aarjaneiro/planetmath huggingface dataset:
deterministic finite automaton

deterministic finite automaton

A

deterministic finite automaton

(or DFA) is a

deterministic automaton

with a finite input

alphabet

and a finite number of states. It can be formally defined as a 5-tuple

(

S

,

Σ

,

δ

,

q

0

,

F

)

, where

•

S

is a non-empty

finite set

of

states

,

•

Σ

is the alphabet (defining what set of input strings the automaton operates on),

•

δ

:

S

×

Σ

→

S

is the

transition function

,

•

q

0

∈

S

is the starting state, and

•

F

⊆

S

is a set of final (or accepting states).

A DFA works exactly like a general automaton:

operation

begins at

q

0

, and movement from state to state is governed by the transition function

δ

. A word is accepted exactly when a final state is reached upon reading the last (rightmost) symbol of the word.

DFAs represent

regular languages

, and can be used to test whether any string in

Σ

*

is in the

language

it represents. Consider the following regular language over the alphabet

Σ

:=

{

𝚊

,

𝚋

}

(represented by the

regular expression

aa*b

):

<

⁢

S

>

::=

a

⁢

A

<

⁢

A

>

::=

b

|

𝚊

A

This language can be represented by the DFA with the following

state diagram

:

Generated on Sat Feb 10 12:43:48 2018 by

LaTeXML
diamond lemma

diamond lemma

The

diamond lemmas

constitute a of results about the

existence

of a unique

normal form

. Diamond lemmas exist in many diverse areas of
mathematics, and as a result their technical

contents

can be quite
different, but they are easily recognisable from their overall
and basic idea — the “

diamond

” condition
from which they inherit their name.

1

Newman’s Diamond Lemma

The basic diamond lemma, that of Newman

[

Newman

]

, is today most
easily presented in of

binary relations

.

Theorem 1

.

Let

X

be a set and let

→

be a binary relation on

X

.
If

→

is

terminating

, then the following conditions are

equivalent

:

1.

For all

a

,

b

,

c

∈

X

such that

a

→

b

and

a

→

c

, then

b

and

c

are

joinable

.

2.

Every

a

∈

X

has a unique normal form.

Condition

1

can be graphically drawn as

\xymatrix@

-

2

⁢

e

⁢

x

⁢

&

⁢

a

⁢

\xy@@ix@

⁢

Generated on Sat Feb 10 12:43:52 2018 by

LaTeXML
Greibach normal form

Greibach normal form

A

formal grammar

G

=

(

Σ

,

N

,

P

,

σ

)

is said to be in

Greibach normal form

if every production has the following form:

A

→

a

⁢

W

where

A

∈

N

(a non-terminal symbol),

a

∈

Σ

(a terminal symbol), and

W

∈

N

*

(a word over

N

).

A formal grammar in Greibach normal form is a

context-free grammar

. Moreover, any context-free language not containing the

empty word

λ

can be generated by a

grammar

in Greibach normal form. And if a context-free language

L

contains

λ

, then

L

can be generated by a grammar that is in Greibach normal form, with the

addition

of the production

σ

→

λ

.

Let

L

be a context-free language not containing

λ

, and let

G

=

(

Σ

,

N

,

P

,

σ

)

be a grammar in Greibach normal form generating

L

. We construct a PDA

M

from

G

based on the following

specifications

:

1.

M

has one state

p

,

2.

the input

alphabet

of

M

is

Σ

,

3.

the

stack alphabet

of

M

is

N

,

4.

the initial stack symbol of

M

is the start symbol

σ

of

G

,

5.

the

start state

of

M

is the only state of

M

, namely

p

6.

there are no final states,

7.

the transition function

T

of

M

takes

(

p

,

a

,

A

)

to the

singleton

{

(

p

,

W

)

}

, provided that

A

→

a

⁢

W

is a production of

G

. Otherwise,

T

⁢

(

p

,

a

,

A

)

=

∅

.

It can be shown that

L

=

L

⁢

(

M

)

, the

language

accepted on empty stack, by

M

. If we further define

T

⁢

(

p

,

λ

,

σ

)

:=

{

(

p

,

λ

)

}

, then

M

accepts

L

∪

{

λ

}

. As a result, any context-free language is accepted by some PDA.

References

1

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

Formal Languages

and Their

Relation

to Automata

, Addison-Wesley, (1969).

Title

Greibach normal form

Canonical

name

GreibachNormalForm

Date of creation

2013-03-22 18:55:22

Last modified on

2013-03-22 18:55:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q42

Classification

msc 68Q45

Synonym

GNF

Related topic

ChomskyNormalForm

Related topic

KurodaNormalForm

Generated on Sat Feb 10 12:44:01 2018 by

LaTeXML
normalizing reduction

normalizing reduction

Definition 1

.

Let

X

be a set and

→

a

reduction

(

binary relation

) on

X

. An

element

x

∈

X

is said to be in

normal form

with respect to

→

if

x

↛

y

for all

y

∈

X

, i.e., if there is no

y

∈

X

such that

x

→

y

. Equivalently,

x

is in normal form with respect to

→

iff

x

∉

dom

⁡

(

→

)

. To be

irreducible

is a common synonym for ‘to be in normal form’.

Denote by

→

*

the

reflexive transitive closure

of

→

. An element

y

∈

X

is said to be

a normal form of

x

∈

X

if

y

is in normal form and

x

→

*

y

.

A reduction

→

on

X

is said to be

normalizing

if every element

x

∈

X

has a normal form.

Examples

.

•

Let

X

be any set. Then no elements in

X

are in normal form with respect to any reduction that is either

reflexive

. If

→

is a

symmetric relation

, then

x

∈

X

is in normal form with respect to

→

iff

x

is not in the domain (or range) of

→

.

•

Let

X

=

{

a

,

b

,

c

,

d

}

and

→

=

{

(

a

,

a

)

,

(

a

,

b

)

,

(

a

,

c

)

,

(

b

,

c

)

,

(

a

,

d

)

}

. Then

c

and

d

are both in normal form. In

addition

, they are both normal forms of

a

, while

d

is a normal form only for

a

. However,

X

is not normalizing because neither

c

nor

d

have normal forms.

•

Let

X

be the set of all positive integers greater than

1

. Define the reduction

→

on

X

as follows:

a

→

b

if there is an element

c

∈

X

such that

a

=

b

⁢

c

. Then it is clear that every

prime number

is in normal form. Furthermore, every element

x

in

X

has

n

normal forms, where

n

is the number of

prime divisors

of

x

. Clearly,

n

≥

1

for every

x

∈

X

. As a result,

X

is normalizing.

Title

normalizing reduction

Canonical

name

NormalizingReduction

Date of creation

2013-03-22 17:57:35

Last modified on

2013-03-22 17:57:35

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q42

Related topic

TerminatingReduction

Related topic

DiamondLemma

Defines

normal form

Defines

irreducible

Defines

normalizing

Generated on Sat Feb 10 12:44:19 2018 by

LaTeXML
pumping lemma (context-free languages)

pumping lemma (context-free languages)

Let

L

be a

context-free language

(a.k.a. type 2

language

). Then there exist
two

integers

m

and

n

such that, if the length of a word

W

is

greater

than

m

, then

W

=

A

⁢

B

⁢

C

⁢

D

⁢

E

where

A

,

B

,

C

,

D

,

E

are

subwords

such that

1.

The length of the subword

B

⁢

C

⁢

D

is less than

n

.

2.

B

⁢

D

is not be empty.

3.

For all integers

k

>

0

, it is the case that

A

⁢

B

k

⁢

C

⁢

D

k

⁢

E

belongs to

L

,
where

exponentiation

denotes repetition of a subword

k

times.

An important use of this lemma is that it allows one to show that a language
is not context-free. (Remember, just because a language happens to be described
in terms of a

context-sensitive grammar

does not automatically preclude the
possibility of describing the same language also by a
context-free language.) The idea is to assume that the language is
context-free, then arrive at a

contradiction

via this lemma.

As an illustrative example, consider the following language, which consists of
but one

terminal symbol

, ‘x’ and which consists of all strings of ‘x’ ’s whose
length is a

perfect square

. Were this a context-free language, there would
exist integers

m

and

n

as above. Choose an integer

h

such that

h

2

>

m

.
Then the word

x

h

2

belongs to our language and the lemma tells us that
it can be written as

A

⁢

B

⁢

C

⁢

D

⁢

E

so as to

satisfy

the conditions enumerated above.
Write

A

=

x

a

,

B

=

x

b

,

C

=

x

c

,

D

=

x

d

,

E

=

x

e

for suitable

nonnegative integers

a

,

b

,

c

,

d

,

e

. Then we have

a

+

b

+

c

+

d

+

e

=

k

2

;
by condition ,

b

+

d

>

0

and, by condition ,

a

+

k

⁢

b

+

c

+

k

⁢

d

+

e

would
have to be a perfect square because

A

⁢

B

k

⁢

C

⁢

D

k

⁢

E

would be a word of the
language. This, however, leads to a contradiction:

h

2

+

k

⁢

(

b

+

d

)

could not possibly be a perfect square for all

k

unless

b

+

d

=

0

.

As an exercise, the reader may consider constructing a context-sensitive
grammar for this language and posting it as an attachment to this entry
(at which time this

sentence

will come down).

Title

pumping lemma

(context-free languages)

Canonical name

PumpingLemmacontextfreeLanguages

Date of creation

2013-03-22 16:21:10

Last modified on

2013-03-22 16:21:10

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 68Q42

Synonym

pumping lemma

Related topic

PumpingLemmaRegularLanguages

Generated on Sat Feb 10 12:44:41 2018 by

LaTeXML
pumping lemma (regular languages)

pumping lemma (regular languages)

Lemma 1

.

Let

L

be a

regular language

(a.k.a. type 3

language

). Then there exist
an integer

n

such that, if the length of a word

W

is greater
than

n

, then

W

=

A

⁢

B

⁢

C

where

A

,

B

,

C

are

subwords

such that

1.

The length of the subword

B

is less than

n

.

2.

The subword

B

cannot be empty (although one of

A

or

C

might

happen to be empty).

3.

For all integers

k

>

0

, it is the case that

A

⁢

B

k

⁢

C

belongs to

L

,
where

exponentiation

denotes repetition of a subword

k

times.

An important use of this lemma is to show that a language
is not

regular

. (Remember, just because a language happens to be described
in terms of an irregular

grammar

does not automatically preclude the
possibility of describing the same language also by a
regular grammar.) The idea is to assume that the language is
regular, then arrive at a

contradiction

via this lemma.

An example of such a use of this lemma is afforded by the language

L

=

{

0

p

⁢

1

q

⁢

0

p

⁢

∣

p

,

q

>

⁢

0

}

.

Let

n

be the number whose

existence

is guaranteed by the lemma.
Now, consider the word

W

=

0

n

+

1

⁢

1

n

+

1

⁢

0

n

+

1

. There must
exist subwords

A

,

B

,

C

such that

W

=

A

⁢

B

⁢

C

and

B

must be of length less than

n

. The only possibilities are the following

1.

A

=

0

u

,

B

=

0

v

,

C

=

0

n

+

1

-

u

-

v

⁢

1

n

+

1

⁢

0

n

+

1

2.

A

=

0

n

+

1

-

u

,

B

=

0

u

⁢

1

v

,

C

=

1

n

+

1

-

v

⁢

0

n

+

1

3.

A

=

0

n

+

1

⁢

1

v

,

B

=

1

u

,

C

=

1

n

+

1

-

u

-

v

⁢

0

n

+

1

4.

A

=

0

n

+

1

⁢

1

n

+

1

-

v

,

B

=

1

v

⁢

0

u

,

C

=

0

n

+

1

-

u

5.

A

=

0

n

+

1

⁢

1

n

+

1

⁢

0

u

,

B

=

0

v

,

C

=

0

n

+

1

-

u

-

v

In these cases,

A

⁢

B

2

⁢

C

would have the following form:

1.

A

⁢

B

2

⁢

C

=

0

n

+

1

+

v

⁢

1

n

+

1

⁢

0

n

+

1

2.

A

⁢

B

2

⁢

C

=

0

n

+

1

⁢

1

v

⁢

0

u

⁢

1

n

+

1

⁢

0

n

+

1

3.

A

⁢

B

2

⁢

C

=

0

n

+

1

⁢

1

n

+

1

+

u

⁢

0

n

+

1

4.

A

⁢

B

2

⁢

C

=

0

n

+

1

⁢

1

n

+

1

⁢

0

u

⁢

1

v

⁢

0

n

+

1

5.

A

⁢

B

2

⁢

C

=

0

n

+

1

⁢

1

n

+

1

⁢

0

n

+

1

+

u

It is easy to see that, in each of these five cases,

A

⁢

B

2

⁢

C

∉

L

.
Hence

L

cannot be a regular language.

Title

pumping lemma

(regular languages)

Canonical name

PumpingLemmaregularLanguages

Date of creation

2013-03-22 16:21:17

Last modified on

2013-03-22 16:21:17

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 68Q42

Synonym

pumping lemma

Related topic

PumpingLemma

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

LaTeXML
convolution

convolution

Let

Σ

be an alphabet,

#

a symbol not in

Σ

.

Let

x

1

⁢

x

2

⁢

…

⁢

x

|

x

|

,

y

1

⁢

y

2

⁢

…

⁢

y

|

y

|

,

z

1

⁢

z

2

⁢

…

⁢

z

|

z

|

,

…

be

n

words over

Σ

*

. Let

l

denote the maximum length.

The

convolution

of these words is

(

x

1

,

y

1

,

…

)

⁢

(

x

2

,

y

2

,

…

)

⁢

…

⁢

(

x

l

,

y

l

,

…

)

where for any

i

>

|

w

|

, the

w

i

is

#

. This is a new word in

(

(

Σ

∪

{

#

}

)

n

)

*

.

The convolution of

x

,

y

,

z

,

…

is sometimes denoted conv(

x

,

y

,

z

,

…

), or

x

⋆

y

⋆

z

⋆

…

Example

The convolution of

a

⁢

n

⁢

d

,

f

⁢

i

⁢

s

⁢

h

,

b

⁢

e

is

(

a

,

f

,

b

)

⁢

(

n

,

i

,

e

)

⁢

(

d

,

s

,

#

)

⁢

(

#

,

h

,

#

)

Notes

This definition bears no to the notion of convolution (

http://planetmath.org/Convolution

) of functions.

Title

convolution

Canonical name

Convolution1

Date of creation

2013-03-22 14:16:51

Last modified on

2013-03-22 14:16:51

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 68Q45

Classification

msc 68Q70

Classification

msc 68R15

Related topic

Language

Related topic

KleeneStar

Generated on Sat Feb 10 12:45:19 2018 by

LaTeXML
metalinear language

metalinear language

Recall that a

linear grammar

is a

formal grammar

G

=

(

Σ

,

N

,

P

,

σ

)

whose productions are of the form

A

→

x

, where

A

is a terminal symbol, and

x

is a word over

Σ

, with at most one

occurrence

of a non-terminal symbol.

The

concept

of a linear grammar can be generalized: define a

k

-linear grammar as a formal grammar

G

=

(

Σ

,

N

,

P

,

σ

)

such that every production in

P

has one of the three following forms:

•

A

→

u

,

•

A

→

u

⁢

B

⁢

v

,

•

σ

→

W

,

where

A

,

B

are non-terminal symbols,

u

,

v

are terminal words, and

W

is a word over

Σ

with no more than

k

occurrences of non-terminal symbols, and none of which is the start symbol

σ

. Any

k

-linear grammar is

context-free

.

A

language

is said to be

k

-linear if it can be generated by a

k

-linear grammar. Note that a language is

1

-linear iff it is linear.

A language is said to be

metalinear

if it is

k

-linear for some positive integer

k

. In other words, if

ℒ

⁢

(

k

)

denotes the family of

k

-linear languages, then the family

ℒ

⁢

(

∞

)

of metalinear langauges is

ℒ

⁢

(

∞

)

=

⋃

{

ℒ

⁢

(

k

)

∣

k

≥

1

}

.

It is easy to see we have the following

inclusions

ℛ

⊆

ℒ

⁢

(

k

)

⊆

⋯

⊆

ℒ

⁢

(

k

)

⊆

⋯

⊆

ℒ

⁢

(

∞

)

⊆

ℱ

where

ℛ

and

ℱ

are the families of

regular

and context-free languages respectively.

In fact, it can be shown that all of the inclusions above are strict, providing us with an

infinite chain

of families of languages between the

regular languages

and the context-free languages.

References

1

A. Salomaa,

Formal Languages

, Academic Press, New York (1973).

Title

metalinear language

Canonical name

MetalinearLanguage

Date of creation

2013-03-22 18:57:09

Last modified on

2013-03-22 18:57:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q45

Classification

msc 68Q70

Related topic

LinearGrammar

Related topic

LinearLanguage

Defines

k

-linear language

Generated on Sat Feb 10 12:46:33 2018 by

LaTeXML
Post correspondence problem

Post correspondence problem

Let

Σ

be an

alphabet

. As usual,

Σ

+

denotes the set of all non-empty words over

Σ

. Let

P

⊂

Σ

+

×

Σ

+

be finite. Call a

finite sequence

(

u

1

,

v

1

)

,

…

,

(

u

n

,

v

n

)

of pairs in

P

a

correspondence

in

P

if

u

1

⁢

⋯

⁢

u

n

=

v

1

⁢

⋯

⁢

v

n

.

The word

u

1

⁢

⋯

⁢

u

n

is called a

match

in

P

.

For example, if

Σ

=

{

a

,

b

}

, and

P

=

{

(

b

,

b

2

)

,

(

a

2

,

a

)

,

(

b

2

⁢

a

,

b

3

)

,

(

a

⁢

b

2

,

a

2

⁢

b

)

}

. Then

(

a

2

,

a

)

,

(

a

⁢

b

2

,

a

2

⁢

b

)

,

(

b

2

⁢

a

,

b

3

)

,

(

a

⁢

b

2

,

a

2

⁢

b

)

,

(

b

,

b

2

)

is a correspondence in

P

.

On the other hand, there are no correspondences in

{

(

a

⁢

b

,

a

)

,

(

a

,

b

⁢

a

2

)

}

.

The

Post correspondence problem

asks the following:

Is there an

algorithm

(

Turing machine

or any other

equivalent

computing models) such that when an arbitrary

P

is given as an input, returns

1

if there exists a correspondence in

P

and

0

otherwise.

The problem is named after E. Post because he proved

Theorem 1

.

The Post correspondence problem is unsolvable (no such algorithms exist).

Title

Post correspondence problem

Canonical name

PostCorrespondenceProblem

Date of creation

2013-03-22 19:09:59

Last modified on

2013-03-22 19:09:59

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q45

Generated on Sat Feb 10 12:46:43 2018 by

LaTeXML
restricted homomorphism

restricted homomorphism

Let

h

be a

homomorphism

over an

alphabet

Σ

. Let

L

be a

language

over

Σ

. We say that

h

is

k

-restricted

on

L

if

1.

there is a letter

b

∈

Alpha

⁡

(

L

)

such that no word in

L

begins with

b

and contains more than

k

-

1

consecutive

occurrences

of

b

in it,

2.

for any

a

∈

Alpha

⁡

(

L

)

,

h

⁢

(

a

)

=

{

λ

if

⁢

a

=

b

a

otherwise.

Here,

Alpha

⁡

(

L

)

is the set of all letters in

Σ

that occur in words of

L

.

It is easy to see that any

k

-

restricted homomorphism

on

L

is a

k

-

linear erasing

on

L

, for if

u

∈

L

is a non-empty word, then we may write

u

=

v

1

⁢

b

m

1

⁢

v

2

⁢

b

m

2

⁢

⋯

⁢

v

n

⁢

b

m

n

, where each

0

<

m

i

≤

k

-

1

, and each

v

i

is a non-empty word not containing any occurrences of

b

. Then

|

u

|

=

|

v

1

⁢

⋯

⁢

v

n

-

1

|

+

∑

i

=

1

n

m

i

≤

|

h

⁢

(

u

)

|

+

n

⁢

(

k

-

1

)

≤

|

h

⁢

(

u

)

|

+

(

k

-

1

)

⁢

|

h

⁢

(

u

)

|

=

k

⁢

|

h

⁢

(

u

)

|

.

Note that

n

≤

|

h

⁢

(

u

)

|

since

1

≤

|

v

i

|

for each

i

=

1

,

…

,

n

. A

k

-linear erasing is in general not a

k

-restricted homomorphism, an example of which is the following:

L

=

{

a

,

a

⁢

b

}

*

and

h

:

{

a

,

b

}

→

{

a

,

b

}

given by

h

⁢

(

a

)

=

a

2

and

h

⁢

(

b

)

=

λ

. Then

h

is a

1

-linear erasing, but not a

1

-restricted homomorphism, on

L

.

A family

ℱ

of languages is said to be

closed under

restricted homomorphism

if for every

L

∈

ℱ

, and every

k

-restricted homomorphism

h

on

L

,

h

⁢

(

L

)

∈

ℱ

. By the previous paragraph, we see that if

ℱ

is closed under linear erasing, it is closed under restricted homomorphism. The

converse

of this is not necessarily true.

References

1

A. Salomaa,

Formal Languages

, Academic Press, New York (1973).

Title

restricted homomorphism

Canonical

name

RestrictedHomomorphism

Date of creation

2013-03-22 18:59:11

Last modified on

2013-03-22 18:59:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q45

Synonym

k

-restricted homomorphism

Synonym

k-restricted homomorphism

Related topic

LinearErasing

Generated on Sat Feb 10 12:46:57 2018 by

LaTeXML
standard enumeration

standard enumeration

The

standard enumeration

of

{

0

,

1

}

*

is the

sequence

of strings

s

0

=

λ

,

s

1

=

0

,

s

2

=

1

,

s

3

=

00

,

s

4

=

01

,

⋯

in

lexicographic order

.

The

characteristic function

of a

language

A

is

χ

A

:

ℕ

→

{

0

,

1

}

such that

χ

A

⁢

(

n

)

=

{

1

,

if

⁢

s

n

∈

A

0

,

if

⁢

s

n

∉

A

.

The

characteristic sequence

of a language

A

(also denoted as

χ

A

) is the

concatenation

of the values of the characteristic function in the natural order.

Title

standard enumeration

Canonical name

StandardEnumeration

Date of creation

2013-03-22 13:33:12

Last modified on

2013-03-22 13:33:12

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

21

Author

mathcam (2727)

Entry type

Definition

Classification

msc 68Q45

Classification

msc 03B65

Synonym

lexicographic enumeration

Defines

characteristic function

Defines

characteristic sequence

Generated on Sat Feb 10 12:47:09 2018 by

LaTeXML
concept lattice

concept lattice

Let

G

and

M

be sets whose

elements

we call

objects

and

attributes

respectively. Let

I

⊆

G

×

M

. We say that object

g

∈

G

has attribute

m

∈

M

iff

(

g

,

m

)

∈

I

. The triple

(

G

,

M

,

I

)

is called a

context

. For any set

X

⊆

G

of objects, define

X

′

:=

{

m

∈

M

∣

(

x

,

m

)

∈

I

⁢

for all

⁢

x

∈

G

}

.

In other words,

X

′

is the set of all attributes that are common to all objects in

X

. Similarly, for any set

Y

⊆

M

of attributes, set

Y

′

:=

{

g

∈

G

∣

(

g

,

y

)

∈

I

⁢

for all

⁢

y

∈

M

}

.

In other words,

Y

′

is the set of all objects having all the attributes in

M

. We call a pair

(

X

,

Y

)

⊆

G

×

M

a

concept

of the context

(

G

,

M

,

I

)

provided that

X

′

=

Y

and

Y

′

=

X

.

If

(

X

,

Y

)

is a concept, then

X

is called the

extent

of the concept and

Y

the

intent

of the concept.

Given a context

(

G

,

M

,

I

)

. Let

𝔹

⁢

(

G

,

M

,

I

)

be the set of all concepts of

(

G

,

M

,

I

)

. Define a

binary relation

≤

on

𝔹

⁢

(

G

,

M

,

I

)

by

(

X

1

,

Y

1

)

≤

(

X

2

,

Y

2

)

iff

X

1

⊆

X

2

. Then

≤

makes

𝔹

⁢

(

G

,

M

,

I

)

a

lattice

, and in fact a

complete lattice

.

𝔹

⁢

(

G

,

M

,

I

)

together with

≤

is called the

concept latice

of the context

(

G

,

M

,

I

)

.

Title

concept lattice

Canonical name

ConceptLattice

Date of creation

2013-03-22 19:22:34

Last modified on

2013-03-22 19:22:34

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q55

Classification

msc 68P99

Classification

msc 08A70

Classification

msc 06B23

Classification

msc 03B70

Classification

msc 06A15

Defines

object

Defines

attribute

Defines

context

Defines

concept

Defines

extent

Defines

intent

Generated on Sat Feb 10 12:47:29 2018 by

LaTeXML
system state

system state

A

system state

, or simply

state

, is a function

s

:

L

→

V

from the set of labels

L

to the set of values

V

.

Title

system state

Canonical

name

SystemState

Date of creation

2013-03-22 15:31:37

Last modified on

2013-03-22 15:31:37

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 68Q65

Classification

msc 68P20

Synonym

state

Related topic

SystemDefinitions

Generated on Sat Feb 10 12:47:31 2018 by

LaTeXML