Mealy machine

Builds on Finite-state machine Alphabet String

Definition

A Mealy machine is a tuple $M = (Q, \Sigma, \Gamma, \delta, \lambda, q_0)$ where:

$(Q, \Sigma, \delta, q_0)$ is a finite-state machine, with finite set of states $Q$ , input alphabet $\Sigma$ , transition function $\delta \colon Q \times \Sigma \to Q$ and initial state $q_0$ ;
$\Gamma$ is an alphabet, the output alphabet;
$\lambda \colon Q \times \Sigma \to \Gamma$ is the output function.

On an input string $a_1 a_2 \cdots a_n$ with $a_i \in \Sigma$ , let $q_i = \delta(q_{i-1}, a_i)$ for $1 \le i \le n$ . The output of $M$ is the string $\lambda(q_0, a_1)\, \lambda(q_1, a_2) \cdots \lambda(q_{n-1}, a_n) \in \Gamma^*$ .

Remarks

Example: let $\Sigma = \Gamma = \set{0, 1}$ and $Q = \set{s, 0, 1}$ with $q_0 = s$ . Put $\delta(q, a) = a$ for all $q$ and $a$ , $\lambda(s, a) = 0$ , and $\lambda(q, a) = 1$ if $q = a$ and $0$ otherwise, for $q \in \set{0, 1}$ . The state after reading a nonempty word is its last bit, so on $0010$ the output is $0100$ : a $1$ marks each bit equal to the bit before it. Non-example: consider a machine that writes $1$ exactly when the number of $1$ 's read so far is a perfect square. It is not a Mealy machine, because no finite set of states can store the information it needs. The output word always has the same length as the input word.

Reading path

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Alphabet concept
Finite-state machine concept
String concept
Mealy machine definition

Tag	`CCNRC91FJ2`
URI	`m2://CCNRC91FJ2`
Kind	definition
Writing	W0 stub
MSC	68Q45
Defines	Mealy machine
Uses	Finite-state machine, Alphabet, String
Changed	3 November 2026

Mealy machine

Definition

Remarks

Reading path

Further reading

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