## 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