THREE COMPLEXITY FUNCTIONS

International audienceFor an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes. Given a language L on a finite alphabet A, the complexity function p L (n) counts for every n the number of factors of length n of L; this is a very useful notion, both inside word combinatorics and for the study of symbolic dynamical systems, see for example the survey [7]; of particular interest are the infinite words which are determined by the complexity of their language, those words for which p L (n) ≤ n for at least one n are ultimately periodic [15], while the Sturmian words, of complexity n + 1 for all n, are natural codings of rotations, see [6, 16], or Chapter 6 of [17], and Section 4 below. Note that the complexity is exponential when the language has positive topological entropy, and has not been widely used for that range of languages. To study further the combinatorial properties of infinite words, the notion of maximal pattern complexity, denoted by p

On Infinite Words Determined by Indexed Languages

We characterize the infinite words determined by indexed languages. An infinite language $L$ determines an infinite word $\alpha$ if every string in $L$ is a prefix of $\alpha$. If $L$ is regular or context-free, it is known that $\alpha$ must be ultimately periodic. We show that if $L$ is an indexed language, then $\alpha$ is a morphic word, i.e., $\alpha$ can be generated by iterating a morphism under a coding. Since the other direction, that every morphic word is determined by some indexed language, also holds, this implies that the infinite words determined by indexed languages are exactly the morphic words. To obtain this result, we prove a new pumping lemma for the indexed languages, which may be of independent interest.Comment: Full version of paper accepted for publication at MFCS 201

The Complexity of Infinite Computations In Models of Set Theory

We prove the following surprising result: there exist a 1-counter B\"uchi automaton and a 2-tape B\"uchi automaton such that the \omega-language of the first and the infinitary rational relation of the second in one model of ZFC are \pi_2^0-sets, while in a different model of ZFC both are analytic but non Borel sets. This shows that the topological complexity of an \omega-language accepted by a 1-counter B\"uchi automaton or of an infinitary rational relation accepted by a 2-tape B\"uchi automaton is not determined by the axiomatic system ZFC. We show that a similar result holds for the class of languages of infinite pictures which are recognized by B\"uchi tiling systems. We infer from the proof of the above results an improvement of the lower bound of some decision problems recently studied by the author