ϵ-transition

ϵ

-transition

Let

A

=

(

S

,

Σ

,

δ

,

I

,

F

)

be an

automaton

. Recall that a transition of

A

is a triple written in the form

p

⟶

α

q

,

where

q

is a next state of the

configuration

(

p

,

α

)

. In other words,

q

∈

δ

⁢

(

p

,

α

)

. Here,

α

is a symbol in

Σ

.

The notion of transitions can be generalized so that

p

⟶

a

q

iff

q

∈

δ

⁢

(

p

,

a

)

, where

a

∈

Σ

*

. The definition of the extended transition function

δ

dictates that

p

⟶

λ

q

implies

p

=

q

. This means that if the

empty word

is fed into an automaton at any state

p

, the next state stays the same. In other words, the automaton does nothing.

An

ϵ

-transition, informally, is a transition in an “automaton” that changes the initial state when an empty word is read. This means, notationally, that

p

⟶

λ

q

where

p

is not necessarily

q

. The double quotes around the word automaton is to signify the fact that when

ϵ

-transitions are considered, the

machine

is no longer an automaton strictly speaking. The “

ϵ

” here refers to the empty word

λ

, which is sometimes denoted by

ϵ

.

The

consequence

of adding

ϵ

-transitions is that the set of next states is potentially enlarged whenever a word is read, because at any point during the reading, a next state could result from an

alphabet

, or it could come from any of the next states by inserting several empty words after the alphabet. Outwardly, the

insertions

of empty words into a word does not change the word itself. However, the possibility of accepting the word is increased. So the question is: does this “automaton with

ϵ

-transitoins” offers more computing power than a traditional automaton? Interestingly, the answer is no, and we will demonstrate this fact in this article.

First, we need to define formally what

ϵ

-transitions and automata with

ϵ

-transitions are.

Definitions

. There are several

concepts

that need to be formalized:

1.

An

automaton with

ϵ

-transitions

, or

ϵ

-automaton

for short, is a sextuple

E

:=

(

S

,

Σ

,

δ

,

I

,

F

,

ϵ

)

such that

ϵ

∉

Σ

, and

E

ϵ

:=

(

S

,

Σ

ϵ

,

δ

,

I

,

F

)

, where

Σ

ϵ

:=

Σ

∪

{

ϵ

}

, is an automaton, called the automaton

associated with

E

.

2.

An

ϵ

-transition

is a transition of

E

ϵ

of the form

p

⟶

ϵ

q

.

3.

We define the

language

L

⁢

(

E

)

accepted by an

ϵ

-automaton

E

to be the set of all words in

Σ

*

that can be obtained from words accepted by

E

ϵ

by deleting any occurrences of

ϵ

. Formally,

L

⁢

(

E

)

:=

h

⁢

(

L

⁢

(

E

ϵ

)

)

,

where

h

:

Σ

ϵ

*

→

Σ

*

is the

monoid homomorphism

given by

h

|

Σ

=

1

and

h

⁢

(

ϵ

)

=

λ

. It is easy to see, by

induction

, that

h

⁢

(

a

)

=

α

1

⁢

α

2

⁢

⋯

⁢

α

n

iff

a

=

ϵ

i

0

⁢

α

1

⁢

ϵ

i

1

⁢

α

2

⁢

ϵ

i

2

⁢

⋯

⁢

ϵ

i

n

-

1

⁢

α

n

⁢

ϵ

i

n

,

where

α

j

∈

Σ

,

j

=

1

,

2

,

…

,

n

.

4.

We say that an

ϵ

-automaton

E

is

equivalent

to an automaton

A

if

L

⁢

(

E

)

=

L

⁢

(

A

)

. It is not hard to see that every

ϵ

-automaton is equivalent to an automaton (proof here (

http://planetmath.org/EveryEpsilonAutomatonIsEquivalentToAnAutomaton

)).

Remark

.

ϵ

-transitions are useful in proving the two properties of

regular languages

: 1. if two languages are

regular

, so is their

juxtaposition

(see proof here (

http://planetmath.org/JuxtapositionOfAutomata

)), and 2. if a language is regular, so is its

Kleene star

(see proof here (

http://planetmath.org/KleeneStarOfAnAutomaton

)).

Title

ϵ

-transition

Canonical name

epsilontransition

Date of creation

2013-03-22 18:03:24

Last modified on

2013-03-22 18:03:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

20

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D05

Classification

msc 68Q45

Synonym

ϵ

-automaton

Synonym

epsilon-automaton

Synonym

epsilon-transition

Synonym

automaton with epsilon-transitions

Defines

automaton with

ϵ

-transitions

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