## What is a Mealy machine?
A Mealy machine is a [[Finite-state automata]] where the output values are determined by its current state and current inputs. It is the closest definition to a deterministic [[Finite-state transducer]].
## 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 Mealy machine has 6 main variables associated with its definition (a sextuple): ($\Sigma$, $S$, $\Gamma$, $\delta$, $\omega$, $s_0$).
- $\Sigma$ 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 $S$; 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*.
Some formulations also allow transition and output functions to be combined as a single function:
$
\delta: S \times \Sigma \rightarrow S \times \Gamma
$
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
$
## Examples of basic Mealy machines
Our example from [[Finite-state transducer]]s fits perfectly here as our transition and output function are coalesced as a single function.
```typescript
// expiry represents our arbitrary output (in seconds)
type expiry = float;
// expiry here is used in a constructor as an arbitrary output
type trafficLightStatus =
| Red(expiry)
| Amber(expiry)
| Green(expiry)
| FlashingRed(expiry)
// elapsed here is used in a constructor as an arbitrary input
type input =
| ExpireTime
| Error
| Restart
let transition = (state, input) =>
switch (state, input) {
| (Red(expiry), ExpireTime) => Green(60.0)
| (Red(expiry), Error) => FlashingRed(30.0)
| (Green(expiry), ExpireTime) => Amber(60.0)
| (Green(expiry), Error) => FlashingRed(30.0)
| (Amber(expiry), ExpireTime) => Red(60.0)
| (Amber(expiry), Error) => FlashingRed(30.0)
| (FlashingRed(expiry), Restart) => Red(60.0)
| _ => state
};
```
## Differences between
### With formal [[Finite-state transducer]]s
Mealy machines are a type of generator and are not used in processing language. As such, they do not have a concept of a final state.
### With [[Moore Machine]]s
oth Mealy and Moore machines are generator-type state machines and can be used to parse [regular language](https://en.wikipedia.org/wiki/Regular_language). The outputs on a Mealy machine depend on **both the state and inputs**, whereas a Moore machine have their outputs **synchronously change with the state.**
> Every Moore machine can be converted to a Mealy machine and every Mealy machine can be converted to a Moore machine. Moore machine and Mealy machine are equivalent.
## Reference
- https://en.wikipedia.org/wiki/Mealy_machine
- https://www.cs.ucdavis.edu/~rogaway/classes/120/spring13/eric-transducers
- https://unstop.com/blog/difference-between-mealy-and-moore-machine