A Moore machine $M$ is a five-tuple $(S,\Sigma,\Delta,\sigma,\lambda)$, where

1. 1.

   $S$ is an alphabet whose elements are called states,

2. 2.

   $\Sigma$ is an alphabet whose elements are called input symbols,

3. 3.

   $\Delta$ is an alphabet whose elements are called output symbols,

4. 4.

   $\sigma:S\times\Sigma\to S$ is a function called the state transition function,

5. 5.

   $\lambda:S\to\Delta$ is a function called the output function.

Given an a pair $(s,a)$ of state $s$ and input symbol $a$, the value $\sigma(s,a)$ is the next state of $(s,a)$ and $\lambda(s)$ is the output of $(s,a)$.

A Moore machine can be thought of as a Mealy machine, where that the output function does not depend on the input. Suppose $M=(S,\Sigma,\Delta,\sigma,\lambda)$ is a Moore machine, then $M^{{\prime}}=(S,\Sigma,\Delta,\sigma,\lambda^{{\prime}})$ where $\lambda^{{\prime}}:S\times\Sigma\to\Delta$, given by $\lambda^{{\prime}}(s,a)=\lambda(s)$, is a Mealy machine.

Like a Mealy machine, a Moore machine also has two visual presentations: via a flow table, which describes the transitions to the next states and the outputs, or via a state diagram.

For example, consider a Moore machine $M$ where $S=\{s,t\}$, $\Sigma=\{a,b\}$, and $\Delta=\{x,y\}$. The state transition function $\sigma$ and the output function $\lambda$ are defined by the following table:

The first column represents all the states of $M$; the second and third columns are the next states corresponding to the input symbols specified on the top row; and the last column shows the corresponding output symbols.

In addition, one can visualize a Moore machine $M$ by way its state diagram $G_{M}$, which is just a (labeled) directed graph. $G_{M}$ is constructed as follows: the vertices of $G_{M}$ represent the states of $M$, and each is labeled $s|x$, where $s$ is some state of $M$, and $x=\lambda(s)$. An edge from vertices $(s,x)$ to $(t,y)$ is constructed iff there is an input $a$ such that $\sigma(s,a)=t$, in which case the edge is labeled $a$. In the example above, the state diagram $G_{M}$ of $M$ is the following

Let $M$ be a Moore machine. Since $M$ is a special type of a Mealy machine, modification of the transition and output functions of $M$ to handle input words could be defined much the same way as if we were treating $M$ as a Mealy machine. But the downside is is that no output can correspond to the empty input word $\epsilon$. Because the output function $\lambda$ of $M$ is input-independent, a slightly different approach to modification gets rid of the problem:

First, define the extension of the transition function $\sigma$ of $M$ as if it were a Mealy machine:

Next, define a function $\beta:S\times\Sigma^{*}\to\Delta$ as follows:

Call $\beta$ the output function on input words for $M$. Notice that $\beta$ is not an extension of $\lambda$, since $\beta$ is input-dependent, whereas $\lambda$ is not. Furthermore, $\beta(s,\epsilon)=\lambda(s)$.

The approach above shows that the mechanism of a Moore machine for handling inputs/outputs is different from that of a Mealy machine, even though a Moore machine is really just a Mealy machine.