semi-Thue system

semi-Thue system

A

semi-Thue system

𝔖

is a pair

(

Σ

,

P

)

where

Σ

is an

alphabet

and

P

is a non-empty finite

binary relation

on

Σ

*

, the

Kleene star

of

Σ

.

Elements of

P

are variously called

defining relations

,

productions

, or

rewrite rules

, and

𝔖

itself is also known as a

rewriting system

. If

(

x

,

y

)

∈

P

, we call

x

the

antecedent

, and

y

the

consequent

. Instead of writing

(

x

,

y

)

∈

P

or

x

⁢

P

⁢

y

, we usually write

x

→

y

.

Let

𝔖

=

(

Σ

,

P

)

be a semi-Thue system. Given a word

u

over

Σ

, we say that a word

v

over

Σ

is

immediately derivable

from

u

if there is a defining relation

x

→

y

such that

u

=

r

⁢

x

⁢

s

and

v

=

r

⁢

y

⁢

s

,

for some words

r

,

s

(which may be empty) over

Σ

. If

v

is immediately derivable from

u

, we write

u

⇒

v

.

Let

P

′

be the set of all pairs

(

u

,

v

)

∈

Σ

*

×

Σ

*

such that

u

⇒

v

. Then

P

⊆

P

′

, and

If

u

⇒

v

, then

w

⁢

u

⇒

w

⁢

v

and

u

⁢

w

⇒

v

⁢

w

for any word

w

.

Next, take the

reflexive transitive closure

P

′′

of

P

′

. Write

a

⇒

*

b

for

(

a

,

b

)

∈

P

′′

. So

a

⇒

*

b

means that either

a

=

b

, or there is a

finite chain

a

=

a

1

,

…

,

a

n

=

b

such that

a

i

⇒

a

i

+

1

for

i

=

1

,

…

,

n

-

1

. When

a

⇒

*

b

, we say that

b

is

derivable

from

a

.

Concatenation

preserves

derivability:

a

⇒

*

b

and

c

⇒

*

d

imply

a

⁢

c

⇒

*

b

⁢

d

.

Example

. Let

𝔖

be a semi-Thue system over the alphabet

Σ

=

{

a

,

b

,

c

}

, with the set of defining relations given by

P

=

{

a

b

→

b

c

,

b

c

→

c

b

}

. Then words

a

⁢

c

3

⁢

b

,

a

2

⁢

c

2

⁢

b

and
as

b

⁢

c

4

are all derivable from

a

2

⁢

b

⁢

c

2

:

•

a

2

⁢

b

⁢

c

2

⇒

a

⁢

(

b

⁢

c

)

⁢

c

2

⇒

a

⁢

c

⁢

(

b

⁢

c

)

⁢

c

⇒

a

⁢

c

2

⁢

(

c

⁢

b

)

=

a

⁢

c

3

⁢

b

,

•

a

2

⁢

b

⁢

c

2

⇒

a

2

⁢

(

c

⁢

b

)

⁢

c

⇒

a

2

⁢

c

⁢

(

c

⁢

b

)

=

a

2

⁢

c

2

⁢

b

, and

•

a

2

⁢

b

⁢

c

2

⇒

a

⁢

(

b

⁢

c

)

⁢

c

2

⇒

(

b

⁢

c

)

⁢

c

⁢

c

2

=

b

⁢

c

4

.

Under

𝔖

, we see that if

v

is derivable from

u

, then they have the same length:

|

u

|

=

|

v

|

. Furthermore, if we denote

|

a

|

u

the number of

occurrences

of letter

a

in a word

u

, then

|

a

|

v

≤

|

a

|

u

,

|

c

|

v

≥

|

c

|

u

, and

|

b

|

v

=

|

b

|

u

. Also, in order for a word

u

to have a non-trivial word

v

(non-trivial in the sense that

u

≠

v

) derivable from it,

u

must have either

a

⁢

b

or

b

⁢

c

as a subword. Therefore, words like

a

3

or

c

3

⁢

b

4

⁢

a

2

have no non-trivial derived words from them.

Remarks

.

1.

Given a semi-Thue system

𝔖

=

(

Σ

,

P

)

, one can associate a subset

A

of

Σ

*

whose

elements

we call

axioms

of

𝔖

. Any word

v

that is derivable from an axiom

a

∈

A

is called a

theorem

(of

𝔖

). If

v

is a theorem, we write

A

⊢

𝔖

v

. The set of all theorems is written

L

𝔖

⁢

(

A

)

, and is called the

language

(over

Σ

) generated by

A

.

2.

Let

𝔖

and

A

be defined as above, and

T

any alphabet. Call the elements of

T

∩

Σ

the

terminals

of

𝔖

. The set

L

𝔖

⁢

(

A

)

∩

T

*

is called the

language generated by

A

over

T

, and written

L

𝔖

⁢

(

A

,

T

)

. It is easy to see that

L

𝔖

⁢

(

A

,

T

)

=

L

𝔖

⁢

(

A

,

T

∩

Σ

)

.

3.

A language

L

over an alphabet

Σ

is said to be

generable by a semi-Thue system

if there is a semi-Thue system

𝔖

and a

finite set

of axioms

A

of

𝔖

such that

L

=

L

𝔖

⁢

(

A

,

Σ

)

.

4.

Semi-Thue systems are “

equivalent

” to formal grammars in the following sense:

a language is generable by a formal grammar iff it is semi-Thue generable.

The idea is to turn every defining relation

x

→

y

in

P

into a production

S

⁢

x

⁢

T

→

S

⁢

y

⁢

T

, where

S

and

T

are non-terminals or

variables

. As such, a production of the form

S

⁢

x

⁢

T

→

S

⁢

y

⁢

T

is sometimes called a

semi-Thue production

.

5.

Given a semi-Thue system

𝔖

=

(

Σ

,

P

)

, the

word problem

for

𝔖

asks whether or not for any pair of words

u

,

v

over

Σ

, one can determine in a finite number of steps (an

algorithm

) that

u

⇒

*

v

. If such an algorithm exists, we say that the word problem for

𝔖

is

solvable

. It turns out there exists a semi-Thue system such that the word problem for it is unsolvable.

6.

The word problem for a specific

𝔖

is the same as finding an algorithm to determine whether

v

is a theorem based on a

singleton

axiom

{

u

}

for arbitrary words

u

,

v

.

7.

The word problem for semi-Thue systems

asks whether or not, given

any

semi-Thue system

𝔖

, the word problem for

𝔖

is solvable. From the previous remark, we see the word problem for semi-Thue systems is unsolvable.

References

1

M. Davis,

Computability and Unsolvability

. Dover Publications, New York (1982).

2

H. Hermes,

Enumerability, Decidability, Computability: An

Introduction

to the Theory of

Recursive Functions

. Springer, New York, (1969).

Title

semi-Thue system

Canonical

name

SemiThueSystem

Date of creation

2013-03-22 17:33:16

Last modified on

2013-03-22 17:33:16

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

21

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q42

Classification

msc 03D40

Classification

msc 20M35

Classification

msc 03D03

Synonym

rewriting rule

Synonym

rewrite rule

Synonym

rewriting system

Synonym

semi-Thue generable

Related topic

FormalGrammar

Related topic

LabelledStateTransitionSystem

Defines

antecedent

Defines

consequent

Defines

immediately derivable

Defines

derivable

Defines

defining relation

Defines

word problem for semi-Thue systems

Defines

semi-Thue production

Defines

generable by a semi-Thue system

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