Thue system

Thue system

A

semi-Thue system

𝔖

=

(

Σ

,

R

)

is said to be a

Thue system

if

R

is a

symmetric relation

on

Σ

*

. In other words, if

x

→

y

is a defining relation in

R

, then so is

y

→

x

.

Like a semi-Thue system, we can define the

concepts

of immediately derivable and

derivable

pairs. Let

R

′

and

R

′′

be the respective

collections

of these pairs. Since

R

is

symmetric

, so is

R

′

and consequently

R

′′

. Similarly, the notations are: for

elements

(

a

,

b

)

∈

R

′

, we write

a

⇒

b

, and for elements

(

c

,

d

)

∈

R

′′

, we write

a

⇒

*

b

.

If we regard

Σ

*

as a

free monoid

with

concatenation

⋅

as

multiplication

and the

empty word

λ

as the

multiplicative identity

, then

⇒

*

is a

congruence relation

on

Σ

*

: it is an

equivalence relation

and respects concatenation, meaning that if

a

⇒

*

b

and

c

⇒

*

d

, then

a

⁢

c

⇒

*

b

⁢

d

. Therefore, we can take the quotent

Σ

*

/

⇒

*

and the resulting set of

equivalence classes

is again a monoid with

[

λ

]

as the multiplicative identity. It is a monoid generated by

[

a

]

whenever

a

∈

Σ

with

relations

[

u

]

=

[

v

]

whenever

u

→

v

is a defining relation in

R

. Thus, two elements are in the same equivalence class if one is derivable from another. Let us denote this monoid by

[

Σ

]

𝔖

.

Now let

𝔖

=

(

Σ

,

R

)

be a Thue system. Then

𝔖

is called a

group system

if there exists an

involution

-

1

on

Σ

given by

a

↦

a

-

1

, and that for every

a

∈

Σ

,

a

⁢

a

-

1

→

λ

is a defining relation in

R

. Since

-

1

is an involution, if

b

is the symbol in

Σ

such that

b

=

a

-

1

, then

b

-

1

=

a

. So

a

-

1

⁢

a

=

b

⁢

a

=

b

⁢

b

-

1

→

λ

also. In fact, it is not hard to see that for a group system

𝔖

,

[

Σ

]

𝔖

is the

group with

generators

[

a

]

whenever

a

∈

Σ

and with relators

[

u

]

⁢

[

v

]

-

1

whenever

u

→

v

is a defining relation in

R

. Every

non-trivial element

in

[

Σ

]

𝔖

has an expression

[

a

1

]

p

1

⁢

⋯

⁢

[

a

n

]

p

n

, where each

a

i

is a letter in

Σ

such that it is distinct from its neighbors (

a

i

≠

a

i

+

1

), and

p

i

are non-zero

integers

. This expression is unique in the sense that it is “

reduced

”. See

reduced words

for more detail.

Remark

. Like the word problem for semi-Thue systems, the

word problem

for Thue systems and group systems can be similarly posed. It can be shown that the word problem for Thue systems and group systems are both unsolvable. As a result, the corresponding word problems for

semigroups

and for groups are also 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).

3

H.R. Lewis, C.H. Papadimitriou

Elements of the Theory of Computation

. Prentice-Hall, Englewood Cliffs, New Jersey (1981).

4

P.S. Novikov,

On the

algorithmic

unsolvability of the word problem in

group theory

,
Trudy Mat. Inst. Steklov 44, 1-143 (1955).

Title

Thue system

Canonical name

ThueSystem

Date of creation

2013-03-22 17:33:19

Last modified on

2013-03-22 17:33:19

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03D03

Classification

msc 03D40

Classification

msc 68Q42

Classification

msc 20M35

Related topic

SemigroupWithInvolution

Defines

group system

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