Séries formelles et algèbres syntactiques

AbstractThe notion of the syntactic monoid is well known to be very important for formal languages, and in particular for rational languages; examples of that importance are Kleene's theorem, Schützenberger's theorem about aperiodic monoid and Eilenberg's theorem about varieties. We introduce here, for formal power series, a similar object: to each formal power series we associate its syntactic algebra. The Kleene-Schützenberger theorem can then be stated in the following way: a series is rational if and only if its syntactic algebra has finite dimension. A rational central series (this means that the coefficient of a word depends only on its conjugacy class) is a linear combination of characters if and only if its syntactic algebra is semisimple. Fatou properties of rational series in one variable are extended to series in several variables and a special case of the rationality of the Hadamard quotient of two series is positively answered. The correspondence between pseudovarieties of finite monoids and varieties of rational languages, as studied by Eilenberg, is extended between pseudovarieties of finite dimensional algebras and varieties of rational series. We study different kinds of varieties that are defined by closure properties and prove a theorem similar to Schützenberger's theorem on aperiodic monoids

Rational stochastic languages

The goal of the present paper is to provide a systematic and comprehensive study of rational stochastic languages over a semiring K \in {Q, Q +, R, R+}. A rational stochastic language is a probability distribution over a free monoid \Sigma^* which is rational over K, that is which can be generated by a multiplicity automata with parameters in K. We study the relations between the classes of rational stochastic languages S rat K (\Sigma). We define the notion of residual of a stochastic language and we use it to investigate properties of several subclasses of rational stochastic languages. Lastly, we study the representation of rational stochastic languages by means of multiplicity automata.Comment: 35 page

Algebraic and context-free subsets of subgroups

We study the relation between the structure of algebraic and context-free subsets of a group G and that of a finite index subgroup H. Using these results, we prove that a kind of Fatou property, previously studied by Berstel and Sakarovitch in the context of rational subsets and by Herbst in the context of algebraic subsets, holds for context-free subsets if and only if the group is virtually free. We also exhibit a counterexample to a question of Herbst concerning this property for algebraic subsets.Comment: minor change

On transductions of formal power series over complete semirings

AbstractRational and pushdown transductions of formal languages are generalized to formal power series with coefficients in a complete semiring. A characterization similar to Nivat's Theorem is given. Commutativity requirements for the coefficients are especially studied