equivalent machines

Let M1=(S1,Σ,Δ,σ1,λ1) and M2=(S2,Σ,Δ,σ2,λ2) be two Mealy machines.

Definition. A state s1 of M1 is said to be equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to a state s2 of M2 iff

λ1(s1,u)=λ2(s2,u) for all non-empty input words u over Σ.

We write for the equivalence.

In the same way, can be defined on two Moore machines. However, where as u is restricted to be non-empty in the Mealy case, input words in Moore case are arbitrary, that is: s1s2 iff

β1(s1,u)=β2(s2,u) for all input words u over Σ.

Here, βi is the modifications of the original output function λi.

Suppose now that M=M1=M2. Then is an equivalence relation on the state alphabet S of M. Given M, form a new machine M=(S,Σ,Δ,σ,λ) such that

  1. 1.

    S:=S/={[s]sS} is the set of equivalence classesMathworldPlanetmath of on S,

  2. 2.


  3. 3.


It is easy to verify that M is indeed a well-defined machine. Furthermore, if M is Moore, so is M.

Call a machine (Mealy or Moore) reduced if s1s2 implies s1=s2. Then the machine M constructed from M above is a reduced machine.

Suppose M1 and M2 are both Mealy machines such that they have the same input and output alphabets.

Definition. M1 is said to be equivalent to M2 if every state of M1 is equivalent to a state of M2, and vice versa. When M1 is equivalent to M2, we write M1M2.

Equivalence between two Moore machines is similarly defined. It is clear that is an equivalence relation on the class of Mealy machines (or Moore machines) over the same input and output alphabets.

In additionPlanetmathPlanetmath, it is also possible to define “equivalence” between machines of different types. However, in the literature, the word “similar” is used instead of “equivalent”:

Definition. Let M1 be a Mealy machine and M2 a Moore machine. M1 is said to be similar to M2 if every state s1 of M1, there is a state s2 of M2 such that

λ(s1,u)=β(s2,u) for all non-empty input words u over Σ.

The definition of a Moore machine similar to a Mealy machine is analogous. Note that in the definition of similarity, input words are restricted to non-empty words only.

Two machines of different types are similar if one is similar to another, and vice versa. We write M1M2 if M1 and M2 are similar.

The conceptMathworldPlanetmath of similarity may be broadened to machines of the same type. In the Mealy case, similarity is the same as equivalence. In the Moore case, the two notions are different. Equivalence is more restrictive in that the empty wordPlanetmathPlanetmathPlanetmath is required in the definition. Two Moore machines may be similar without being equivalent. In fact, we have the following facts:

Proposition 1.

Two Mealy machines similar to a Moore machine are equivalent. There exist two inequivalent Moore machines similar to a Mealy machine.

Title equivalent machines
Canonical name EquivalentMachines
Date of creation 2013-03-22 19:02:39
Last modified on 2013-03-22 19:02:39
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 03D05
Classification msc 68Q45
Related topic MealyMachine
Related topic MooreMachine
Defines equivalent
Defines similar
Defines reduced machine