A Decision Problem for Ultimately Periodic Sets in Nonstandard Numeration Systems

Abstract

Abstract. Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0, 1} without two consecutive 1. Given a set X of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not X is a finite union of arithmetic progressions. We obtain a decision procedure under some hypothesis about the considered numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract numeration systems built on an infinite regular language.

Syntactic Complexity of Ultimately Periodic Sets of Integers and Application to a Decision Procedure

“…Lemma 4.1. Let b ≥ 2 and m = db n q be given as in (3). Let X ⊆ N be a periodic set of (minimal) period m. For any words u, v ∈ A * b of length at least η, we have…”

“…See also [2] and in particular [5] for a first order logic approach. Recently this decision problem was settled positively in [3] for a large class of numeration systems based on linear recurrence sequences. Considering this decision problem for any abstract numeration system turns out to be equivalent to the so-called ω-HD0L ultimate periodicity decision problem, see again [4,15].…”

Syntactic Complexity of Ultimately Periodic Sets of Integers

“…See also [2] and in particular [5] for a first order logic approach. Recently this decision problem was settled positively in [3] for a large class of numeration systems based on linear recurrence sequences. Considering this decision problem for any abstract numeration system turns out to be equivalent to the so-called ω-HD0L ultimate periodicity decision problem, see again [4], or [10].…”

State Complexity of Testing Divisibility

“…. , m − 1} k given by(3). The system H k x ≡ b u (mod m) has a solution denoted by x u .Define γ 0 , .…”

“…denoted by τ. w r = (δ U (q 0 , w), val U (w), val U (w0)) τ(r, 0) τ(r, 1) ε, 0, 10 3 10 r 0 = (q 0 , 0, 0) r 0 r 1 1 r 1 = (q 1 , 1, 2) r 2 10, 10100 r 2 = (q 0 , 2, 0) r 3 r 4 100 r 3 = (q 0 , 0, 2) r 5 r 6 101 r 4 = (q 1 , 1, 1) r 7 1000, (10) 3 r 5 = (q 0 , 2, 2) r 8 r 9 1001 r 6 = (q 1 , 0, 1) r 10 1010, (100) 2 r 7 = (q 0 , 1, 2) r 2 r 11 10 4 , 10 4 10 r 8 = (q 0 , 2, 1) r 12 r 13 10 3 1 r 9 = (q 1 , 0, 0) r 0 10010, 10 7 r 10 = (q 0 , 1, 1) r 7 r 14 10101 r 11 = (q 1 , 0, 2) r 5 10 5 r 12 = (q 0 , 1, 0) r 15 r 16 10 4 1 r 13 = (q 1 , 2, 2) r 8 100101 r 14 = (q 1 , 2, 1) r 12 10 6 r 15 = (q 0 , 0, 1) r 10 r 17 10 5 1…”