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See detailOn the number of abelian bordered words (with an example of automatic theorem-proving)
Goc, Daniel; Rampersad, Narad; Rigo, Michel ULg et al

in International Journal of Foundations of Computer Science (2014), 8

In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible ... [more ▼]

In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in $\mathbb{Z}$ not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a $3$-letter alphabet, the connection with paths in the triangular lattice is made. Finally, we characterize the lengths of the abelian unbordered factors occurring in the Thue--Morse word using some kind of automatic theorem-proving provided by a logical characterization of the $k$-automatic sequences. [less ▲]

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See detailEnumeration and decidable properties of automatic sequences
Charlier, Emilie ULg; Rampersad, Narad; Shallit, Jeffrey

in International Journal of Foundations of Computer Science (2012), 23(5), 1035-1066

We show that various aspects of k-automatic sequences — such as having an unbordered factor of length n — are both decidable and effectively enumerable. As a consequence it follows that many related ... [more ▼]

We show that various aspects of k-automatic sequences — such as having an unbordered factor of length n — are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems. [less ▲]

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See detailLogical characterization of recognizable sets of polynomials over a finite field
Rigo, Michel ULg; Waxweiler, Laurent

in International Journal of Foundations of Computer Science (2011), 22(7), 1549-1563

The ring of integers and the ring of polynomials over a finite field share a lot of properties. Using a bounded number of polynomial coefficients, any polynomial can be decomposed as a linear combination ... [more ▼]

The ring of integers and the ring of polynomials over a finite field share a lot of properties. Using a bounded number of polynomial coefficients, any polynomial can be decomposed as a linear combination of powers of a non-constant polynomial P playing the role of the base of the numeration. Having in mind the theorem of Cobham from 1969 about recognizable sets of integers, it is natural to study $P$-recognizable sets of polynomials. Based on the results obtained in the Ph.D. thesis of the second author, we study the logical characterization of such sets and related properties like decidability of the corresponding first-order theory. [less ▲]

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See detailComputing Convex Hulls by Automata Iteration
Cantin, François ULg; Legay, Axel; Wolper, Pierre ULg

in International Journal of Foundations of Computer Science (2009), 20(4), 647-667

This paper considers the problem of computing the real convex hull of a finite set of n-dimensional integer vectors. The starting point is a finite-automaton representation of the initial set of vectors ... [more ▼]

This paper considers the problem of computing the real convex hull of a finite set of n-dimensional integer vectors. The starting point is a finite-automaton representation of the initial set of vectors. The proposed method consists in computing a sequence of automata representing approximations of the convex hull and using extrapolation techniques to compute the limit of this sequence. The convex hull can then be directly computed from this limit in the form of an automaton-based representation of the corresponding set of real vectors. The technique is quite general and has been implemented. [less ▲]

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