linear bounded automaton

linear bounded automaton

A

linear bounded automaton

, or LBA for short, is a restricted form of a

non-deterministic Turing machine

with a single tape and a single tape head, such that, given an input word on the tape, the tape head can only scan and rewrite symbols on the cells occupied by the initial input word.

Formally, a

linear bounded automaton

is a 9-tuple

M

=

(

Q

,

Σ

,

Γ

,

δ

,

q

0

,

B

,

F

,

\Yleft

,

\Yright

)

, where

1.

Q

is the state

alphabet

2.

Σ

is the input alphabet

3.

Γ

is the tape alphabet and

Σ

⊂

Γ

4.

\Yleft

,

\Yright

∈

Σ

are the left and right end markers.

5.

δ

is a

function

from

Q

×

Γ

to

P

⁢

(

Q

×

Γ

×

{

L

,

R

}

)

, called the

next move function

6.

q

0

∈

Q

is the

start state

7.

B

∈

Γ

-

Σ

is the blank symbol

8.

F

⊂

Q

is the set of final states

The

interpretation

of

(

q

,

b

,

D

)

∈

δ

⁢

(

p

,

a

)

is the same as that of a non-deterministic TM: if the

M

is in state

p

and the tape head is reading a tape cell containing

a

, then it replaces

a

by

b

in that tape cell, and move in the direction of

D

(

D

=

L

for left, and

D

=

R

for right), and changes its state to

q

. In

addition

, the next move function

δ

has the following

properties

:

•

if

(

q

,

b

,

D

)

∈

δ

⁢

(

p

,

\Yleft

)

, then

b

=

\Yleft

and

D

=

R

, and

•

if

(

q

,

b

,

D

)

∈

δ

⁢

(

p

,

\Yright

)

, then

b

=

\Yright

and

D

=

L

.

•

if

(

q

,

b

,

D

)

∈

δ

⁢

(

p

,

a

)

and

a

∉

{

\Yleft

,

\Yright

}

, then

b

∉

{

\Yleft

,

\Yright

}

.

In other words, if the tape head is reading the left end marker

\Yleft

, then if it has a next move, it can only move right without replacing the end marker

\Yleft

. Similarly, if it is reading the right end marker

\Yright

, then it can only move left if it has any next move at all, without replacing

\Yright

. Furthermore, no symbol can be replaced by an end marker unless the symbol itself is the end marker.

An LBA is also known as a non-deterministic LBA, or NLBA for short. An LBA is said to be

deterministic

(abbreviated as DLBA) if

δ

⁢

(

p

,

a

)

is at most a

singleton

for every pair

(

p

,

a

)

∈

Q

×

Γ

.

A

configuration

of

M

is a triple

(

p

,

u

,

i

)

, where

p

∈

Q

is the current state of

M

,

u

∈

{

\Yleft

}

⁢

Γ

*

⁢

{

\Yright

}

is the

content

of the tape (including the end markers), and

i

a non-negative integer, the position of the tape head, where

i

=

0

is the position of

\Yleft

, the left end marker.

Define a

binary relation

⇒

on the set of all configurations of

M

: for

a

i

,

b

∈

Γ

,

(

p

,

a

0

⁢

⋯

⁢

a

i

-

1

⁢

a

⁢

a

i

+

1

⁢

⋯

⁢

a

n

+

1

,

i

)

⇒

(

q

,

a

0

⁢

⋯

⁢

a

i

-

1

⁢

b

⁢

a

i

+

1

⁢

⋯

⁢

a

n

+

1

,

j

)

,

where

i

=

1

,

…

,

n

and

n

a non-negative integer, iff any one of the following holds:

(

q

,

b

,

L

)

∈

δ

⁢

(

p

,

a

i

)

⁢

and

⁢

j

=

i

-

1

or

(

q

,

b

,

R

)

∈

δ

⁢

(

p

,

a

i

)

⁢

and

⁢

j

=

i

+

1

.

Notice that

a

0

=

\Yleft

and

a

n

+

1

=

\Yright

. If

i

=

0

, then the former case can not happen, and if

i

=

n

+

1

, then the latter case can not happen.

Take the

reflexive transitive closure

⇒

*

of

⇒

, and set

L

(

M

)

:=

{

u

∈

Σ

*

∣

(

q

0

,

\Yleft

u

\Yright

,

0

)

⇒

*

(

q

,

\Yleft

v

\Yright

,

j

)

for some

q

∈

F

}

the

language

accepted by

M

.

Remarks

•

For a given input word

u

, the workspace

w

⁢

(

u

)

of an LBA is defined as the portion of the tape between the end markers (including the end markers). In the

definition

above, we see that

|

w

⁢

(

u

)

|

=

|

u

|

+

2

, where

|

⋅

|

is the

length function

. One can enlarge the workspace so that

|

w

⁢

(

u

)

|

=

r

⁢

|

u

|

+

s

, where

r

,

s

are positive integers with

r

≥

1

and

s

≥

2

. This is the reason for the name “linear bounded” in LBA. However, the computational power of an LBA with enlarged workspace is not increased: the language accepted by such an LBA can be accepted by an LBA defined above.

•

A language is

context-sensitive

iff it can be accepted by an LBA.

•

Every

context-free language

can be accepted by a DLBA, but not

conversely

.

•

It is an

open question

whether every context-sensitive language can be accepted by a DLBA.

References

1

J.E. Hopcroft, J.D. Ullman,

Formal Languages

and Their

Relation

to Automata

, Addison-Wesley, (1969).

Title

linear bounded automaton

Canonical

name

LinearBoundedAutomaton

Date of creation

2013-03-22 18:57:40

Last modified on

2013-03-22 18:57:40

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q45

Synonym

LBA

Synonym

NLBA

Synonym

DLBA

Synonym

deterministic LBA

Related topic

ContextSensitiveGrammar

Related topic

ContextSensitiveLanguage

Defines

deterministic linear bounded automaton

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