State-Identification Problems for Finite-State Transducers

A well-established theory exists for testing finite-state machines, in particular Moore and Mealy machines. A fundamental class of problems handled by this theory is state identification: we are given a machine with known state space and transition relation but unknown initial state, and we are asked to find experiments which permit to identify the initial or final state of the machine, called distinguishing and homing experiments, respectively. In this paper, we study state-identification for finite-state transducers. The latter are a generalization of Mealy machines where outputs are sequences rather than symbols. Transducers permit to model systems where inputs and outputs are not synchronous, as is the case in Mealy machines. It is well-known that every deterministic and minimal Mealy machine admits a homing experiment. We show that this property fails for transducers, even when the latter are deterministic and minimal. We provide answers to the decidability question, namely, checking whether a given transducer admits a particular type of experiment. First, we show how the standard successor-tree algorithm for Mealy machines can be turned into a semi-algorithm for transducers. Second, we show that the state-identification problems are undecidable for finite-state transducers in general. Finally, we identify a sub-class of transducers for which these problems are decidable. A transducer in this sub-class can be transformed into a Mealy machine, to which existing methods apply.