equivalent machines

equivalent machines

Let

M

1

=

(

S

1

,

Σ

,

Δ

,

σ

1

,

λ

1

)

and

M

2

=

(

S

2

,

Σ

,

Δ

,

σ

2

,

λ

2

)

be two

Mealy machines

.

Definition

. A state

s

1

of

M

1

is said to be

equivalent

to a state

s

2

of

M

2

iff

λ

1

⁢

(

s

1

,

u

)

=

λ

2

⁢

(

s

2

,

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:

s

1

≈

s

2

iff

β

1

⁢

(

s

1

,

u

)

=

β

2

⁢

(

s

2

,

u

)

⁢

for all input words

⁢

u

⁢

over

⁢

Σ

.

Here,

β

i

is the modifications of the original output

function

λ

i

.

Suppose now that

M

=

M

1

=

M

2

. 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.

S

′

:=

S

/

≈

=

{

[

s

]

∣

s

∈

S

}

is the set of

equivalence classes

of

≈

on

S

,

2.

σ

′

⁢

(

[

s

]

,

a

)

=

[

σ

⁢

(

s

,

a

)

]

,

3.

λ

′

⁢

(

[

s

]

,

a

)

=

λ

⁢

(

s

,

a

)

.

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

s

1

≈

s

2

implies

s

1

=

s

2

. Then the machine

M

′

constructed from

M

above is a reduced machine.

Suppose

M

1

and

M

2

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

Definition

.

M

1

is said to be

equivalent

to

M

2

if every state of

M

1

is equivalent to a state of

M

2

, and vice versa. When

M

1

is equivalent to

M

2

, we write

M

1

≈

M

2

.

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

addition

, 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

M

1

be a Mealy machine and

M

2

a Moore machine.

M

1

is said to be

similar

to

M

2

if every state

s

1

of

M

1

, there is a state

s

2

of

M

2

such that

λ

⁢

(

s

1

,

u

)

=

β

⁢

(

s

2

,

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

M

1

∼

M

2

if

M

1

and

M

2

are similar.

The

concept

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 word

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

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