Corollary 4 ([14]). A recognisable set of numbers is L-recognisable in any rational abstract **numeration** **system** L.
If Corollary 4 requires formally no proof after the characterisation of rational abstract **numeration** systems given by Theorem 1, it is interesting to further in- vestigate the construction which, given L and a recognisable set of numbers X computes an automaton which recognises the set hXi L . The computation method used in the preceding section (which is not the mere application of the general result that yields Corollary 4) allows to establish easily the following statement. Proposition 5. Let L be a rational language over A ∗ recognised by a deterministic

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The 3 2 -number system introduced by Akiyama, Frougny and Sakarovitch (2008) has a numeration language which is not context-free.. AIM : To provide a unified approach for representing re[r]

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Compared to the integer bases, new technicalities have to be taken into account to generalize Pascal triangles to a larger class of **numeration** systems. The **numeration** systems occurring in this paper essentially have two properties. The first one is that the language of the **numeration** **system** comes from a particular automaton. The second one is the Bertrand condition which allows to delete or add ending zeroes to valid representations.

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A. Muchnik showed that Problem 2 turns out to be decidable for any linear **numeration** **system** U for which both rep U (N) and addition are rec- ognizable by finite automata [Muc03]. Still, it is a difficult question to characterize **numeration** systems U for which addition is computable by a finite automaton. 1 For details in this area, as already mentioned on page 24, see [BH97, Fro92], in which the authors mainly considered positional nu- meration systems defined by a linear recurrence relation whose characteristic polynomial is the minimal polynomial of a Pisot number. In [Fro97] the se- quentiality of the successor function, i.e., the action of adding 1, is studied. If addition is computable by a finite automaton, so the successor function is, but the converse does not hold in general. In particular, some examples of linear **numeration** systems for which addition is not computable by a finite automaton are given in [Fro97]: for example, the sequence (U i ) i≥0 defined by U i+4 = 3U i+3 + 2U i+2 + 3U i for all i ∈ N with any integer initial con- ditions satisfying 1 = U 0 < U 1 < U 2 < U 3 . So the decision techniques from [Ler05, Muc03] cannot be applied to that **numeration** **system**. Nev- ertheless, as we shall see in Example 3.5.14, our decision procedure can be applied to this **system**. Also, note that, in the extended framework of ab- stract **numeration** systems, one can exhibit systems such that multiplication by a constant does not preserve S-recognizability. For a discussion on this topic, see Theorems 2.2.1 and 2.2.10, and Propositions 2.5.10 and 2.5.11 in Chapter 2, which led to the proof of Theorem 2.6.1 on page 43. Therefore the powerful tools from logic discussed above cannot be applied in that context either.

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with z ℓ > z ℓ−1 > · · · > z 1 ≥ 0. Fraenkel [4] called this **system** combinatorial **numeration**
**system** and referred to Lehmer [8]. Even if this seems to be a folklore result, the only proof that we were able to trace out goes back to Katona [6] who developed diﬀerent arguments to obtain the same decomposition.

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◦ É. Charlier, M. Le Gonidec, and M. Rigo, Representing real numbers in a generalized **numeration** **system**, J. Comput. **System** Sci. 77 (2011), no. 4, 743–759.
◦ A. Cobham, Uniform tag sequences, Math. Systems Theory 6 (1972), 164–192.
◦ K. Culik, J. Karhumäki, and A. Lepistö, Alternating iteration of morphisms and the Kolakovski sequence, in Lindenmayer systems, 93–106, Springer, Berlin, (1992).

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as considered for instance in [9]. A set X ⊆ N is said to be B ` -recognizable if rep ` (X) is a
regular language over the alphabet Σ ` , i.e., accepted by a finite automaton. This one-to-one
correspondence between the words of B ` and the integers can be extended to any infinite
regular language L over a totally ordered alphabet (Σ, <). This leads to the general notion of abstract **numeration** **system**.

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` ] ∈ N d+1 : ∃m, [ m n ] ∈ G f ∧ ` < m}.
Since for all n ∈ N d , f (n) = Card{` ∈ N : [ n ` ] ∈ X}, the result follows from Theorem 53.
Remark 65. Even though both families of S-automatic sequences and (S, N)-regular sequences are closed under sum, product and product by a constant, it is no longer the case of the family of (S, S 0 )-synchronized sequences. For instance, the sequence N → N, n 7→ n is (S, S)-synchronized for any abstract **numeration** **system** S. However, the sequence N → N, n 7→ 2n is not (S, S)-synchronized in general. For example, it is not for the unary **system** S = (c ∗ , c).

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or not X is ultimately periodic, i.e., whether or not X is a finite union of arithmetic progressions ?
J. Honkala showed in [1] that Problem 1 turns out to be decidable for the usual integer base b ≥ 2 **numeration** **system** defined by U n = b U n−1 for n ≥ 1.

In the next section, Theorem ?? gives a decision procedure for Problem 1 when- ever U is a linear **numeration** **system** such that N is U -recognizable and satisfying a relation like (1) with a k = ±1 (the main reason for this assumption is that 1
and −1 are the only two integers invertible modulo n for all n ≥ 2). In the last section, we consider the same decision problem but restated in the framework of abstract **numeration** systems [7]. We apply successfully the same kind of techniques to a large class of abstract **numeration** systems (for instance, an example consisting of two copies of the Fibonacci **system** is considered). The corresponding decision procedure is given by Theorem ??. All along the paper, we try whenever it is pos- sible to state results in their most general form, even if later on we have to restrict ourselves to particular cases. For instance, results about the admissible preperiods do not require any extra assumption.

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Currently little is known about the automata accepting other kind of **numeration** languages. In the first part of this paper we study the structure of these automata for a wide class of **numeration** systems. In Section 2 we review the needed background concerning **numeration** systems. Then in Section 3 we provide several examples in order to illustrate the different types of automata that can arise from these **numeration** systems. In Section 4 we describe the conditions under which such automata can have more than one strongly connected component and the form of any such additional strongly connected component. In the case where the **numeration** **system** has a dominant root β > 1 (see the next section for the definition), we are able to provide a more specific description of the structure. For instance, we show that for any automaton A arising

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Figure 7: The conjectured functions H 3 , H 4 , H 5 , H 7 .
The graphs of H 3 , H 4 , H 5 , H 7 have been depicted in Figure 7 on the interval [0, 1). Such
a result is in the line of (1).
If we leave the k-regular setting and try to replace the Fibonacci sequence with another linear recurrent sequence, the situation seems to be more intricate. For the Tribonacci **numeration** **system** T = (T (n)) n>0 built on the language of words over {0, 1} avoiding three consecutive ones, we conjecture that a result similar to Theorem 2 should hold for the corresponding summatory function A T . Computing the first values of A T (T (n)), the

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Theorem
Let ℓ be a positive integer. For the abstract **numeration** **system** S = (a ∗ 1 . . . a ∗ ℓ , {a 1 < . . . < a ℓ }),
multiplication by λ > 1 preserves S-recognizability if and only if one of the following condition is satisfied :

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systems, Honkala has proved that one can decide whether or not X is ultimately periodic [14]. A shorter proof of this result was given in [2]. The same decidability question was answered positively in [8, 3] for a wide class of linear **numeration** systems containing the Fibonacci **numeration** **system**. Furthermore, in [7], as an application, for the Fibonacci **numeration** **system** F , we show that the number of states of the trim minimal automaton accepting 0 ∗ rep

We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand **numeration** systems. These automatic sequences can be viewed as general- izations of the more typical k-automatic sequences and Pisot-automatic sequences. We show that, like k-automatic sequences, Parry-automatic sequences have sublinear factor complexity while there exist Bertrand-automatic sequences with superlinear factor complexity. We prove that the set of Parry-automatic sequences with respect to a fixed Parry **numeration** **system** is not closed under taking images by uniform substitutions or periodic deletion of letters. These closure properties hold for k-automatic sequences and Pisot-automatic sequences, so our re- sult shows that these properties are lost when generalizing to Parry **numeration** systems and beyond. Moreover, we show that a multidimensional sequence is U -automatic with respect to a positional **numeration** **system** U with regular language of **numeration** if and only if its U -kernel is finite.

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ABSTRACT **NUMERATION** SYSTEMS
DEFINITION (P. LECOMTE, M.R. ’01)
An abstract **numeration** **system** is a triple S = ( L , Σ, <) where L is a regular language over a totally ordered alphabet (Σ, <) . Enumerating the words of L with respect to the genealogical ordering induced by < gives a one-to-one correspondence

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An abstract **numeration** **system** (ANS) is a triple S = (L, Σ, <) where L is an innite regular language over a totally ordered alphabet (Σ, <) .
By enumerating the words of L w.r.t. the radix order < rad induced by < , we dene a bijection :

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Theorem
Let U be a **numeration** **system** satisfying (H1), (H2) and (H3), and such that the gcd of the coefficients of the recurrence relation of U is larger than 1. Assume there is a computable positive integer D such that for all ultimately periodic sets X of period π X = mX · p µ 1 1 · · · p

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RESPONSE TIME
To measure the response time, the beam is tuned to lose 4µAe in SSC1. We cut the beam by the chopper and we set the threshold to 2µAe. The beam is then sent through the cyclotron, and the safety **system** detects the loss and stops the beam.

This section addresses the allocation of the **system** specifi- cation to the **system** segments. The requirement derivation addressed in the previous section is the result of the **System** Design Activity. Figure 6 shows that the derivation relation between the GSRD specification and the segment require- ments and interfaces is indeed generated through the Design Definition File (DDF) and Justification and Performance Budget File (resp. DJF and PBF). The DJF Annex A provides for each GSRD requirement a short summary of the DDF and DJF/PBF objects that contributes to meeting the **system** re- quirement. DOORS scripts exploit the DDJF traceability to automatically reconstruct the derivation links between the GSRD and Segment Requirements (X-Traceability).

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