On the decidability of the valuedness problem for two-way finite transducers

A transducer is infinite-valued if the maximal number of different outputs for an input string is not bounded by any constant. For one-way finite transducers, sufficient and necessary conditions exist in terms of the structure of a transducer to characterize whether the transducer is infinite-valued or not, yielding the decidability result of the finite-valuedness problem. As crossing sequences in two-way automata often play similar roles as states in their one-way counterparts, we consider analogous criteria in the setting of crossing sequences to characterize the infinite-valuedness of two-way finite transducers. The characterization leads to a decidability proof for the valuedness problem of two-way finite transducers.