## What is finite-state transducer? It is essentially a [[Finite-state automata]] that has both inputs and outputs. In [Turing machines](https://en.wikipedia.org/wiki/Turing_machine), these inputs and outputs are referred to as 2 separate tapes: - **input tape**: a set of strings or related data. - **output tape**: a set of relations generated from the input tape. This differs from a finite-state automaton which only has 1 (input) tape. ## Mathematical Model As per the general classification noted on [UC Davis outline on transducers](https://www.cs.ucdavis.edu/~rogaway/classes/120/spring13/eric-transducers) (formatted with similar variables to [[Finite-state automata]]s), a deterministic finite-state machine has 7 main variables associated with its definition (a septuple): ($\Sigma$, $S$, $\Gamma$, $\delta$, $\omega$, $s_0$, \_$F$). - $\Sigma$ is the _input alphabet_ (a finite non-empty set of symbols) -> our events; - $S$ is a finite non-empty set of states; - $\Gamma$ is the *output alphabet*; - $\delta$ is the state-transition function: $\delta: S \times \Sigma \rightarrow S$ - $\omega$ is the output-transition function: $\omega: S \times \Sigma \rightarrow \Gamma$ - $s_0$ is an _initial state_, an element of $S$; - $F$ is the set of *final states* and is a subset of $S$; and - $\delta \subseteq S \times (\Sigma \cup \{\epsilon\}) \times (\Gamma \cup \{\epsilon\}) \times S$ (where ε is the [empty string](https://en.wikipedia.org/wiki/Empty_string 'Empty string')) is the *transition relation*. Given any initial state in $s_0$, to transition our state to the next state with our output alphabet, our transition would be: $\delta: s_0 \times \Sigma \rightarrow S$ $\omega: s_0 \times \Sigma \rightarrow \Gamma$ $\delta \subseteq s_0 \times (\Sigma \cup \{\epsilon\}) \times (\Gamma \cup \{\epsilon\}) \times S$ If we were to reach a final state starting from a known set of states, our transition would look pretty similar: $\delta: S \times \Sigma \rightarrow F$ $\omega: S \times \Sigma \rightarrow \Gamma$ $\delta \subseteq S \times (\Sigma \cup \{\epsilon\}) \times (\Gamma \cup \{\epsilon\}) \times F$ Transducers that have a final state are used to recognize languages and have their use cases in [[Natural-Language Processing]]. ### Simplified meaning There's actually quite a few ways to write up a transducer, at it is not always simply a beefed-up state machine. To oversimplify, we'll model it closer to a regular state machine: typescript type State = string type Input = string type Output = {...} // const transition = (state: State, input: Input): Output => ...  Outputs that are product types of itself and the next state of a transition is referred to as a [[Mealy machine]]. ## Examples of basic transducers Although we've mentioned before that [[Reducers]] are single state machines, the canonical method of creating one mentioned in [Redux](https://redux.js.org/) and [React](https://reactjs.org/docs/hooks-reference.html#usereducer) are pretty much transducers as they can return state or objects (as **notions** of outputs). typescript // https://reactjs.org/docs/hooks-reference.html#usereducer // our outputs here is the { count: number } object const initialState = { count: 0 } function reducer(state, action) { switch (action.type) { case 'increment': return { count: state.count + 1 } case 'decrement': return { count: state.count - 1 } default: throw new Error() } } function Counter() { const [state, dispatch] = useReducer(reducer, initialState) return ( <> Count: {state.count} <button onClick={() => dispatch({ type: 'decrement' })}>-</button> <button onClick={() => dispatch({ type: 'increment' })}>+</button> </> ) }  We can take our example from [[Finite-state automata]] and wrap it in a way that encapsulates both state, inputs and outputs (in Rescript): typescript // elapsed represents our arbitrary output type elapsed = float; // elapsed here is used in a constructor as an arbitrary output type taskStatus = | NotStarted | Running(elapsed) | Paused(elapsed) | Done(elapsed); // elapsed here is used in a constructor as an arbitrary input type input = | Start | Pause | Resume | Finish | Tick(elapsed); let transition = (state, input) => switch (state, input) { | (NotStarted, Start) => Running(0.0) | (Running(elapsed), Pause) => Paused(elapsed) | (Running(elapsed), Finish) => Done(elapsed) | (Paused(elapsed), Resume) => Running(elapsed) | (Paused(elapsed), Finish) => Done(elapsed) | (Running(elapsed), Tick(tick)) => Running(elapsed +. tick) | _ => state };  ## Reference - https://en.wikipedia.org/wiki/Turing_machine - https://t-pl.io/ddd-aggregates-processes-state-machines-and-transducers - https://en.wikipedia.org/wiki/Finite-state_transducer - https://reactjs.org/docs/hooks-reference.html#usereducer - https://www.cs.ucdavis.edu/~rogaway/classes/120/spring13/eric-transducers - https://dl.acm.org/doi/10.5555/972695.972698 - https://web.stanford.edu/~laurik/publications/ciaa-2000/fst-in-nlp/fst-in-nlp.html