## Section: New Results

### A Kleene Theorem for Forest Languages

Participant : Lutz Straßburger.

Kleene's theorem on the coincidence of the rational and the recognizable languages in a free monoid [47] is a fundamental result of theoretical computer science. In [20] we present a generalization of Kleene's theorem to forest languages, which are a generalization of tree languages. However, our result is not a generalization of the result by Thatcher and Wright on tree languages [64] . We proposed an alternative approach to the standard notion of rational (or regular) expression for tree languages. The main difference is that in our new notion we have only one concatenation operation and only one star-operation, instead of many different ones. This is achieved by considering forests instead of trees over a ranked alphabet, or, algebraicly speaking, by considering cartesian categories instead of term-algebras. The main result is that in the free cartesian category the rational languages and the recognizable languages coincide.