Rational relations and rational series

On Rational Series and Rational Languages

We study the connections between rational series with coeecients in a semiring and their languages.

Rational relations having a rational trace on each finite intersection of rational relations

We consider the family of rational relations on words, i.e., relations recognized by multitape automata, which have rational trace on any finite intersection of rational relations. We prove that this family consists of finite unions of relations which are of two types: stars of tuples possibly with an extra prefix and suffix and relations with all but one singleton set components.

Rational tree relations

We investigate forests and relations on trees generated by grammars in which the non-terminals represent relations. This introduces some synchronization between productions. We show that these sets are also solutions of systems of equations, that they are described by rational expressions involving union, substitution and iterated substitution, and that they are preserved by residuals. We show ...

Uniformization of Rational Relations

Axiomatizing rational power series

Iteration semirings are Conway semirings satisfying Conway’s group identities. We show that the semirings Nrat〈〈Σ∗〉〉 of rational power series with coefficients in the semiring N of natural numbers are the free partial iteration semirings. Moreover, we characterize the semirings Nrat ∞ 〈〈Σ∗〉〉 as the free semirings in the variety of iteration semirings defined by three additional simple identitie...