  A finite automaton is called bideterministic if it is both deterministic and
codeterministic -- that is, if it is deterministic and its transpose is
deterministic as well. The study of such automata in a weighted setting is
initiated. All trim bideterministic weighted automata over integral domains and
over positive semirings are proved to be minimal. On the contrary, it is
observed that this property does not hold over commutative rings in general:
non-minimal trim bideterministic weighted automata do exist over all semirings
that are not zero-divisor free, and over many such semirings, these automata
might not even admit equivalents that are both minimal and bideterministic. The
problem of determining whether a given rational series is realised by a
bideterministic automaton is shown to be decidable over fields and over
tropical semirings. An example of a positive semiring over which this problem
becomes undecidable is given as well.
