Mahsa Shirmohammadi - Minimal probabilistic automata have to make irrational choices

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In this talk, we answer the question of whether, given a probabilistic automaton (PA) with rational transition probabilities, there always exists a minimal equivalent PA that also has rational transition probabilities. The PA and its minimal equivalent PA accept all finite words with equal probabilities.

We approach this problem by exhibiting a connection with a longstanding open question concerning nonnegative matrix factorization (NMF). NMF is the problem of decomposing a given nonnegative n×m matrix M into a product of a nonnegative n×d matrix W and a nonnegative d×m matrix H. In 1993, Cohen and Rothblum posed the problem of whether a rational matrix M always has an NMF of minimal inner dimension d whose factors W and H are also rational. We answer this question negatively, by exhibiting a matrix for which W and H require irrational entries. As a corollary we obtain a negative answer to the question on rationality of minimal probabilistic automata.

This talk is based on two papers in ICALP 2016 and SODA 2017.