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Profinite methods in automata theory
 INVITED LECTURE AT STACS
, 2009
"... This survey paper presents the success story of the topological approach to automata theory. It is based on profinite topologies, which are built from finite topogical spaces. The survey includes several concrete applications to automata theory. ..."
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Cited by 11 (4 self)
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This survey paper presents the success story of the topological approach to automata theory. It is based on profinite topologies, which are built from finite topogical spaces. The survey includes several concrete applications to automata theory.
Lagrange’s theorem for Hopf monoids in species
, 2012
"... Abstract. Following Radford’s proof of Lagrange’s theorem for pointed Hopf algebras, ..."
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Abstract. Following Radford’s proof of Lagrange’s theorem for pointed Hopf algebras,
On the decidability of semigroup freeness
, 2008
"... This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X ⊆ S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of two freeness problems have been clos ..."
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This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X ⊆ S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoid. In 1991, Klarner, Birget and Satterfield proved the undecidability of the freeness problem over threebythree integer matrices. Both results led to the publication of many subsequent papers. The aim of the present paper is threefold: (i) to present general results concerning freeness problems, (ii) to study the decidability of freeness problems over various particular semigroups (special attention is devoted to multiplicative matrix semigroups), and (iii) to propose precise, challenging open questions in order to promote the study of the topic. 1
A Hopfpower Markov chain on compositions
"... Abstract. In a recent paper, Diaconis, Ram and I constructed Markov chains using the coproductthenproduct map of a combinatorial Hopf algebra. We presented an algorithm for diagonalising a large class of these “Hopfpower chains”, including the GilbertShannonReeds model of riffleshuffling of a ..."
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Abstract. In a recent paper, Diaconis, Ram and I constructed Markov chains using the coproductthenproduct map of a combinatorial Hopf algebra. We presented an algorithm for diagonalising a large class of these “Hopfpower chains”, including the GilbertShannonReeds model of riffleshuffling of a deck of cards and a rockbreaking model. A very restrictive condition from that paper is removed in my thesis, and this extended abstract focuses on one application of the improved theory. Here, I use a new technique of lumping Hopfpower chains to show that the Hopfpower chain on the algebra of quasisymmetric functions is the induced chain on descent sets under riffleshuffling. Moreover, I relate its right and left eigenfunctions to GarsiaReutenauer idempotents and ribbon characters respectively, from which I recover an analogous result of Diaconis and Fulman (2012) concerning the number of descents under riffleshuffling. Résumé. Dans un récent article avec Diaconis et Ram, nous avons construit des chaînes de Markov en utilisant une composition du coproduit et produit d’une algébre de Hopf combinatoire. Nous avons présenté un algorithme pour diagonaliser une large classe de ces “chaînes de Hopf puissance”, en particulier nous avons diagonalisé le modèle de GilbertShannonReeds de mélange de cartes en “riffle shuffle ” (couper en deux, puis intercaler) et un modèle de cassage de pierres. Dans mon travail de thèse, nous supprimons une condition très restrictive de cet article, et ce papier se concentre sur
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"... For a fixed base, John H. Conway’s RATS sequences are generated by iterating the following procedure on an initial integer: Reverse the digits of the integer, Add the reversal to the original, Then Sort the resulting digits in increasing order. For example, 334+433=767, which gets sorted into 677. I ..."
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For a fixed base, John H. Conway’s RATS sequences are generated by iterating the following procedure on an initial integer: Reverse the digits of the integer, Add the reversal to the original, Then Sort the resulting digits in increasing order. For example, 334+433=767, which gets sorted into 677. In base 10, Conway discovered the curious sequence: 12333334444, 55666667777, 123333334444, 556666667777,.... Although the sequence is not periodic, it does display some periodiclike behavior which we refer to as “quasiperiodic.” Conway conjectured that all RATS sequences in base 10 are either eventually periodic, or they eventually lead to the previously mentioned quasiperiodic sequence. In this thesis, we study RATS sequences in various bases. In particular, we prove an ErdősKac type result for the periods of RATS sequences in base 3; we establish a connection between RATS sequences in general bases and Lyndon words; and we construct infinite families of bases for which there exist RATS sequences having certain prescribed periodicity properties, e.g., we show that there are infinitely many bases for which we can construct quasiperiodic RATS sequences all of a similar type. In the final chapter, we consider a similar iteration process, the reverseadd process. We present data and heuristic arguments on a problem of D.H. Lehmer asking whether every sequence obtained by this process contains a palindrome. ii To Amber
Hopf algebras and Markov chains: Two . . .
, 2012
"... The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural “rockbreaking” process, and Markov chains on simplicial complexes. ..."
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The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural “rockbreaking” process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rockbreaking, an explicit description of the quasistationary distribution and sharp rates to absorption follow.
A formalism of the object compounds viewed as information processing support
"... Abstract—This paper aims to create a background for information processing support. We introduce formal classes called object compounds. By object compounds, we refer to the formalism needed for the construction of (biological inspired equivalent) classes of molecules, compounds or complex like obj ..."
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Abstract—This paper aims to create a background for information processing support. We introduce formal classes called object compounds. By object compounds, we refer to the formalism needed for the construction of (biological inspired equivalent) classes of molecules, compounds or complex like objects. KeywordsNaturalComputing; Soft Computing; formal languages; information processing I.