Bideterministic Weighted Automata

Abstract

A deterministic finite automaton is called bideterministic if 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 positive semirings are proved to be minimal. On the contrary, it is observed that this property does not hold over finite commutative rings in general. Moreover, it is shown that the problem of determining whether a given rational series is realised by a bideterministic automaton is decidable over fields as well as over tropical semirings.