Direct and dual laws for automata with multiplicities

We present here theoretical results coming from the implementation of the package called AMULT (automata with multiplicities in several noncommutative variables). We show that classical formulas are ``almost every time'' optimal, characterize the dual laws preserving rationality and also relators that are compatible with these laws

Extending the scalars of minimizations

In the classical theory of formal languages, finite state automata allow to recognize the words of a rational subset of $\Sigma^*$ where $\Sigma$ is a set of symbols (or the alphabet). Now, given a semiring (\K,+,.), one can construct \K-subsets of $\Sigma^*$ in the sense of Eilenberg , that are alternatively called noncommutative formal power series for which a framework very similar to language theory has been constructed Particular noncommutative formal power series, which are called rational series, are the behaviour of a family of weighted automata (or \K-automata). In order to get an efficient encoding, it may be interesting to point out one of them with the smallest number of states. Minimization processes of \K-automata already exist for \K being:\\ {\bf a)} a field ,\\ {\bf b)} a noncommutative field ,\\ {\bf c)} a PID .\\ When \K is the bolean semiring, such a minimization process (with isomorphisms of minimal objects) is known within the category of deterministic automata. Minimal automata have been proved to be isomorphic in cases {\bf (a)} and {\bf (b)}. But the proof given for (b) is not constructive. In fact, it lays on the existence of a basis for a submodule of \K^n. Here we give an independent algorithm which reproves this fact and an example of a pair of nonisomorphic minimal automata. Moreover, we examine the possibility of extending {\bf (c)}. To this end, we provide an {\em Effective Minimization Process} (or {\em EMP}) which can be used for more general sets of coefficients

The mechanics of shuffle products and their siblings

We carry on the investigation initiated in [15] : we describe new shuffle products coming from some special functions and group them, along with other products encountered in the literature, in a class of products, which we name $\varphi$-shuffle products. Our paper is dedicated to a study of the latter class, from a combinatorial standpoint. We consider first how to extend Radford's theorem to the products in that class, then how to construct their bi-algebras. As some conditions are necessary do carry that out, we study them closely and simplify them so that they can be seen directly from the definition of the product. We eventually test these conditions on the products mentioned above

Combinatorics of ϕ\phi-deformed stuffle Hopf algebras

In order to extend the Sch\"utzenberger's factorization to general perturbations, the combinatorial aspects of the Hopf algebra of the $\phi$-deformed stuffle product is developed systematically in a parallel way with those of the shuffle product

A Three Parameter Hopf Deformation of the Algebra of Feynman-like Diagrams

We construct a three-parameter deformation of the Hopf algebra \LDIAG. This is the algebra that appears in an expansion in terms of Feynman-like diagrams of the {\em product formula} in a simplified version of Quantum Field Theory. This new algebra is a true Hopf deformation which reduces to \LDIAG for some parameter values and to the algebra of Matrix Quasi-Symmetric Functions (\MQS) for others, and thus relates \LDIAG to other Hopf algebras of contemporary physics. Moreover, there is an onto linear mapping preserving products from our algebra to the algebra of Euler-Zagier sums

Combinatorial Deformations of Algebras: Twisting and Perturbations

The framework used to prove the multiplicative law deformation of the algebra of Feynman-Bender diagrams is a \textit{twisted shifted dual law} (in fact, twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameters is a perturbation of the shuffle coproduct of Hoffman type which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions

Algebraic Elimination of epsilon-transitions

We present here algebraic formulas associating a k-automaton to a k-epsilon-automaton. The existence depends on the definition of the star of matrices and of elements in the semiring k. For this reason, we present the theorem which allows the transformation of k-epsilon-automata into k-automata. The two automata have the same behaviour.Comment: 13 decembre 200