How do finite transducers process input and produce output?

In automata theory, a finite transducer is an automaton equipped with both input and output capabilities, often conceptualized as a "black box" that processes input symbols and emits output symbols.12

Formally, a finite transducer is defined as a quintuple $(S,Q,O,M,N)$, where:
* S, Q, and O are finite, nonempty sets representing inputs, internal states, and outputs, respectively.1
* M is a function that maps the current state and input to a new state ($Q\times S\to Q$).1
* N is a function that maps the current state and input to an output ($Q\times S\to O$).1

As the machine operates, it transitions between states based on the input received while simultaneously generating an output determined by its current state and that input. These functions can be extended to process sequences of inputs. Transducers can also be interconnected in series, where the output of one machine serves as the input to another, or in parallel, where inputs are processed simultaneously by multiple systems.13