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Cobham’s theorem for substitutions
 J. Eur. Math. Soc
"... Abstract. The seminal theorem of Cobham has given rise during the last 40 years to a lot of works around nonstandard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the socalled sub ..."
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Abstract. The seminal theorem of Cobham has given rise during the last 40 years to a lot of works around nonstandard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the socalled substitutive sequences. Let α and β be two multiplicatively independent Perron numbers. Then, a sequence x ∈ AN, where A is a finite alphabet, is both αsubstitutive and βsubstitutive if and only if x is ultimately periodic. 1.
1.1 Extended Iterated Pushdown Automata
"... 2 The theory of successor extended by several predicates In [ER66], Elgot and Rabin devise a method for constructing unary predicates P such that the MSO theory of 〈N, +1, P 〉 is decidable (here +1 denotes the successor relation). Further results in this direction have been established in ([Sie70, S ..."
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2 The theory of successor extended by several predicates In [ER66], Elgot and Rabin devise a method for constructing unary predicates P such that the MSO theory of 〈N, +1, P 〉 is decidable (here +1 denotes the successor relation). Further results in this direction have been established in ([Sie70, Sem84, Mae99, CT02, FS06]). This kind of problem takes place in the more general perspective of studying “weak ” arithmetical theories, which possess interesting decidability properties ([Bès01]). We present here a method allowing to define infinite sequences of monadic predicates P1,..., Pn, such that the MSO theory of 〈N, +1, (Pi)i∈N 〉 is decidable. To our knowledge the only one result dealing with several predicates have been given in [Hos71] in the special case where Pi = {n2i}n∈N. This work extends the one we made in [FS06], where the method consisted of consider integer sequences computed by Iterated Pushdown Automata. These automata have been introduced in [Aho69] as a generalization of Pushdown Automata and have been more studied, see e.g. [Mas76, Dam82, Eng83, ES84, EV86, DG86], or more recently [Cau02, KNU02]. We obtain here more powerful results by the same method but by using a novel class of automata. The new feature of the automata here considered is that transitions are ”controlled ” by some predicates. This allows to obtain two main improvements: first, results are extended to several predicates, and second, these predicates belong to a largest class. In particular, in [FS06], every predicates are included in the one studied in [CT02], and are then Residualy Ultimately Periodics”. Here we go out this class by showing, e.g., that structures 〈N, +1, n ⌊ √ n⌋ 〉 and 〈N, +1, n⌊log(n)⌋ 〉 have a decidable MSOtheory. In particular, we build predicates Pi that can have very slow ”growth”; i.e., the function associating to k the kth element of Pi can be comparable to ⌊nlogn⌋, ⌊nlog(logn)) ⌋ or even ⌊nlog ∗ (n)⌋.