Multi-dimensional sets recognizable in all abstract numeration systems

نویسندگان

  • Émilie Charlier
  • Anne Lacroix
  • Narad Rampersad
چکیده

numeration systems An abstract numeration system is a triple S = (L,Σ, <) where L is an infinite regular language over the totally ordered alphabet (Σ, <) By enumerating words of L in the radix order (induced by <), we define a one-to-one correspondence between N and L repS : N → L : n 7→ (n + 1)th word of L valS = rep −1 S : L → N Abstract numeration systems Example S = (a∗b∗, {a, b}, a < b) L = a∗b∗ N ε 0 a 1 b 2 aa 3 ab 4 bb 5 aaa 6 aab 7 abb 8 .. .. A set X ⊆ N is S-recognizable if the language repS(X ) is accepted by a finite automaton.numeration systems Example S = (a∗b∗, {a, b}, a < b) L = a∗b∗ N ε 0 a 1 b 2 aa 3 ab 4 bb 5 aaa 6 aab 7 abb 8 .. .. A set X ⊆ N is S-recognizable if the language repS(X ) is accepted by a finite automaton. Abstract numeration systems Remarknumeration systems Remark ◮ The numeration system in base k is an abstract numeration system built on the language L = {1, 2, . . . , k − 1}Σk ∪ {ε} Abstract numeration systems Remarknumeration systems Remark ◮ The set {n : n ∈ N} is never k-recognizable but is S-recognizable for S = (ab ∪ ac, {a, b, c}, a < b < c). L = a∗b∗ ∪ a∗c∗ N ε 0 a 1 b 2 c 3 aa 4 ab 5 ac 6 bb 7 cc 8 aaa 9 .. .. Abstract numeration systemsnumeration systems Theorem (Lecomte, Rigo, 2001) Ultimately periodic sets are S-recognizable for all ANS S built on a regular language. Abstract numeration systemsnumeration systems Theorem (Lecomte, Rigo, 2001) Ultimately periodic sets are S-recognizable for all ANS S built on a regular language. Corollary A set X ⊆ N is S-recognizable for all ANS S if and only if X is ultimately periodic. Corollary A set X ⊆ N is S-recognizable for all ANS S if and only if X is 1-recognizable.

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عنوان ژورنال:
  • RAIRO - Theor. Inf. and Applic.

دوره 46  شماره 

صفحات  -

تاریخ انتشار 2012