Mealy and Moore Machines in TOC

Moore and Mealy Machines are Transducers that help in producing outputs based on the input of the current state or previous state. In this article we are going to discuss Moore Machines and Mealy Machines, the difference between these two machines as well as Conversion from Moore to Mealy and Conversion from Mealy to Moore Machines.

Moore Machines

Moore Machines are finite state machines with output value and its output depends only on the present state. It can be defined as (Q, q0, ∑, O, δ, λ) where:

Q is a finite set of states.
q0 is the initial state.
∑ is the input alphabet.
O is the output alphabet.
δ is the transition function which maps Q×∑ → Q.
λ is the output function which maps Q → O.

Figure 1: Moore Machines

In the Moore machine shown in Figure 1, the output is represented with each input state separated by /. The length of output for a Moore Machine is greater than input by 1.

Input: 11
Transition: δ (q0,11)=> δ(q2,1)=>q2
Output: 000 (0 for q0, 0 for q2 and again 0 for q2)

Mealy Machines

Mealy machines are also finite state machines with output value and their output depends on the present state and current input symbol. It can be defined as (Q, q0, ∑, O, δ, λ’) where:

Q is a finite set of states.
q0 is the initial state.
∑ is the input alphabet.
O is the output alphabet.
δ is the transition function which maps Q×∑ → Q.
‘λ’ is the output function that maps Q×∑→ O.

Figure 2:Mealy Machines

In the mealy machine shown in Figure 1, the output is represented with each input symbol for each state separated by /. The length of output for a mealy machine is equal to the length of input.

Input:11
Transition: δ (q0,11)=> δ(q2,1)=>q2
Output: 00 (q0 to q2 transition has Output 0 and q2 to q2 transition also has Output 0)

NOTE: If there are n inputs in the Mealy machine then it generates n outputs while if there are n inputs in the Moore machine then it generates n + 1 outputs.

Moore Machines vs Mealy Machines

Aspect	Moore Machines	Mealy Machines
Output	Outputs depend only on the current state.	Outputs depend on the current state and input.
Number of States	Tends to require more states due to separate output behavior.	Might require fewer states as outputs are tied to transitions.
Response Time	Slower response to input changes as outputs update on state changes.	Faster response to input changes due to immediate output updates.
Complexity	Can be simpler due to separation of output behavior.	Can be more complex due to combined state-input cases.

For more differences, refer to Difference Between Moore and Mealy Machines.

Conversion From Mealy to Moore Machine

Let us take the transition table of the mealy machine shown in Figure 2.

	Input=0		Input=1
Present State	Next State	Output	Next State	Output
q0	q1	0	q2	0
q1	q1	0	q2	1
q2	q1	1	q2	0

Table 1

Step 1. First, find out those states which have more than 1 output associated with them. q1 and q2 are the states which have both output 0 and 1 associated with them.

Step 2 Create two states for these states. For q1, two states will be q10 (a state with output 0) and q11 (a state with output 1). Similarly, for q2, two states will be q20 and q21.

Step 3: Create an empty Moore machine with a newly generated state. For more machines, Output will be associated with each state irrespective of inputs.

	Input=0	Input=1
Present State	Next State	Next State	Output
q0
q10
q11
q20
q21

Table 2

Step 4: Fill in the entries of the next state using the mealy machine transition table shown in Table 1. For q0 on input 0, the next state is q10 (q1 with output 0). Similarly, for q0 on input 1, the next state is q20 (q2 with output 0). For q1 (both q10 and q11) on input 0, the next state is q10. Similarly, for q1(both q10 and q11), next state is q21. For q10, the output will be 0 and for q11, the output will be 1. Similarly, other entries can be filled.

	Input=0	Input=1
Present State	Next State	Next State	Output
q0	q10	q20	0
q10	q10	q21	0
q11	q10	q21	1
q20	q11	q20	0
q21	q11	q20	1

Table 3

This is the transition table of the Moore machine shown in Figure 1.

Conversion From Moore Machine to Mealy Machine

Let us take the Moore machine of Figure 1 and its transition table is shown in Table 3.

Step 1. Construct an empty mealy machine using all states of the Moore machine as shown in Table 4.

	Input=0		Input=1
Present State	Next State	Output	Next State	Output
q0
q10
q11
q20
q21

Table 4

Step 2: Next state for each state can also be directly found from Moore machine transition Table as:

	Input=0		Input=1
Present State	Next State	Output	Next State	Output
q0	q10		q20
q10	q10		q21
q11	q10		q21
q20	q11		q20
q21	q11		q20

Table 5

Step 3: As we can see output corresponds to each input in the Moore machine transition table. Use this to fill the Output entries.

Example: Output corresponding to q10, q11, q20 and q21 are 0, 1, 0 and 1 respectively.

	Input=0		Input=1
Present State	Next State	Output	Next State	Output
q0	q10	0	q20	0
q10	q10	0	q21	1
q11	q10	0	q21	1
q20	q11	1	q20	0
q21	q11	1	q20	0

Table 6

Step 4: As we can see from Table 6, q10 and q11 are similar to each other (same value of next state and Output for different Inputs). Similarly, q20 and q21 are also similar. So, q11 and q21 can be eliminated.

	Input=0		Input=1
Present State	Next State	Output	Next State	Output
q0	q10	0	q20	0
q10	q10	0	q21	1
q20	q11	1	q20	0

Table 7

This is the same mealy machine shown in Table 1. So we have converted Mealy to Moore machine and converted back Moore to Mealy.

Note: Number of statAes in the mealy machine can’t be greater than number of states in moore machine.

Example: The Finite state machine is described by the following state diagram with A as starting state, where an arc label is x / y and x stands for 1-bit input and y stands for 2-bit output.