Channel Synthesis for Finite Transducers

We investigate how two agents can communicate through a noisy medium modeled as a finite non deterministic transducer. The sender and the receiver are also described by finite transducers which can respectively encode and decode binary messages. When the communication is reliable, we call the encoder/decoder pair a channel. We study the channel synthesis problem which, given a transducer, asks whether or not such sender and receiver exist and builds them if the answer is positive. To that effect we introduce the structural notion of encoding state in a transducer which is a necessary condition for the existence of a channel. It is not, however, a sufficient condition. In fact, we prove that the problem is undecidable. Nonetheless, we obtain a synthesis procedure when the transducer is functional. We discuss these results in relation to security properties.