Computational Aspects of Disjunctive Sequences

Abstract

An infinite binary sequence is disjunctive if every binary word occurs as a subword in the sequence. For a computational analysis of disjunctive sequences we can identify an infinite 0-1-sequence either with its prefix set or with its corresponding set, where a set A of binary words corresponds to a sequence α if α is the characteristic sequence of A. Most of the previous investigations of disjunctive sequences have dealt with prefix sets. Here, following the more common point of view in computability theory, we focus our investigations on the sets corresponding to disjunctive sequences.

We analyze the computational complexity and the Chomsky complexity of sets corresponding to disjunctive sequences. In particular, we show that no such set is regular but that there are linear languages with disjunctive characteristic sequences. Moreover, we discuss decidability questions. Here our main results are that the disjunctivity problem for Chomsky-0-grammars is Π$^{\rm 0}_{\rm 3}$-complete while the corresponding problem for linear or context free or context sensitive grammars is Π$^{\rm 0}_{\rm 2}$-complete. The latter implies that, for any language class C which contains the linear languages, the class of the languages in C corresponding to disjunctive sequences is not recursively presentable.