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 contextsensitive, 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 context-free 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]. 1 Preliminaries Let Σ be a finite set and denote by Σ∗ and Σ, respectively, the sets of all (finite) strings and (one-way infinite) sequences over Σ. For x in Σ we define the following two sets: S(x) = {u ∈ Σ∗ | x = vuy, v ∈ Σ∗, y ∈ Σ}, and P (x) = {u ∈ Σ∗ | x = uy, y ∈ Σ}, that is, S(x) is the set of all finite substrings of x, and P (x) is the set of all finite prefixes of x. For a language L ⊆ Σ∗ define Sf (L) = {v ∈ Σ∗ | uvw ∈ L, u,w ∈ Σ∗}. Note that Sf is similar to S, but for languages of finite strings rather than for infinite sequences. Similarly, we define Pf (L) = {u ∈ Σ∗ | uw ∈ L, w ∈ Σ∗}. ∗This paper has been completed during the second author’s visit at the University of Auckland in 1995. The first author has been supported by AURC A18/XXXXX/62090/F3414030, 1994. †Computer Science Department, The University of Auckland, Private Bag 92109, Auckland, New Zealand, e-mail: [email protected]. ‡Department of Computer Science, The University of Western Ontario, London, Ontario, Canada N6A 5B7, e-mail: [email protected]. For a finite string u ∈ Σ∗, |u| denotes the length of u. For a language L ⊆ Σ∗, card(L) denotes the cardinality of L. Lemma 1.1. For each x ∈ Σ, S(x) = Sf (P (x)). Proof. If u belongs to S(x), then x = vuy, for some v ∈ Σ∗ and y ∈ Σ. So, vu is in P (x) and, therefore, u ∈ Sf (P (x)). Conversely, let w ∈ Sf (P (x)), i.e. uwv = z, for some u, v ∈ Σ∗ and z ∈ P (x). Since z ∈ P (x), it follows that x = zx′, for some x′ ∈ Σ, that is x = uwvx′ = uwx′′, where x′′ = vx′. Consequently, w ∈ S(x). 2 For every language L ⊆ Σ∗ define the density function DL by DL(n) = card(L ∩ Σ), where Σ denotes the set of all strings of length n over Σ. If a language L has a constant density, i.e., DL = O(1), then it is called a slender language, which was termed in [1]. The following results have been proven in [15]. Lemma 1.2. A regular language R over Σ has a density O(n), k ≥ 0 if and only if R can be represented as a finite union of regular expressions of the following form: xy∗ 1z1 · · · y∗ t zt, where x, y1, z1, . . . , yt, zt ∈ Σ∗ and 0 ≤ t ≤ k + 1. Lemma 1.3. Let R be a regular language, R′ = Sf (R) and let k be a non-negative integer. Then DR(n) = O(n) if and only if DR′(n) = O(n). Several of the subsequent proofs depend on the following result, which has been proved in [11] (see also [16]). Lemma 1.4. Let L ⊆ Σ∗ be a context-free language. Then L is slender, i.e., DL(n) = O(1), if and only if L is a finite union of languages of the form: {u1u2u3u4u5 | i ≥ 0}, where u1, u2, u3, u4, u5 ∈ Σ∗. 2 How Complex Are Disjunctive Sequences? A sequence x ∈ Σ is disjunctive [13] provided it contains all possible finite strings over Σ as its substrings, i.e. S(x) = Σ∗. At the top, disjunctive sequences x can be random, non-random but non-recursive, recursive, but arbitrarily complex. At the bottom, the complexity of a sequence x will be measured by the complexity of the language P (x) consisting of all prefixes of x; these languages can be context-sensitive, but not context-free. Chaitin’s Omega Number [7] is Borel normal in any base and, therefore, disjunctive in any base. More generally, by Theorem 3.6 in [3], every random sequence is Borel normal and, hence, disjunctive. All these sequences are non-recursive; they form a class of measure one [4]. Non-random and non-recursive disjunctive sequences have been constructed in [13]. Having disposed of the non-recursive case we turn our attention to recursive disjunctive sequences. First we rely on Rabin’s Theorem (see, for instance, Theorem 3.5 in [2]) to construct arbitrarily complex recursive disjunctive sequences: