aarjaneiro/planetmath huggingface dataset:
alternative treatment of concatenation

alternative treatment of concatenation

It is possible to define words and

concatenation

in terms of

ordered sets

. Let

A

be a set, which we shall call our

alphabet

. Define a

word

on

A

to be a map
from a

totally ordered set

into

A

. (In order to have words in the usual sense, the
ordered set should be finite but, as the definition presented here does not require
this condition, we do not impose it.)

Suppose that we have totally ordered sets

(

u

,

<

)

and

(

v

,

≺

)

and words

f

:

u

→

A

and

g

:

v

→

A

. Let

u

⁢

∐

v

denote the

disjoint union

of

u

and

v

and let

p

:

u

→

u

⁢

∐

v

and

q

:

u

→

u

⁢

∐

v

be the

canonical maps

. Then
we may define an order

≪

on

u

⁢

∐

v

as follows:

•

If

x

∈

u

and

y

∈

u

, then

p

⁢

(

x

)

≪

p

⁢

(

y

)

if and only if

x

<

y

.

•

If

x

∈

u

and

y

∈

v

, then

p

⁢

(

x

)

≪

q

⁢

(

y

)

.

•

If

x

∈

v

and

y

∈

v

, then

q

⁢

(

x

)

≪

q

⁢

(

y

)

if and only if

x

≺

y

.

We define the

concatenation

of

f

and

g

, which will be denoted

f

∘

g

, to be
map from

u

⁢

∐

v

to

A

defined by the following conditions:

•

If

x

∈

u

, then

(

f

∘

g

)

⁢

(

p

⁢

(

x

)

)

=

f

⁢

(

x

)

.

•

If

y

∈

u

, then

(

f

∘

g

)

⁢

(

q

⁢

(

x

)

)

=

g

⁢

(

x

)

.

Title

alternative treatment of concatenation

Canonical name

AlternativeTreatmentOfConcatenation

Date of creation

2013-03-22 17:24:10

Last modified on

2013-03-22 17:24:10

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 68Q70

Classification

msc 20M35

Related topic

Word

Generated on Sat Feb 10 12:47:45 2018 by

LaTeXML
commutative language

commutative language

Let

Σ

be an

alphabet

and

u

a word over

Σ

. Write

u

as a

product

of symbols in

Σ

:

u

=

a

1

⁢

⋯

⁢

a

n

where

a

i

∈

Σ

. A of

u

is a word of the form

a

π

⁢

(

1

)

⁢

⋯

⁢

a

π

⁢

(

n

)

,

where

π

is a

permutation

on

{

1

,

…

,

n

}

. The set of all permutations of

u

is denoted by

com

⁡

(

u

)

. If

Σ

=

{

b

1

,

…

,

b

m

}

, it is easy to see that

com

⁡

(

u

)

has

n

!

n

1

!

⁢

⋯

⁢

n

m

!

elements

, where

n

i

=

|

u

|

b

i

, the number of

occurrences

of

b

i

in

u

.

Define a

binary relation

∼

on

Σ

*

by:

u

∼

v

if

v

is a permutation of

u

. Then

∼

is a

congruence relation

on

Σ

*

with respect to

concatenation

. In fact,

Σ

*

/

∼

is a

commutative monoid

.

A

language

L

over

Σ

is said to be

commutative

if for every

u

∈

L

, we have

com

⁡

(

u

)

⊆

L

. Two

equivalent

characterization

of a commutative language

L

are:

•

If

u

=

v

⁢

x

⁢

y

⁢

w

∈

L

, then

v

⁢

y

⁢

x

⁢

w

∈

L

.

•

Ψ

-

1

∘

Ψ

⁢

(

L

)

⊆

L

, where

Ψ

is the

Parikh mapping

over

Σ

(under some

ordering

).

The first equivalence comes from the fact that if

v

⁢

y

⁢

x

⁢

w

is just a permutation of

v

⁢

x

⁢

y

⁢

w

, and that every permutation on

{

1

,

…

,

n

}

may be expressed as a product of

transpositions

. The second equivalence is the realization of the fact that

com

⁡

(

u

)

is just the set

{

v

∣

|

v

|

a

=

|

u

|

a

,

a

∈

Σ

}

.

We have just seen some examples of commutative closed langauges, such as

com

⁡

(

u

)

for any word

u

, and

Ψ

-

1

∘

Ψ

⁢

(

L

)

, for any language

L

.

Given a language

L

, the smallest commutative language containing

L

is called the

commutative closure

of

L

. It is not hard to see that

Ψ

-

1

∘

Ψ

⁢

(

L

)

is the

commutative closure

of

L

.

For example, if

L

=

{

(

a

⁢

b

⁢

c

)

n

∣

n

≥

0

}

, then

Ψ

-

1

∘

Ψ

⁢

(

L

)

=

{

w

∣

|

w

|

a

=

|

w

|

b

=

|

w

|

c

}

.

Remark

. The above example illustrates the fact that the families of

regular languages

and

context-free languages

are not

closed under

commutative closures. However, it can be shown that the families of

context-sensitive languages

and

type-0 languages

are closed under commutative closures.

References

1

M. Ito,

Algebraic Theory of

Automata

and Languages

, World Scientific, Singapore (2004).

Title

commutative language

Canonical name

CommutativeLanguage

Date of creation

2013-03-22 18:56:56

Last modified on

2013-03-22 18:56:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q70

Defines

commutative

Defines

commutative closure

Generated on Sat Feb 10 12:48:05 2018 by

LaTeXML
Fine and Wilf’s theorem on words

Fine and Wilf’s theorem on words

Let

w

be a word on an alphabet

A

and let

|

w

|

be its length.
A period of

w

is a value

p

>

0

such that

w

i

+

p

=

w

i

for every

i

∈

{

1

,

2

,

…

,

|

w

|

-

p

}

.
This is the same as saying that

w

=

u

k

⁢

r

for some

u

,

r

∈

A

∗

and

k

∈

ℕ

such that

|

u

|

=

p

and

r

is a prefix of

u

.

Fine and Wilf’s

theorem

gives a condition on the length of the periods a word can have.

Theorem 1

Let

w

be a word on an alphabet

A

having periods

p

and

q

.
If

|

w

|

≥

p

+

q

-

gcd

⁡

(

p

,

q

)

,
then

w

has period

gcd

⁡

(

p

,

q

)

.
The value

p

+

q

-

gcd

⁡

(

p

,

q

)

is the smallest one that makes the theorem true.

As a

counterexample

showing that the condition on

|

w

|

is necessary,
the word

a

⁢

a

⁢

a

⁢

b

⁢

a

⁢

a

⁢

a

has periods

4

and

6

, but not

2

=

gcd

⁡

(

4

,

6

)

.
This can happen because its length is

7

<

4

+

6

-

2

=

8

.

Observe that Theorem

1

can be restated as follows
(cf.

[

1

]

).

Theorem 2

Let

u

and

w

be words over an alphabet

A

.
Suppose

u

h

and

w

k

, for some

h

and

k

,
have a common prefix of length

|

u

|

+

|

w

|

-

gcd

⁡

(

|

u

|

,

|

w

|

)

.

Then there exists

z

∈

A

∗

of length

gcd

⁡

(

|

u

|

,

|

w

|

)

such that

u

,

w

∈

z

∗

.
The value

|

u

|

+

|

w

|

-

gcd

⁡

(

|

u

|

,

|

w

|

)

is also the smallest one that makes the theorem true.

In fact, Theorem

1

clearly implies Theorem

2

.
Now, suppose Theorem

2

is true.
Suppose

w

has periods

p

and

q

and length at least

p

+

q

-

gcd

⁡

(

p

,

q

)

:
write

w

=

u

k

⁢

r

=

v

h

⁢

s

with

|

u

|

=

p

,

|

v

|

=

q

,

r

prefix of

u

,

s

prefix of

v

.
Let

M

be a common multiple of

p

and

q

greater

than

p

,

q

, and

|

w

|

:
then

u

M

/

p

and

v

M

/

q

have the common prefix

w

,
so they also have a common prefix of length

p

+

q

-

gcd

⁡

(

u

,

v

)

.
Then

u

and

v

are powers of a word

z

of length

gcd

⁡

(

p

,

q

)

,
and it is easy to see that

w

=

z

m

⁢

t

for some

m

>

0

and some prefix

t

of

z

.

References

1

M. Lothaire.

Combinatorics on words.

Cambridge University Press 1997.

Title

Fine and Wilf’s theorem on words

Canonical

name

FineAndWilfsTheoremOnWords

Date of creation

2013-03-22 18:00:39

Last modified on

2013-03-22 18:00:39

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

18

Author

Ziosilvio (18733)

Entry type

Theorem

Classification

msc 68Q70

Classification

msc 68Q45

Classification

msc 68R15

Generated on Sat Feb 10 12:48:18 2018 by

LaTeXML
monad

monad

A

monad

over a

category

𝒞

is a triple

(

T

,

η

,

μ

)

, where

T

is an

endofunctor

of

𝒞

,

η

is a

natural transformation

from the

identity functor

on

𝒞

, and

μ

is a natural transformations from

T

∘

T

to

T

, such that the following two properties hold:

•

μ

∘

(

μ

∘

T

)

≡

μ

∘

(

T

∘

μ

)

•

μ

∘

(

T

∘

η

)

≡

id

𝒞

≡

μ

∘

(

η

∘

T

)

These laws are illustrated in the following

diagrams

.

Generated on Sat Feb 10 12:48:41 2018 by

LaTeXML
Myhill-Nerode theorem

Myhill-Nerode theorem

Let

L

be a

language

on the finite

alphabet

A

and let

𝒩

L

be its

Nerode equivalence

.
The following are

equivalent

.

1.

L

is recognized by a

deterministic finite automaton

.

2.

A

∗

/

𝒩

L

is finite.

Moreover, the number of classes of

𝒩

L

is the smallest number of states of a DFA recognizing

L

.

Proof.

First, suppose

A

∗

/

𝒩

L

=

{

q

0

=

[

λ

]

𝒩

L

,

…

,

q

k

-

1

}

=

Q

,

where

λ

is the

empty word

on

A

.
Construct a DFA

𝒜

=

⟨

Q

,

A

,

q

0

,

δ

,

F

⟩

(called the

Nerode

automaton

for

L

)
with

δ

:

Q

×

A

→

Q

defined by

δ

⁢

(

q

,

a

)

=

[

w

⁢

a

]

𝒩

L

,

w

∈

q

,

(1)

and

F

=

{

q

∈

Q

∣

∃

w

∈

L

∣

w

∈

q

}

.

(2)

Then

δ

is well defined
because

w

1

⁢

𝒩

L

⁢

w

2

implies

w

1

⁢

u

⁢

𝒩

L

⁢

w

2

⁢

u

.
It is also straightforward that

𝒜

recognizes

L

.

On the other hand,
let

𝒜

=

⟨

Q

,

A

,

q

0

,

δ

,

F

⟩

be a DFA that recognizes

L

.
Extend

δ

to

Q

×

A

∗

by putting

δ

⁢

(

q

,

λ

)

=

q

and

δ

⁢

(

q

,

a

⁢

w

)

=

δ

⁢

(

δ

⁢

(

q

,

a

)

,

w

)

for every

q

∈

Q

,

a

∈

A

,

w

∈

A

∗

.
Define

f

:

Q

→

A

∗

/

𝒩

L

∪

{

∅

}

as

f

⁢

(

q

)

=

{

[

w

]

𝒩

L

if

⁢

δ

⁢

(

q

0

,

w

)

=

q

∅

if

⁢

δ

⁢

(

q

0

,

w

)

≠

q

⁢

∀

w

∈

A

∗

(3)

Then

f

is well defined.
In fact, suppose

q

1

=

q

2

=

q

:
then either

f

⁢

(

q

1

)

=

f

⁢

(

q

2

)

=

∅

,
or there are

w

1

,

w

2

∈

A

∗

such that

δ

⁢

(

q

0

,

w

1

)

=

δ

⁢

(

q

0

,

w

2

)

=

q

.
But in the latter case,

δ

⁢

(

q

0

,

w

1

⁢

u

)

=

δ

⁢

(

q

0

,

w

2

⁢

u

)

=

δ

⁢

(

q

,

u

)

for any

u

∈

A

∗

,
hence

w

1

⁢

𝒩

L

⁢

w

2

since

𝒜

recognizes

L

.
Finally, for any

w

∈

A

∗

we have

[

w

]

𝒩

L

=

f

⁢

(

δ

⁢

(

q

0

,

w

)

)

,

so every class of

𝒩

L

has a

preimage

according to

f

;
consequently,

|

Q

|

≥

|

A

∗

/

𝒩

L

|

.

□

Title

Myhill-Nerode theorem

Canonical name

MyhillNerodeTheorem

Date of creation

2013-03-22 18:52:13

Last modified on

2013-03-22 18:52:13

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

14

Author

Ziosilvio (18733)

Entry type

Theorem

Classification

msc 68Q70

Classification

msc 20M35

Synonym

Myhill-Nerode theorem for languages

Generated on Sat Feb 10 12:48:44 2018 by

LaTeXML
Nerode equivalence

Nerode equivalence

Let

S

be a

semigroup

and let

X

⊆

S

.
The

relation

s

1

𝒩

X

s

2

⇔

∀

t

∈

S

(

s

1

t

∈

X

⇔

s

2

t

∈

X

)

(1)

is an

equivalence relation

over

S

,
called the

Nerode equivalence

of

X

.

As an example,
if

S

=

(

ℕ

,

+

)

and

X

=

{

n

∈

ℕ

∣

∃

k

∈

ℕ

∣

n

=

3

k

}

,

then

m

⁢

𝒩

X

⁢

n

iff

m

mod

3

=

n

mod

3

.

The Nerode equivalence is right-invariant,

i.e.

, if

s

1

⁢

𝒩

X

⁢

s

2

then

s

1

⁢

t

⁢

𝒩

X

⁢

s

2

⁢

t

for any

t

.
However, it is usually not a

congruence

.

The Nerode equivalence is

maximal

in the following sense:

•

if

η

is a right-invariant equivalence over

S

and

X

is union of classes of

η

,

•

then

s

⁢

η

⁢

t

implies

s

⁢

𝒩

X

⁢

t

.

In fact, let

r

∈

S

:
since

s

⁢

η

⁢

t

and

η

is right-invariant,

s

⁢

r

⁢

η

⁢

t

⁢

r

.
However,

X

is union of classes of

η

,
therefore

s

⁢

r

and

t

⁢

r

are either both in

X

or both outside

X

.
This is true for all

r

∈

S

, thus

s

⁢

𝒩

X

⁢

t

.

The Nerode equivalence is linked to the

syntactic congruence

by the following fact, whose proof is straightforward:

s

1

≡

X

s

2

⁢

iff

⁢

l

⁢

s

1

⁢

𝒩

X

⁢

l

⁢

s

2

⁢

∀

l

∈

S

.

Title

Nerode equivalence

Canonical name

NerodeEquivalence

Date of creation

2013-03-22 18:52:11

Last modified on

2013-03-22 18:52:11

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

4

Author

Ziosilvio (18733)

Entry type

Definition

Classification

msc 68Q70

Classification

msc 20M35

Defines

maximality property of Nerode equivalence

Generated on Sat Feb 10 12:48:46 2018 by

LaTeXML
quotient of languages

quotient of languages

Let

L

1

,

L

2

be two

languages

over some

alphabet

Σ

. The

quotient

of

L

1

by

L

2

is defined to be the set

L

1

/

L

2

:=

{

u

∈

Σ

*

∣

u

⁢

v

∈

L

1

⁢

for some

⁢

v

∈

L

2

}

.

L

1

/

L

2

is sometimes written

L

1

⁢

L

2

-

1

. The quotient so defined is also called the

right quotient

, for one can similarly define the

left quotient

of

L

1

by

L

2

:

L

1

\

L

2

:=

{

u

∈

Σ

*

∣

v

⁢

u

∈

L

1

⁢

for some

⁢

v

∈

L

2

}

.

L

1

\

L

2

is sometimes written

L

2

-

1

⁢

L

1

.

Below are some examples of quotients:

•

If

L

1

=

{

a

n

⁢

b

n

⁢

c

n

∣

n

≥

0

}

and

L

2

=

{

b

,

c

}

*

, then

–

L

1

/

L

2

=

{

a

m

⁢

b

n

∣

m

≥

n

≥

0

}

–

L

2

/

L

1

=

L

2

–

L

1

\

L

2

=

{

λ

}

, the

singleton

consisting the

empty word

–

L

2

\

L

1

=

L

2

•

for any language

L

over

Σ

:

–

L

/

Σ

*

is the language of all prefixes of words of

L

–

Σ

*

/

L

=

Σ

*

–

L

\

Σ

*

is the language of all suffixes of words of

L

–

Σ

*

\

L

=

Σ

*

•

λ

∈

L

/

L

∩

L

\

L

, and if

λ

∈

L

, then

L

⊆

L

/

L

∩

L

\

L

.

Here are some basic

properties

of quotients:

1.

L

1

⊆

(

L

1

/

L

2

)

⁢

L

2

∩

L

2

⁢

(

L

1

\

L

2

)

.

2.

(

L

1

/

L

2

)

⁢

L

2

⊆

(

L

1

⁢

L

2

)

/

L

2

, and

L

2

⁢

(

L

1

\

L

2

)

⊆

(

L

2

⁢

L

1

)

\

L

2

.

3.

right and left quotients are related via

reversal

:

(

L

1

\

L

2

)

R

=

{

u

R

∣

v

u

∈

L

1

for some

v

∈

L

2

}

=

{

u

R

∣

(

v

u

)

R

∈

L

1

R

for some

v

R

∈

L

2

R

}

=

{

u

R

∣

u

R

v

R

∈

L

1

R

for some

v

R

∈

L

2

R

}

=

L

1

R

/

L

2

R

.

A family

ℱ

of languages is said to be

closed under

quotient by a language

L

if for every language

M

∈

ℱ

,

M

/

L

∈

ℱ

. Furthermore,

ℱ

is said to be

closed under quotient

if

M

/

L

∈

ℱ

for any

M

,

L

∈

ℱ

.

Closure

under quotient is also termed closure under right quotient. Closure under left quotient is similarly defined.

It can be shown that the families of

regular

,

context-free

, and

type-0 languages

are closed under quotient (both left and right) by a

regular language

. The family of

context-sensitive languages

does not have this closure property.

Since all of the families mentioned above are closed under reversal, each of the families, except the context-sensitive family, is closed under left quotient by a regular language, according to the second property above.

References

1

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

Formal Languages

and Their

Relation

to

Automata

, Addison-Wesley, (1969).

Title

quotient of languages

Canonical

name

QuotientOfLanguages

Date of creation

2013-03-22 18:56:06

Last modified on

2013-03-22 18:56:06

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q70

Classification

msc 68Q45

Defines

quotient

Defines

left quotient

Defines

right quotient

Generated on Sat Feb 10 12:48:56 2018 by

LaTeXML
reversal

reversal

Let

Σ

be an

alphabet

and

w

a word over

Σ

. The

reversal

of

w

is the word obtained from

w

by “spelling” it backwards. Formally, the

reversal

is defined as a

function

rev

:

Σ

*

→

Σ

*

such that, for any word

w

=

a

1

⁢

⋯

⁢

a

n

, where

a

i

∈

Σ

,

rev

⁡

(

w

)

:=

a

n

⁢

⋯

⁢

a

1

. Furthermore,

rev

⁡

(

λ

)

:=

λ

. Oftentimes

w

R

or

mi

⁡

(

w

)

is used to denote the reversal of

w

.

For example, if

Σ

=

{

a

,

b

}

, and

w

=

a

⁢

a

⁢

b

⁢

a

⁢

b

⁢

b

, then

rev

⁡

(

w

)

=

b

⁢

b

⁢

a

⁢

b

⁢

a

⁢

a

.

Two properties of the reversal are:

•

it fixes all

a

∈

Σ

:

rev

⁡

(

a

)

=

a

.

•

it is

idempotent

:

rev

∘

rev

=

1

, and

•

it reverses

concatenation

:

rev

⁡

(

a

⁢

b

)

=

rev

⁡

(

b

)

⁢

rev

⁡

(

a

)

.

In other words, the reversal is an

antihomomorphism

. In fact, it is the antihomomorphism that fixes every element of

Σ

. Furthermore,

g

is an antihomomorphism iff

g

∘

rev

is a

homomorphism

. By the second property above,

h

is a homomorphism iff

h

∘

rev

is an antihomomorphism.

A word that is fixed by the reversal is called a palindrome. The

empty word

λ

as well as any symbol in the alphabet

Σ

are trivially palindromes. Also, for any word

w

, the words

w

⁢

x

⁢

rev

⁡

(

w

)

and

rev

⁡

(

w

)

⁢

x

⁢

w

are both palindromes, where

x

is either a symbol in

Σ

or the empty word. In fact, every palindrome can be written this way.

The

language

consisting of all palindromes over an alphabet is

context-free

, and not

regular

if

Σ

has more than one symbol. It is not hard to see that the

productions

are

σ

→

λ

,

σ

→

a

and

σ

→

a

⁢

σ

⁢

a

, where

a

ranges over

Σ

.

Reversal of words can be extended to languages: let

L

be a language over

Σ

, then

rev

⁡

(

L

)

:=

{

rev

⁡

(

w

)

∣

w

∈

L

}

.

A family

ℱ

of languages is said to be

closed under

reversal

if for any

L

∈

ℱ

,

rev

⁡

(

L

)

∈

ℱ

. It can be shown that

regular languages

, context-free languages,

context-sensitive languages

, and

type-0 languages

are all closed under reversal.

Title

reversal

Canonical name

Reversal

Date of creation

2013-03-22 18:55:20

Last modified on

2013-03-22 18:55:20

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q70

Classification

msc 68Q45

Synonym

mirror image

Related topic

Palindrome

Generated on Sat Feb 10 12:49:06 2018 by

LaTeXML
syntactic congruence

syntactic congruence

Let

S

be a

semigroup

and let

X

⊆

S

.
The

relation

s

1

≡

X

s

2

iff

∀

l

,

r

∈

S

(

l

s

1

r

∈

X

iff

l

s

2

r

∈

X

)

(1)

is called the

syntactic congruence

of

X

.
The

quotient

S

/

≡

X

is called the

syntactic semigroup

of

X

,
and the natural

morphism

ϕ

:

S

→

S

/

≡

X

is called the

syntactic morphism

of

X

.
If

S

is a monoid, then

S

/

≡

X

is also a monoid,
called the

syntactic monoid

of

X

.

As an example,
if

S

=

(

ℕ

,

+

)

and

X

=

{

n

∈

ℕ

∣

∃

k

∈

ℕ

∣

n

=

3

k

}

,

then

m

≡

X

n

if

m

mod

3

=

n

mod

3

,
and the syntactic monoid is

isomorphic

to the

cyclic group

of order three.

It is straightforward that

≡

X

is an

equivalence relation

and

X

is union of classes of

≡

X

.
To prove that it is a

congruence

, let

s

1

,

s

2

,

t

1

,

t

2

∈

S

satisfy

s

1

≡

X

s

2

and

t

1

≡

X

t

2

.
Let

l

,

r

∈

S

be arbitrary.
Then

l

⁢

s

1

⁢

t

1

⁢

r

∈

X

iff

l

⁢

s

2

⁢

t

1

⁢

r

∈

X

because

s

1

≡

X

s

2

,
and

l

⁢

s

2

⁢

t

1

⁢

r

∈

X

iff

l

⁢

s

2

⁢

t

2

⁢

r

∈

X

because

t

1

≡

X

t

2

.
Then

s

1

⁢

t

1

≡

X

s

2

⁢

t

2

since

l

and

r

are arbitrary.

The syntactic congruence is both left- and right-invariant,

i.e.

, if

s

1

≡

X

s

2

,
then

t

⁢

s

1

≡

X

t

⁢

s

2

and

s

1

⁢

t

≡

X

s

2

⁢

t

for any

t

.

The syntactic congruence is

maximal

in the following sense:

•

if

χ

is a congruence over

S

and

X

is union of classes of

χ

,

•

then

s

⁢

χ

⁢

t

implies

s

≡

X

t

.

In fact, let

l

,

r

∈

S

:
since

s

⁢

χ

⁢

t

and

χ

is a congruence,

l

⁢

s

⁢

r

⁢

χ

⁢

l

⁢

t

⁢

r

.
However,

X

is union of classes of

χ

,
therefore

l

⁢

s

⁢

r

and

l

⁢

t

⁢

r

are either both in

X

or both outside

X

.
This is true for all

l

,

r

∈

S

, thus

s

≡

X

t

.

Title

syntactic congruence

Canonical name

SyntacticCongruence

Date of creation

2013-03-22 18:52:08

Last modified on

2013-03-22 18:52:08

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

6

Author

Ziosilvio (18733)

Entry type

Definition

Classification

msc 68Q70

Classification

msc 20M35

Defines

syntactic semigroup

Defines

syntactic monoid

Defines

syntactic morphism

Defines

maximality property of syntactic congruence

Generated on Sat Feb 10 12:49:28 2018 by

LaTeXML
weak bisimulation

weak bisimulation

Let

M

=

(

S

,

Σ

,

→

)

be a

labelled state transition system

(LTS). Recall that for each label

α

∈

Σ

, there is an associated

binary relation

→

α

on

S

. Single out a label

τ

∈

Σ

, and call it the

silent step

. Define the following

relations

:

1.

Let

⇒

be the

reflexive

and

transitive closures

of

→

τ

. In other words,

p

⇒

q

iff either

p

=

q

, or there is a positive integer

n

>

1

and states

r

1

,

…

,

r

n

such that

p

=

r

1

and

q

=

r

n

and

r

i

→

τ

r

i

+

1

, where

i

=

1

,

…

,

n

-

1

.

2.

Next, for any label

α

that is not the silent step

τ

in

Σ

, define

⇒

α

⁣

:=

⁣

→

α

⁣

∘

⁣

⇒

⁣

∘

⁣

→

α

,

where

∘

denotes the

relational composition

operation

. In other words,

p

⇒

α

q

iff there are states

r

and

s

such that

p

→

α

r

,

r

⇒

s

, and

s

→

α

q

.

3.

Finally, for any label

α

∈

Σ

, let

⇒

(

α

)

:=

{

⇒

if

⁢

α

=

τ

⇒

α

otherwise.

Definition

. Let

M

=

(

S

1

,

Σ

,

→

1

)

and

N

=

(

S

2

,

Σ

,

→

2

)

be two labelled state transition systems, with

τ

∈

Σ

the silent step. A relation

≈

⊆

S

1

×

S

2

is called a

weak simulation

if whenever

p

≈

q

and any labelled transition

p

→

α

1

p

′

, there is a state

q

′

∈

S

2

such that

p

′

≈

q

′

and

p

′

⇒

(

α

)

2

q

.

≈

is a

weak bisimulation

if both

≈

and its

converse

≈

-

1

are weak simulations.

Title

weak bisimulation

Canonical name

WeakBisimulation

Date of creation

2013-03-22 19:30:55

Last modified on

2013-03-22 19:30:55

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q85

Synonym

invisible step

Defines

silent step

Defines

weak simulation

Generated on Sat Feb 10 12:49:48 2018 by

LaTeXML
Boolean domain

Boolean domain

\PMlinkescapephrase

generic

\PMlinkescapephrase

Generic

A

Boolean domain

𝔹

is a generic 2-element set, say,

𝔹

=

{

0

,

1

}

, whose

elements

are interpreted as logical values, typically,

0

=

false

and

1

=

true

.

A

Boolean variable

x

is a variable that takes its value from a Boolean domain, as

x

∈

𝔹

.

Title

Boolean domain

Canonical name

BooleanDomain

Date of creation

2013-03-22 17:58:05

Last modified on

2013-03-22 17:58:05

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

6

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 68R01

Classification

msc 68Q01

Classification

msc 68P01

Classification

msc 03G05

Classification

msc 03C55

Classification

msc 03B05

Related topic

BooleanValuedFunction

Defines

Boolean variable

Generated on Sat Feb 10 12:49:53 2018 by

LaTeXML
logical matrix

logical matrix

A

logical matrix

, in the finite dimensional case, is a

k

-dimensional array with entries from the

Boolean domain

𝔹

=

{

0

,

1

}

. Such a matrix affords a

matrix representation

of a

k

-adic

relation

.

Title

logical matrix

Canonical name

LogicalMatrix

Date of creation

2013-03-22 17:51:18

Last modified on

2013-03-22 17:51:18

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

5

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 68R01

Classification

msc 68P15

Classification

msc 08A02

Classification

msc 05C65

Classification

msc 05B30

Classification

msc 05B20

Classification

msc 03E20

Classification

msc 03B10

Synonym

Boolean matrix

Related topic

RelationTheory

Related topic

RelationComposition2

Related topic

MatrixRepresentationOfRelationComposition

Related topic

TriadicRelation

Defines

logical matrix

Generated on Sat Feb 10 12:49:55 2018 by

LaTeXML