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

Legendre Transform, Hessian Conjecture and Tree Formula

Let $\phi$ be a polynomial over $K$ (a field of characteristic 0) such that the Hessian of $\phi$ is a nonzero constant. Let $\bar\phi$ be the formal Legendre Transform of $\phi$. Then $\bar\phi$ is well-defined as a formal power series over $K$. The Hessian Conjecture introduced here claims that $\bar\phi$ is actually a polynomial. This conjecture is shown to be true when K=\bb{R} and the Hessian matrix of $\phi$ is either positive or negative definite somewhere. It is also shown to be equivalent to the famous Jacobian Conjecture. Finally, a tree formula for $\bar\phi$ is derived; as a consequence, the tree inversion formula of Gurja and Abyankar is obtained.Comment: 9 pages, references are update

Postcondition-preserving fusion of postorder tree transformations

Tree transformations are commonly used in applications such as program rewriting in compilers. Using a series of simple transformations to build a more complex system can make the resulting software easier to understand, maintain, and reason about. Fusion strategies for combining such successive tree transformations promote this modularity, whilst mitigating the performance impact from increased numbers of tree traversals. However, it is important to ensure that fused transformations still perform their intended tasks. Existing approaches to fusing tree transformations tend to take an informal approach to soundness, or be too restrictive to consider the kind of transformations needed in a compiler. We use postconditions to define a more useful formal notion of successful fusion, namely postcondition-preserving fusion. We also present criteria that are sufficient to ensure postcondition-preservation and facilitate modular reasoning about the success of fusion

Inversion Problem, Legendre Transform and Inviscid Burgers' Equations

Let $F(z)=z-H(z)$ with order $o(H(z))\geq 1$ be a formal map from \bC^n to \bC^n and $G(z)$ the formal inverse map of $F(z)$. We first study the deformation $F_t(z)=z-tH(z)$ of $F(z)$ and its formal inverse $G_t(z)=z+tN_t(z)$. (Note that $G_{t=1}(z)=G(z)$ when $o(H(z))\geq 2$.) We show that $N_t(z)$ is the unique power series solution of a Cauchy problem of a PDE, from which we derive a recurrent formula for $G_t(z)$. Secondly, motivated by the gradient reduction obtained by M. de Bondt, A. van den Essen \cite{BE1} and G. Meng \cite{M} for the Jacobian conjecture, we consider the formal maps $F(z)=z-H(z)$ satisfying the gradient condition, i.e. $H(z)=\nabla P(z)$ for some P(z)\in \bC[[z]] of order $o(P(z))\geq 2$. We show that, under the gradient condition, $N_t(z)=\nabla Q_t(z)$ for some Q_t(z)\in \bC[[z, t]] and the PDE satisfied by $N_t(z)$ becomes the $n$-dimensional inviscid Burgers' equation, from which a recurrent formula for $Q_t(z)$ also follows. Furthermore, we clarify some close relationships among the inversion problem, Legendre transform and the inviscid Burgers' equations. In particular the Jacobian conjecture is reduced to a problem on the inviscid Burgers' equations. Finally, under the gradient condition, we derive a binary rooted tree expansion inversion formula for $Q_t(z)$. The recurrent inversion formula and the binary rooted tree expansion inversion formula derived in this paper can also be used as computational algorithms for solutions of certain Cauchy problems of the inviscid Burgers' equations and Legendre transforms of the power series $f(z)$ of $o(f(z))\geq 2$.Comment: Latex, 21 pages. Some misprints have been correcte

An Extension Theorem with an Application to Formal Tree Series

A grove theory is a Lawvere algebraic theory T for which each hom-set T(n,p) is a commutative monoid; composition on the right distributes over all finite sums: (\sum f_i) . h = \sum f_i . h. A matrix theory is a grove theory in which composition on the left and right distributes over finite sums. A matrix theory M is isomorphic to a theory of all matrices over the semiring S = M(1,1). Examples of grove theories are theories of (bisimulation equivalence classes of) synchronization trees, and theories of formal tree series over a semiring S . Our main theorem states that if T is a grove theory which has a matrix subtheory M which is an iteration theory, then, under certain conditions, the fixed point operation on M can be extended in exactly one way to a fixed point operation on T such that T is an iteration theory. A second theorem is a Kleene-type result. Assume that T is an iteration grove theory and M is a sub iteration grove theory of T which is a matrix theory. For a given collection Sigma of scalar morphisms in T we describe the smallest sub iteration grove theory of T containing all the morphisms in M union Sigma

Colored operads, series on colored operads, and combinatorial generating systems

We introduce bud generating systems, which are used for combinatorial generation. They specify sets of various kinds of combinatorial objects, called languages. They can emulate context-free grammars, regular tree grammars, and synchronous grammars, allowing us to work with all these generating systems in a unified way. The theory of bud generating systems uses colored operads. Indeed, an object is generated by a bud generating system if it satisfies a certain equation in a colored operad. To compute the generating series of the languages of bud generating systems, we introduce formal power series on colored operads and several operations on these. Series on colored operads are crucial to express the languages specified by bud generating systems and allow us to enumerate combinatorial objects with respect to some statistics. Some examples of bud generating systems are constructed; in particular to specify some sorts of balanced trees and to obtain recursive formulas enumerating these.Comment: 48 page