Given two weighted automata (or labelled Markov chains), is there a constant c such that the weight of every word in the first automaton is at most c times its weight in the second?

We measure distances between states in labelled Markov chains in the spirit of bisimilarity relation. It turns out that asymmetric distance functions are, in some sense, better.

A pair of states in a labelled Markov chain can be bisimilar or not bisimilar. We extend this equivalence relation to a distance function which upper-bounds the δ parameter in approximate differential privacy.

We find a matrix for which the nonnegative rank over the reals and over the rationals are different. Consequently, state minimization of hidden Markov models may require irrational probabilities.

There exists a pair of 3D polytopes with rational vertices, one inside the other, such that every intermediate polytope with 5 vertices must have a vertex with an irrational coordinate.