2 research outputs found
Numeration Systems: a Link between Number Theory and Formal Language Theory
We survey facts mostly emerging from the seminal results of Alan Cobham
obtained in the late sixties and early seventies. We do not attempt to be
exhaustive but try instead to give some personal interpretations and some
research directions. We discuss the notion of numeration systems, recognizable
sets of integers and automatic sequences. We briefly sketch some results about
transcendence related to the representation of real numbers. We conclude with
some applications to combinatorial game theory and verification of
infinite-state systems and present a list of open problems.Comment: 21 pages, 3 figures, invited talk DLT'201
On the Recognizability of Self-Generating Sets
peer reviewedLet I be a finite set of integers and F be a finite set of maps of the form n->k_i n + l_i with integer coefficients. For an integer base k>=2, we study the k-recognizability of the minimal set X of integers containing I and satisfying f(X)\subseteq X for all f in F. In particular, solving a conjecture of Allouche, Shallit and Skordev, we show under some technical conditions that if two of the constants k_i are multiplicatively independent, then X is not k-recognizable for any k>=2