263,791 research outputs found
On Probability Distributions for Trees: Representations, Inference and Learning
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
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
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
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
Given a formal map of the form 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
- …