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The (left) linear hull of a weighted automaton over a field is a topologicalinvariant. If the automaton is minimal, the linear hull can be used todetermine whether or not the automaton is equivalent to a deterministic one.Furthermore, the linear hull can also be used to determine whether the minimalautomaton is equivalent to an unambiguous one. We show how to compute thelinear hull, and thus prove that it is decidable whether or not a givenautomaton over a number field is equivalent to a deterministic one. In thiscase we are also able to compute an equivalent deterministic automaton. We alsoshow the analogous decidability and computability result for the unambiguouscase. Our results resolve a problem posed in a 2006 survey by Lombardy andSakarovitch.

Computing the linear hull: Deciding Deterministic? and Unambiguous? for weighted automata over fields