
Robustness of Pisotregular sequences
We consider numeration systems based on a dtuple ๐=(U_1,โฆ,U_d) of seque...
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Regularity of Position Sequences
A person is given a numbered sequence of positions on a sheet of paper. ...
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Marginalization in Composed Probabilistic Models
Composition of lowdimensional distributions, whose foundations were lai...
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Analogical Dissimilarity: Definition, Algorithms and Two Experiments in Machine Learning
This paper defines the notion of analogical dissimilarity between four o...
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Efficient Approximation Algorithms for String Kernel Based Sequence Classification
Sequence classification algorithms, such as SVM, require a definition of...
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Automatic sequences: from rational bases to trees
The nth term of an automatic sequence is the output of a deterministic f...
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Comparator automata in quantitative verification
The notion of comparison between system runs is fundamental in formal ve...
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Regular sequences and synchronized sequences in abstract numeration systems
The notion of bregular sequences was generalized to abstract numeration systems by Maes and Rigo in 2002. Their definition is based on a notion of ๐ฎkernel that extends that of bkernel. However, this definition does not allow us to generalize all of the many characterizations of bregular sequences. In this paper, we present an alternative definition of ๐ฎkernel, and hence an alternative definition of ๐ฎregular sequences, which enables us to use recognizable formal series in order to generalize most (if not all) known characterizations of bregular sequences to abstract numeration systems. We then give two characterizations of ๐ฎautomatic sequences as particular ๐ฎregular sequences. Next, we present a general method to obtain various families of ๐ฎregular sequences by enumerating ๐ฎrecognizable properties of ๐ฎautomatic sequences. As an example of the many possible applications of this method, we show that, provided that addition is ๐ฎrecognizable, the factor complexity of an ๐ฎautomatic sequence defines an ๐ฎregular sequence. In the last part of the paper, we study ๐ฎsynchronized sequences. Along the way, we prove that the formal series obtained as the composition of a synchronized relation and a recognizable series is recognizable. As a consequence, the composition of an ๐ฎsynchronized sequence and a ๐ฎregular sequence is shown to be ๐ฎregular. All our results are presented in an arbitrary dimension d and for an arbitrary semiring ๐.
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