Language-theoretic Complexity of Disjunctive Sequences

A sequence over an alphabet Σ is called disjunctive [13] if it contains all possible finite strings over Σ as its substrings. Disjunctive sequences have been recently studied in various contexts, e.g. [12, 9]. They abound in both category and measure senses [5]. In this paper we measure the complexity of a sequence x by the complexity of the language P (x) consisting of all prefixes of x. The languages P (x) associated to disjunctive sequences can be arbitrarily complex. We show that for some disjunctive numbers x the language P (x) is context-sensitive, but no language P (x) associate to a disjunctive number can be context-free. We also show that computing a disjunctive number x by rationals corresponding to an infinite subset of P (x) does not decrease the complexity of the procedure, i.e. if x is disjunctive, then P (x) contains no infinite contextfree language. This result reinforces, in a way, Chaitin’s thesis [6] according to which perfect sets, i.e. sets for which there is no way to compute infinitely many of its members essentially better (simpler/quicker) than computing the whole set, do exist. Finally we prove the existence of the following language-theoretic complexity gap: There is no x ∈ Σ such that P (x) is context-free but not regular. If the set of all finite substrings of a sequence x ∈ Σ is slender, then the set of all prefixes of x is regular, that is P (x) is regular if and only if S(x) is slender. The proofs essentially use some recent results concerning the complexity of languages containing a bounded number of strings of each length [15, 14, 11, 16]. Proceedings of CATS’96 (Computing: The Australasian Theory Symposium), Melbourne, Australia, January 29–January 3