State Complexity of Testing Divisibility

Abstract

Under some mild assumptions, we study the state complexity of the trim minimal automaton accepting the greedy representations of the multiples of m ≥ 2 for a wide class of linear numeration systems. As an example, the number of states of the trim minimal automaton accepting the greedy representations of m N in the Fibonacci system is exactly 2m 2 .

On the enumerating series of an abstract numeration system

“…If the numeration system considered is a positional numeration system (and still a rational one), and under some supplementary hypotheses, then the exact number of states for the minimal automaton of X p,r L can be computed (cf. [6]).…”

Numeration Systems: A Link between Number Theory and Formal Language Theory

“…Consider the Fibonacci numeration system given by the basis F 0 = 1, F 1 = 2 and F n+2 = F n+1 + F n for all n ≥ 0. For this system, 0 * rep F (N) is given by the set of words over {0, 1} avoiding the factor 11 and the set of even numbers is U -recognizable [32] using the DFA shown in Figure 3. To conclude this section, we present a linear numeration basis U such that the set of squares Q = {n 2 | n ∈ N} is U -recognizable.…”

“…given by the set of words over {0, 1} avoiding the factor 11 and the set of even numbers is U -recognizable [32] using the DFA shown in Figure 3. To conclude this section, we present a linear numeration basis U such that the set of squares Q = {n 2 | n ∈ N} is U -recognizable.…”

“…For this state complexity question studied in the wider context of linear numeration systems (cf. Definition 3), see [32]. The base dependence of recognizability shown by Cobham's result strongly motivates the general study of recognizable sets and the introduction of non-standard or exotic numeration systems based on an increasing sequence satisfying a linear recurrence relation.…”

“…For this state complexity question studied in the wider context of linear numeration systems (cf. Definition 3), see [32].…”