263,791 research outputs found

    On Probability Distributions for Trees: Representations, Inference and Learning

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    We study probability distributions over free algebras of trees. Probability distributions can be seen as particular (formal power) tree series [Berstel et al 82, Esik et al 03], i.e. mappings from trees to a semiring K . A widely studied class of tree series is the class of rational (or recognizable) tree series which can be defined either in an algebraic way or by means of multiplicity tree automata. We argue that the algebraic representation is very convenient to model probability distributions over a free algebra of trees. First, as in the string case, the algebraic representation allows to design learning algorithms for the whole class of probability distributions defined by rational tree series. Note that learning algorithms for rational tree series correspond to learning algorithms for weighted tree automata where both the structure and the weights are learned. Second, the algebraic representation can be easily extended to deal with unranked trees (like XML trees where a symbol may have an unbounded number of children). Both properties are particularly relevant for applications: nondeterministic automata are required for the inference problem to be relevant (recall that Hidden Markov Models are equivalent to nondeterministic string automata); nowadays applications for Web Information Extraction, Web Services and document processing consider unranked trees

    A Generalization of the Iteration Theorem for Recognizable Formal Power Series on Trees

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    Berstel and Reutenauer stated the iteration theorem for recognizable formal power series on trees over fields and vector spaces. The key idea of its proof is the existence of pseudo-regular matrices in matrix-products. This theorem is generalized to integral domains and modules over integral domains in this thesis. It only requires the reader to have basic knowledge in linear algebra. Concepts from the advanced linear algebra and abstract algebra are introduced in the preliminary chapter.:1. Introduction 2. Preliminaries 3. Long products of matrices 4. Formal power series on trees 5. The generalized iteration theorem 6. Conclusio

    Bialgebra deformations and algebras of trees

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    Let A denote a bialgebra over a field k and let A sub t = A((t)) denote the ring of formal power series with coefficients in A. Assume that A is also isomorphic to a free, associative algebra over k. A simple construction is given which makes A sub t a bialgebra deformation of A. In typical applications, A sub t is neither commutative nor cocommutative. In the terminology of Drinfeld, (1987), A sub t is a quantum group. This construction yields quantum groups associated with families of trees

    Relating two Hopf algebras built from an operad

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    Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. By another construction, one can also build a group directly from the operad. We then consider its Hopf algebra of functions. We prove that there exists a surjective morphism from the latter Hopf algebra to the former one. This is illustrated by the case of an operad built on rooted trees, the \NAP operad, where the incidence Hopf algebra is identified with the Connes-Kreimer Hopf algebra of rooted trees.Comment: 21 pages, use graphics, 12 figures Version 2 : references added, minor changes. This version has not been corrected after submission. The final and corrected version will appear in IMRN and can be obtained from the author

    D-log and formal flow for analytic isomorphisms of n-space

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    Given a formal map F=(F1...,Fn)F=(F_1...,F_n) of the form z+higherz+\text{higher} order terms, we give tree expansion formulas and associated algorithms for the D-Log of F and the formal flow F_t. The coefficients which appear in these formulas can be viewed as certain generalizations of the Bernoulli numbers and the Bernoulli polynomials. Moreover the coefficient polynomials in the formal flow formula coincide with the strict order polynomials in combinatorics for the partially ordered sets induced by trees. Applications of these formulas to the Jacobian Conjecture are discussed.Comment: Latex, 32 page
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