is L 1 recognizable?” is as follows. It is undecidable, for an arbitrary terminating and confluent term rewriting system R and a recognizable tree language L, whether the set of elements irreducible with respect to R derivable from L is recognizable or not.