Let Σ be a finite alphabet. A set R of regular languages over Σ is called rational if there exists a finite set E of regular languages over Σ, such that R is a rational subset of the finitely generated semigroup (S, ·) = 〈E〉 with E as the set of generators and language concatenation as a product. We prove that for any rational set R and any regular language R ⊆ Σ∗ it is decidable (1) whether R ...

We give a new and hopefully more easily understandable structural proof for the decomposition of a k-valued transducer into k unambiguous functional transducers, a result established by A. Weber. The number of states of the transducers given by our construction outperforms Weber’s one by one exponential. Moreover, we solve a problem left open by Weber, the decomposition of a bounded length-degr...

We study symmetric play in a class of repeated games when players are patient. We show that, while the use of symmetric strategy profiles essentially does not restrict the set of feasible payoffs, the set of equilibrium payoffs is an interesting proper subset of the feasible and individually rational set. We also provide a theory of how rational individuals play these games, identifying particu...

Ulam asked in 1945 if there is an everywhere dense rational set, i.e. a point set in the plane with all its pairwise distances rational. Erdős conjectured that if a set S has a dense rational subset, then S should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erdős’ conjecture for algeb...

Guided by the idea of letters with finite lists of predecessors and successors, the paper develops the Chomsky-Schützenberger theorem for languages of arrows in any free prop with a finite set of generators on the positive integers. The setting of monoidal categories is essential to obtain a well-behaved generalization of rational set, from monoids to monoidal categories.

The paper argues that, although a distinction between a priori and a posteriori knowledge (or justification) can be drawn, it is a superficial one, of little theoretical significance. The point is not that the distinction has borderline cases, for virtually all useful distinctions have such cases. Rather, it is argued by means of an example, the differences even between a clear case of a priori...