Multi-dimensional sets recognizable in all abstract numeration systems

in RAIRO : Informatique Théorique et Applications = Theoretical Informatics and Applications (2012), 46(1), 51-65

We prove that the subsets of N^d that are S-recognizable for all abstract numeration systems S are exactly the 1-recognizable sets. This generalizes a result of Lecomte and Rigo in the one-dimensional ... [more ▼]

We prove that the subsets of N^d that are S-recognizable for all abstract numeration systems S are exactly the 1-recognizable sets. This generalizes a result of Lecomte and Rigo in the one-dimensional setting. [less ▲]

Inverse star, borders, and palstars

in Information Processing Letters (2011), 111

A language L is closed if L = L*. We consider an operation on closed languages, L-*, that is an inverse to Kleene closure. It is known that if L is closed and regular, then L-* is also regular. We show ... [more ▼]

A language L is closed if L = L*. We consider an operation on closed languages, L-*, that is an inverse to Kleene closure. It is known that if L is closed and regular, then L-* is also regular. We show that the analogous result fails to hold for the context-free languages. Along the way we find a new relationship between the unbordered words and the prime palstars of Knuth, Morris, and Pratt. We use this relationship to enumerate the prime palstars, and we prove that neither the language of all unbordered words nor the language of all prime palstars is context-free. [less ▲]

State complexity of testing divisibility

in McQuillan, Ian, Pighizzini, Giovanni (Ed.) Proceedings Twelfth Annual Workshop on Descriptional Complexity of Formal Systems (2010, August)

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 ... [more ▼]

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 mN in the Fibonacci system is exactly 2m^2. [less ▲]

Structure of the minimal automaton of a numeration language and applications to state complexity

in Actes des Journées Montoises d'Informatique Théorique (2010)

We study the structure of automata accepting the greedy representations of N in a wide class of numeration systems. We describe the conditions under which such automata can have more than one strongly ... [more ▼]

We study the structure of automata accepting the greedy representations of N in a wide class of numeration systems. We describe the conditions under which such automata can have more than one strongly connected component and the form of any such additional components. Our characterization applies, in particular, to any automaton arising from a Bertrand numeration system. Furthermore, we show that for any automaton A arising from a system with a dominant root > 1, there is a morphism mapping A onto the automaton arising from the Bertrand system associated with the number . Under some mild assumptions, we also 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 mN in the Fibonacci system is exactly 2m^2. [less ▲]