aarjaneiro/planetmath huggingface dataset:
counting problem

counting problem

If

R

is a

search problem

then

c

R

⁢

(

x

)

=

|

{

y

∣

R

⁢

(

x

)

}

|

is the corresponding counting

function

and

#

⁢

R

=

{

(

x

,

y

)

∣

y

≤

c

R

⁢

(

x

)

}

denotes the corresponding

counting problem

. Note that

c

R

is a search problem while

#

⁢

R

is a

decision problem

, however

c

R

can be

𝒞

Cook

reduced

to

#

⁢

R

(for appropriate

𝒞

) using a

binary search

(the reason

#

⁢

R

is defined the way it is, rather than being the graph of

c

R

, is to make this binary search possible).

Title

counting problem

Canonical name

CountingProblem

Date of creation

2013-03-22 13:02:17

Last modified on

2013-03-22 13:02:17

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q25

Generated on Sat Feb 10 12:41:45 2018 by

LaTeXML
decision problem

decision problem

Let

T

be a

Turing machine

and let

L

⊆

Γ

+

be a

language

. We say

T

decides

L

if for any

x

∈

L

,

T

accepts

x

, and for any

x

∉

L

,

T

rejects

x

.

We say

T

enumerates

L

if:

x

∈

L

⁢

iff

⁢

T

⁢

accepts

⁢

x

For some Turing machines (for

instance

non-deterministic

machines

) these definitions are

equivalent

, but for others they are not. For example, in order for a

deterministic Turing machine

T

to decide

L

, it must be that

T

halts on every input. On the other hand

T

could enumerate

L

if it does not halt on some strings which are not in

L

.

L

is sometimes said to be a

decision problem

, and a Turing machine which decides it is said to solve the decision problem.

The set of strings which

T

accepts is denoted

L

⁢

(

T

)

.

Title

decision problem

Canonical name

DecisionProblem

Date of creation

2013-03-22 13:01:33

Last modified on

2013-03-22 13:01:33

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

12

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q25

Defines

enumerates

Defines

decide

Generated on Sat Feb 10 12:41:47 2018 by

LaTeXML
promise problem

promise problem

A

promise problem

is a

generalization

of a

decision problem

. It is defined by two decision problems

L

1

and

L

2

with

L

1

∩

L

2

=

∅

. A

Turing machine

decides a promise problem if, for any

x

∈

L

1

∪

L

2

, it accepts when

x

∈

L

1

and rejects if

x

∈

L

2

. Behavior is undefined when

x

∉

L

1

∪

L

2

(this is the promise: that

x

is in one of the two sets).

If

L

2

=

Γ

+

∖

L

1

then this is just the decision problem for

L

1

.

Title

promise problem

Canonical name

PromiseProblem

Date of creation

2013-03-22 13:02:27

Last modified on

2013-03-22 13:02:27

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q25

Generated on Sat Feb 10 12:41:52 2018 by

LaTeXML
proof of Stirling’s approximation

proof of Stirling’s approximation

Computing the

Taylor expansion

with

remainder

of the

functions

log

and

x

↦

x

⁢

log

⁡

x

-

x

, we have

(

n

+

1

)

⁢

log

⁡

(

n

+

1

)

-

log

⁡

(

n

+

1

)

=

n

⁢

log

⁡

n

-

n

+

log

⁡

n

+

1

2

⁢

n

+

1

6

⁢

ξ

n

2

log

⁡

(

n

+

1

)

=

log

⁡

n

+

1

n

-

1

2

⁢

η

n

2

where

n

≤

ξ

n

≤

n

+

1

and

n

≤

η

n

≤

n

+

1

.

Summing

the first equation from

1

to

n

-

1

, we have

n

log

n

-

n

=

-

1

+

log

(

n

-

1

)

!

+

1

2

∑

m

=

1

n

-

1

1

m

+

1

6

∑

m

=

1

n

-

1

1

ξ

n

2

.

Title

proof of Stirling’s

approximation

Canonical

name

ProofOfStirlingsApproximation

Date of creation

2014-05-08 22:09:30

Last modified on

2014-05-08 22:09:30

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 68Q25

Classification

msc 30E15

Generated on Sat Feb 10 12:41:54 2018 by

LaTeXML
range problem

range problem

A

range problem

is a weakened form of a

search problem

. It consists of two

functions

f

l

and

f

u

(the lower and

upper bounds

) and a

linear ordering

<

on the ranges of

f

1

and

f

2

. A

Turing machine

solves a range problem if, for any

x

, the

machine

eventually halts with an output

y

such that

f

1

⁢

(

x

)

<

y

<

f

2

⁢

(

x

)

.

For example, given any function

f

with range in

ℝ

and any

g

:

ℕ

→

ℝ

, the

strong range problem

StrongRange

g

⁡

(

f

)

is given by

lower bound

f

⁢

(

x

)

⋅

(

1

-

1

1

-

g

⁢

(

|

x

|

)

)

and upper bound

f

⁢

(

x

)

⋅

(

1

-

1

1

+

g

⁢

(

|

x

|

)

)

(note that

g

is passed the length of

x

, not the value, which need not even be a number).

Title

range problem

Canonical name

RangeProblem

Date of creation

2013-03-22 13:02:25

Last modified on

2013-03-22 13:02:25

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q25

Defines

strong range problem

Defines

strong range

Generated on Sat Feb 10 12:41:56 2018 by

LaTeXML
search problem

search problem

If

R

is a

binary relation

such that

field

⁡

(

R

)

⊆

Γ

+

and

T

is a

Turing machine

, then

T

calculates

f

if:

•

If

x

is such that there is some

y

such that

R

⁢

(

x

,

y

)

then

T

accepts

x

with output

z

such that

R

⁢

(

x

,

z

)

(there may be multiple

y

, and

T

need only find one of them)

•

If

x

is such that there is no

y

such that

R

⁢

(

x

,

y

)

then

T

rejects

x

Note that the of a

partial function

is a binary relation, and if

T

calculates a partial function then there is at most one possible output.

A

R

can be viewed as a

search problem

, and a Turing machine which calculates

R

is also said to solve it. Every search problem has a corresponding

decision problem

, namely

L

⁢

(

R

)

=

{

x

∣

∃

y

⁢

R

⁢

(

x

,

y

)

}

.

This definition may be generalized to

n

-ary

relations

using any suitable

encoding

which allows multiple strings to be compressed into one string (for

instance

by listing them consecutively with a delimiter).

Title

search problem

Canonical name

SearchProblem

Date of creation

2013-03-22 13:01:39

Last modified on

2013-03-22 13:01:39

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q25

Defines

calculate

Generated on Sat Feb 10 12:41:58 2018 by

LaTeXML
Stirling’s approximation

Stirling’s approximation

Stirling’s formula gives an

approximation

for

n

!

, the factorial . It is

n

!

≈

2

⁢

n

⁢

π

⁢

n

n

⁢

e

-

n

We can derive this from the

gamma function

. Note that for large

x

,

Γ

⁢

(

x

)

=

2

⁢

π

⁢

x

x

-

1

2

⁢

e

-

x

+

μ

⁢

(

x

)

(1)

where

μ

⁢

(

x

)

=

∑

n

=

0

∞

(

x

+

n

+

1

2

)

⁢

ln

⁡

(

1

+

1

x

+

n

)

-

1

=

θ

12

⁢

x

with

0

<

θ

<

1

. Taking

x

=

n

and multiplying by

n

, we have

n

!

=

2

⁢

π

⁢

n

n

+

1

2

⁢

e

-

n

+

θ

12

⁢

n

(2)

Taking the approximation for large

n

gives us Stirling’s formula.

There is also a

big-O notation

version of Stirling’s approximation:

n

!

=

(

2

⁢

π

⁢

n

)

⁢

(

n

e

)

n

⁢

(

1

+

𝒪

⁢

(

1

n

)

)

(3)

We can prove this equality starting from (

2

). It is clear that the big-O portion of (

3

) must come from

e

θ

12

⁢

n

, so we must consider the

asymptotic

behavior of

e

.

First we observe that the

Taylor series

for

e

x

is

e

x

=

1

+

x

1

+

x

2

2

!

+

x

3

3

!

+

⋯

But in our case we have

e

to a vanishing

exponent

. Note that if we vary

x

as

1

n

, we have as

n

⟶

∞

e

x

=

1

+

𝒪

⁢

(

1

n

)

We can then (almost) directly plug this in to (

2

) to get (

3

) (note that the factor of 12 gets absorbed by the big-O notation.)

Title

Stirling’s approximation

Canonical name

StirlingsApproximation

Date of creation

2013-03-22 12:00:36

Last modified on

2013-03-22 12:00:36

Owner

drini (3)

Last modified by

drini (3)

Numerical id

22

Author

drini (3)

Entry type

Theorem

Classification

msc 68Q25

Classification

msc 30E15

Classification

msc 41A60

Synonym

Stirling’s formula

Synonym

Stirling’s approximation formula

Related topic

MinkowskisConstant

Related topic

AsymptoticBoundsForFactorial

Generated on Sat Feb 10 12:42:00 2018 by

LaTeXML
computationally indistinguishable

computationally indistinguishable

If

{

D

n

}

n

∈

ℕ

and

{

E

n

}

n

∈

ℕ

are

distribution ensembles

(on

Ω

) then we say they are

computationally indistinguishable

if for any probabilistic,

polynomial time

algorithm

A

and any polynomal function

f

there is some

m

such that for all

n

>

m

:

|

Prob

A

(

D

n

)

=

Prob

A

(

E

n

)

|

<

1

p

⁢

(

n

)

where

Prob

A

⁡

(

D

n

)

is the

probability

that

A

accepts

x

where

x

is chosen according to the

distribution

D

n

.

Title

computationally indistinguishable

Canonical name

ComputationallyIndistinguishable

Date of creation

2013-03-22 13:03:11

Last modified on

2013-03-22 13:03:11

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q30

Generated on Sat Feb 10 12:42:02 2018 by

LaTeXML
distribution ensemble

distribution ensemble

A

distribution ensemble

is a sequence

{

D

n

}

n

∈

ℕ

where each

D

n

is a

distribution

with on some set

Ω

.

Title

distribution ensemble

Canonical

name

DistributionEnsemble

Date of creation

2013-03-22 13:02:43

Last modified on

2013-03-22 13:02:43

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 68Q30

Classification

msc 60A99

Generated on Sat Feb 10 12:42:04 2018 by

LaTeXML
invariance theorem

invariance theorem

The

invariance theorem

states that a

universal Turing machine

provides an
optimal means of description, up to a

constant

. Formally, for every

Turing machine

M

there exists a constant

c

such that for
all binary strings

x

we have

C

U

⁢

(

x

)

≤

C

M

⁢

(

x

)

+

c

.

Here,

C

U

means the

complexity

with respect to the universal
Turing machine and

C

M

means the complexity with respect to

M

.

This follows trivially from the definition of a universal Turing machine,
taking

c

=

l

⁢

(

<

M

>

)

as the length of the

encoding

of

M

.

The invariance theorem holds likewise for prefix and

conditional

complexities.

Title

invariance theorem

Canonical name

InvarianceTheorem

Date of creation

2013-03-22 13:43:54

Last modified on

2013-03-22 13:43:54

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 68Q30

Generated on Sat Feb 10 12:42:05 2018 by

LaTeXML
Kolmogorov complexity

Kolmogorov complexity

Consider flipping a coin 50 times to obtain the binary string
000101000001010100010000010101000100000001010000010.
Can we call this random? The string has rather an abundance of 0s,
and on closer inspection every other bit is 0. We wouldn’t expect even
a biased coin to come up with such a pattern. Still, this string
has probability

2

-

50

, just like any other binary string of the same length,
so how can we call it any less random?

Kolmogorov Complexity

provides an answer to these questions in the form
of a measure of

information content

in individual objects. Objects with low
information content may be considered non-random.
The topic was founded in the 1960s independently by three people:
Ray Solomonoff, Andrei Kolmogorov, and Gregory Chaitin.

See Kolmogorov complexity function and

invariance theorem

for more details.

Title

Kolmogorov complexity

Canonical name

KolmogorovComplexity

Date of creation

2013-03-22 13:43:41

Last modified on

2013-03-22 13:43:41

Owner

tromp (1913)

Last modified by

tromp (1913)

Numerical id

9

Author

tromp (1913)

Entry type

Topic

Classification

msc 68Q30

Synonym

algorithmic information theory

Synonym

algorithmic entropy

Generated on Sat Feb 10 12:42:07 2018 by

LaTeXML
Kolmogorov complexity function

Kolmogorov complexity function

The (plain)

complexity

C

⁢

(

x

)

of a binary string

x

is the length of a shortest

program

p

such that

U

⁢

(

p

)

=

x

, i.e. the (plain)

universal Turing machine

on
input

p

, outputs

x

and halts. The lexicographically least such

p

is
denoted

x

∗

.
The prefix complexity

K

⁢

(

x

)

is defined similarly in terms of the prefix

universal

machine

. When clear from

context

,

x

∗

is also used to denote the
lexicographically least prefix program for

x

.

Plain and prefix

conditional

complexities

C

(

x

|

y

)

,

K

(

x

|

y

)

are defined similarly
but with

U

(

x

|

y

)

=

x

, i.e. the universal machine starts out with

y

written
on its worktape.

Subscripting these

functions

with a

Turing machine

M

, as in

K

M

(

x

|

y

)

, denotes the corresponding complexity in which we use machine

M

in place of the universal machine

U

.

Title

Kolmogorov complexity function

Canonical name

KolmogorovComplexityFunction

Date of creation

2013-03-22 13:43:49

Last modified on

2013-03-22 13:43:49

Owner

tromp (1913)

Last modified by

tromp (1913)

Numerical id

9

Author

tromp (1913)

Entry type

Definition

Classification

msc 68Q30

Synonym

algorithmic entropy

Synonym

information content

Generated on Sat Feb 10 12:42:08 2018 by

LaTeXML
natural numbers identified with binary strings

natural numbers identified with binary strings

It is convenient to identify a

natural number

n

with the

n

th binary
string in

lexicographic order

:

0

ϵ

1

0

2

1

3

00

4

01

5

10

6

11

7

000

…

…

The more common

binary notation

for numbers fails to be a

bijection

because of leading zeroes. Yet, there is a close

relation

: the

n

th
binary string is the result of stripping the leading 1 from the binary
notation of

n

+

1

.

With this correspondence in place, we can talk about such things as
the length

l

⁢

(

n

)

of a number

n

, which can be seen to equal

⌊

log

⁡

(

n

+

1

)

⌋

.

Title

natural numbers identified with binary strings

Canonical

name

NaturalNumbersIdentifiedWithBinaryStrings

Date of creation

2013-03-22 13:43:44

Last modified on

2013-03-22 13:43:44

Owner

tromp (1913)

Last modified by

tromp (1913)

Numerical id

7

Author

tromp (1913)

Entry type

Definition

Classification

msc 68Q30

Generated on Sat Feb 10 12:42:10 2018 by

LaTeXML
one-way function

one-way function

A function

f

is a

one-way function

if for any probabilistic,

polynomial time

computable function

g

and any

polynomial function

p

there is

m

such that for all

n

>

m

:

Pr

⁡

[

f

⁢

(

g

⁢

(

f

⁢

(

x

)

)

)

=

f

⁢

(

x

)

]

<

1

p

⁢

(

n

)

where

x

has length

n

and all numbers of length

n

are equally likely.

That is, no probabilistic, polynomial time function can effectively compute

f

-

1

.

Note that, since

f

need not be injective, this is a stricter requirement than

Pr

[

g

(

f

(

x

)

)

)

=

x

]

<

1

p

⁢

(

n

)

since not only is

g

⁢

(

f

⁢

(

x

)

)

(almost always) not

x

, it is (almost always) no value such that

f

⁢

(

g

⁢

(

f

⁢

(

x

)

)

)

=

f

⁢

(

x

)

.

Title

one-way function

Canonical name

OnewayFunction

Date of creation

2013-03-22 13:03:14

Last modified on

2013-03-22 13:03:14

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q30

Synonym

one way function

Synonym

one-way

Synonym

one way

Generated on Sat Feb 10 12:42:12 2018 by

LaTeXML
pseudorandom

pseudorandom

A

distribution ensemble

{

D

n

}

n

∈

ℕ

is

pseudorandom

if it is

computationally indistinguishable

from the ensemble

{

U

n

}

n

∈

ℕ

where each

U

n

is the

uniform distribution

on the

support

of

D

n

. That is, no reasonable procedure can make meaningful predictions about what element will be chosen.

Title

pseudorandom

Canonical name

Pseudorandom

Date of creation

2013-03-22 13:02:36

Last modified on

2013-03-22 13:02:36

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

8

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q30

Classification

msc 60A99

Synonym

pseudorandom distribution ensemble

Synonym

pseudorandom ensemble

Generated on Sat Feb 10 12:42:13 2018 by

LaTeXML
pseudorandom generator

pseudorandom generator

Let

G

be a

deterministic

polynomial-time

function

from

ℕ

<

ω

to

ℕ

<

ω

with

stretch function

l

:

ℕ

→

ℕ

, so that if

x

has length

n

then

G

⁢

(

x

)

has length

l

⁢

(

n

)

. Then let

G

n

be the

distribution

on strings of length

l

⁢

(

n

)

defined by the output of

G

on a randomly selected string of length

n

selected by the

uniform distribution

.

Then we say

G

is

pseudorandom generator

if

{

G

n

}

n

∈

ℕ

is

pseudorandom

.

In effect,

G

translates a random input of length

n

to a pseudorandom output of length

l

⁢

(

n

)

. Assuming

l

⁢

(

n

)

>

n

, this expands a random sequence (and can be applied multiple times, since

G

n

can be replaced by the distribution of

G

⁢

(

G

⁢

(

x

)

)

).

Title

pseudorandom generator

Canonical name

PseudorandomGenerator

Date of creation

2013-03-22 13:03:16

Last modified on

2013-03-22 13:03:16

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q30

Defines

stretch function

Generated on Sat Feb 10 12:42:15 2018 by

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Chomsky normal form

Chomsky normal form

A

grammar

is said to be of

Chomsky normal form

if every production
has either of the two forms

A

→

B

⁢

C

or

A

→

a

where

A

,

B

,

C

are non-terminal symbols, and

a

is a terminal symbol.

Grammars of this sort are

context-free

, hence they describe context-free
languages. Moreover, given any context-free language not containing the

empty word

λ

, there exists a
Chomsky normal form grammar which describes it.

Title

Chomsky normal form

Canonical name

ChomskyNormalForm

Date of creation

2013-03-22 16:21:13

Last modified on

2013-03-22 16:21:13

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 68Q42

Classification

msc 68Q45

Related topic

GreibachNormalForm

Related topic

KurodaNormalForm

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Chomsky-Schützenberger theorem

Chomsky-Schützenberger theorem

An important

characterization

of

context-free languages

is captured in what is known as the Chomsky-Schützenberger

theorem

. It shows the intimate

connection

between context-free and

parenthesis languages

.

Theorem 1

(Chomsky-Schützenberger).

A langauge

L

over an

alphabet

Σ

is context-free iff for some

n

≥

0

, there there is a

homomorphism

h

:

Σ

n

*

→

Σ

*

such that

L

=

h

⁢

(

𝐏𝐚𝐫𝐞𝐧

𝒏

∩

R

)

,

where

Paren

𝐧

is the parenthesis language over

Σ

n

, and

R

is a

regular language

(over

Σ

n

).

Note that the “if” part is the

trivial

consequence

of the following facts: parenthesis languages are context-free; any homomorphic image of a context-free language is context-free; any

intersection

of a context-free language with a regular language is context-free.

References

1

N. Chomsky, M.P. Schützenberger,

The Algebraic Theory of Context-Free Languages, Computer Programming and

Formal Systems

, North-Holland, Amsterdam (1963).

2

D. C. Kozen,

Automata

and Computability

, Springer, New York (1997).

Title

Chomsky-Schützenberger theorem

Canonical name

ChomskySchutzenbergerTheorem

Date of creation

2013-03-22 18:55:40

Last modified on

2013-03-22 18:55:40

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 68Q42

Classification

msc 68Q45

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Church-Rosser property

Church-Rosser property

Let

→

be a

reduction

(a

binary relation

) on a set

S

, and let

↔

*

be the

reflexive

transitive

symmetric closure

of

→

. The reduction

→

is said to have the

Church-Rosser property

provided that

a

↔

*

b

implies that

a

and

b

are

joinable

, for any

a

,

b

∈

S

.

In terms of diagrams, the Church-Rosser property means the following, for any

a

,

b

∈

S

, if

a

↔

x

1

↔

x

2

↔

⋯

↔

x

n

↔

b

where

u

↔

v

means

u

→

v

or

u

←

v

(

:=

v

→

u

), then there is some

x

∈

S

such that

a

→

a

1

⁢

⋯

→

a

p

→

x

←

b

q

←

⋯

←

b

1

←

b

.

Remark

. It can be shown that

→

has the Church-Rosser property iff it is

confluent

.

References

1

F. Baader, T. Nipkow,

Term Rewriting and All That

, Cambridge University Press (1998).

Title

Church-Rosser property

Canonical

name

ChurchRosserProperty

Date of creation

2013-03-22 17:47:28

Last modified on

2013-03-22 17:47:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q42

Related topic

Confluence

Generated on Sat Feb 10 12:42:53 2018 by

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confluence

confluence

Call a

binary relation

→

on a set

S

a

reduction

. Let

→

*

be the

reflexive transitive closure

of

→

. Two

elements

a

,

b

∈

S

are said to be

joinable

if there is an element

c

∈

S

such that

a

→

*

c

and

b

→

*

c

. Graphically, this means that

\xymatrix

⁢

@

⁢

R

-

=

10

⁢

p

⁢

t

⁢

a

⁢

\ar

⁢

[

d

⁢

r

]

*

⁢

&

⁢

c

⁢

b

⁢

\ar

⁢

[

u

⁢

r

]

*

where the starred arrows represent

a

→

a

1

→

⋯

→

a

n

→

c

and

b

→

b

1

→

⋯

→

b

m

→

c

respectively (

m

,

n

are non-negative integers). The case

m

=

0

(or

n

=

0

) means

a

→

c

(or

b

→

c

).

Definition

.

→

is said to be

confluent

if whenever

x

→

*

a

and

x

→

*

b

, then

a

,

b

are joinable. In other words,

→

is confluent iff

→

*

has the

amalgamation property

. Graphically, this says that, whenever we have a diagram

\xymatrix

⁢

@

⁢

R

-

=

10

⁢

p

⁢

t

⁢

&

⁢

a

⁢

x

⁢

\ar

⁢

[

u

⁢

r

]

*

⁢

\ar

⁢

[

d

⁢

r

]

*

⁢

&

⁢

&

⁢

b

then it can be “completed” into a “

diamond

”:

\xymatrix

⁢

@

⁢

R

-

=

10

⁢

p

⁢

t

⁢

&

⁢

a

⁢

\ar

⁢

[

d

⁢

r

]

*

⁢

x

⁢

\ar

⁢

[

u

⁢

r

]

*

⁢

\ar

⁢

[

d

⁢

r

]

*

⁢

&

⁢

&

⁢

c

⁢

&

⁢

b

⁢

\ar

⁢

[

u

⁢

r

]

*

⁢

&

Remark

. A more general

property

than

confluence

, called

semi-confluence

is defined as follows:

→

is

semi-confluent

if for any triple

x

,

a

,

b

∈

S

such that

x

→

a

and

x

→

*

b

, then

a

,

b

are joinable. It turns out that this seemingly weaker notion is actually

equivalent

to the stronger notion of confluence. In

addition

, it can be shown that

→

is confluent iff

→

has the

Church-Rosser property

.

References

1

F. Baader, T. Nipkow,

Term Rewriting and All That

, Cambridge University Press (1998).

Title

confluence

Canonical

name

Confluence

Date of creation

2013-03-22 17:47:24

Last modified on

2013-03-22 17:47:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q42

Related topic

ChurchRosserProperty

Related topic

AmalgamationProperty

Related topic

NormalizingReduction

Related topic

TerminatingReduction

Defines

confluent

Defines

joinable

Defines

semi-confluent

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