Decision algorithms for Fibonacci-automatic Words, I: Basic results

We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) “Fibonacci-automatic”. This class includes, for example, the famous Fibonacci word f = 01001010 · · · , the fixed point of the morphism 0 → 01 and 1 → 0. We then recover many results about the Fibonacci word from the literature (and improve some of them), such as assertions about the occurrences in f of squares, cubes, palindromes, and so forth. 1 Decidability As is well-known, the logical theory Th(N,+), sometimes called Presburger arithmetic, is decidable [43, 44]. Büchi [10] showed that if we add the function Vk(n) = k , for some fixed integer k ≥ 2, where e = max{i : k |n}, then the resulting theory is still decidable. This theory is powerful enough to define finite automata; for a survey, see [9]. As a consequence, we have the following theorem (see, e.g., [50]): Theorem 1. There is an algorithm that, given a proposition phrased using only the universal and existential quantifiers, indexing into one or more k-automatic sequences, addition, subtraction, logical operations, and comparisons, will decide the truth of that proposition. Here, by a k-automatic sequence, we mean a sequence a computed by deterministic finite automaton with output (DFAO) M = (Q,Σk,∆, δ, q0, κ). Here Σk := {0, 1, . . . , k− 1} is the input alphabet, ∆ is the output alphabet, and outputs are associated with the states given by the map κ : Q → ∆ in the following manner: if (n)k denotes the canonical expansion of n in base k, then a[n] = κ(δ(q0, (n)k)). The prototypical example of an automatic sequence School of Computer Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada; [email protected], [email protected] . Computer Science and Artificial Intelligence Laboratory, The Stata Center, MIT Building 32, 32 Vassar Street, Cambridge, MA 02139 USA; [email protected] .