Post system

Post system

Introduction

A

Post canonical system

(or

Post system

for short)

𝔓

is a triple

(

Σ

,

X

,

P

)

, such that

1.

Σ

is an

alphabet

,

2.

X

is an alphabet,

disjoint

from

Σ

, whose

elements

we call

variables

, and

3.

P

is a non-empty finite

binary relation

on

Σ

*

, the

Kleene star

of

P

, such that for every

(

u

,

v

)

∈

P

,

–

u

∈

Σ

*

⁢

X

⁢

Σ

*

, and

–

if

S

is a variable occurring in

v

, then it

occurs in

u

.

An element

(

x

,

y

)

∈

P

is called a

production

of

𝔓

,

x

is its

antecedent

, and

y

the

consequent

.

(

x

,

y

)

∈

P

is often written

x

→

y

.

The last condition basically says that in a production

x

→

y

,

x

must contain at least one variable, and

y

can not contain any variables that are not already occurring in

x

. Put it more concretely, a production in a Post canonical system has the form

a

1

⁢

S

1

⁢

a

2

⁢

S

2

⁢

⋯

⁢

a

n

⁢

S

n

⁢

a

n

+

1

→

b

1

⁢

S

ϕ

⁢

(

1

)

⁢

b

2

⁢

S

ϕ

⁢

(

2

)

⁢

⋯

⁢

b

m

⁢

S

ϕ

⁢

(

m

)

⁢

b

m

+

1

(1)

where

a

i

and

b

j

are fixed words on

Σ

, while

S

k

are variables, with

0

<

n

,

0

≤

m

,

m

≤

n

, and

ϕ

is a

function

(not necessarily

bijective

) on the set

{

1

,

…

,

n

}

.

Examples

. Let

Σ

=

{

a

,

b

,

c

}

and

X

=

{

S

,

U

,

V

,

W

}

. Then

(

Σ

,

X

,

P

)

with

P

consisting of

a

⁢

S

⁢

b

2

→

b

⁢

a

,

c

⁢

V

⁢

a

⁢

W

⁢

a

⁢

U

⁢

b

→

a

⁢

W

⁢

U

,

a

3

⁢

c

⁢

U

⁢

b

⁢

S

⁢

W

→

S

⁢

a

⁢

b

⁢

U

,

b

⁢

V

⁢

a

→

a

⁢

V

2

⁢

c

is a Post canonical system, while

(

Σ

,

X

,

Y

)

with

Y

consisting of

a

⁢

b

2

→

b

⁢

a

,

c

⁢

V

⁢

a

⁢

W

⁢

a

⁢

U

⁢

b

→

a

⁢

W

⁢

U

,

a

⁢

U

⁢

b

⁢

S

⁢

c

2

⁢

W

→

S

⁢

c

⁢

a

⁢

V

,

a

→

S

is not, for the following reasons:

•

the antecedents in the first and fourth productions do not contain a variable

•

the consequents in the third and fourth productions contain variables (

V

in the third, and

S

in the fourth) which do not occur in the corresponding antecedents.

Normal systems

. A Post canonical system

𝔓

=

(

Σ

,

X

,

P

)

is called a

Post normal system

, or

normal system

for short, if each production has the form

a

⁢

S

→

S

⁢

b

(called a

normal production

), where

a

,

b

are words on

Σ

and

S

is a variable.

Languages generated by a Post system

Let us fix a Post system

𝔓

=

(

Σ

,

X

,

P

)

. A word

v

is said to be

immediately derivable

from a word

u

if there is a production of the form (1) above, such that

u

=

a

1

⁢

u

1

⁢

a

2

⁢

u

2

⁢

⋯

⁢

a

n

⁢

u

n

⁢

a

n

+

1

and

v

=

b

1

⁢

a

ϕ

⁢

(

1

)

⁢

b

2

⁢

a

ϕ

⁢

(

2

)

⁢

⋯

⁢

b

m

⁢

a

ϕ

⁢

(

m

)

⁢

b

m

+

1

,

where

a

i

are words (not variables). This means that if we can write a word

u

in the form of an antecedent of a production by replacing all the variables with words, then we can “produce”, or “derive” a word

v

in the form of the corresponding consequent, replacing the corresponding variables with the corresponding words. When

v

is immediately derivable from

u

, we write

u

⇒

v

. Using the example above, with the production

c

⁢

V

⁢

a

⁢

W

⁢

a

⁢

U

⁢

b

→

a

⁢

W

⁢

U

, we see that

•

c

⁢

a

4

⁢

b

=

c

⁢

a

⁢

a

⁢

a

⁢

a

⁢

b

⇒

a

3

if we set

V

=

λ

and

W

=

U

=

a

, or

•

c

⁢

a

4

⁢

b

=

c

⁢

a

⁢

a

⁢

a

⁢

a

⁢

b

⇒

a

2

if we set

V

=

a

and exactly one of

W

,

U

=

a

and the other

λ

.

A word

v

is

derivable

from a word

u

if there is a

finite sequence

of words

u

1

,

…

,

u

n

such that

u

=

u

1

⇒

u

2

⇒

⋯

⇒

u

n

=

v

.

When

v

is derivable from

u

, we write

u

⇒

*

v

. Again, following from the example above, we see that

c

2

⁢

a

⁢

b

⁢

a

⁢

b

2

⇒

*

a

⁢

c

, since

c

2

⁢

a

⁢

b

⁢

a

⁢

b

2

=

c

⁢

c

⁢

a

⁢

b

⁢

a

⁢

b

⁢

b

⇒

a

⁢

b

2

⇒

b

⁢

a

⇒

a

⁢

c

.

Given a finite subset

A

of words on

Σ

, let

T

A

be the set of all words derivable from words in

A

. Elements of

A

are called

axioms

of

𝔓

and elements of

T

A

theorems

(of

𝔓

derived from axioms of

A

). Intuitively, we see that the Post system

𝔓

is a

language

generating

machine

that creates the language

T

A

via a set

A

of axioms. In general, we say that a language

M

over an alphabet

Σ

is

generable by a Post system

if there is a Post system

𝔓

=

(

Σ

1

,

X

,

P

)

such that

Σ

⊆

Σ

1

, a

finite set

A

of axioms on

Σ

1

such that

M

=

T

A

∩

Σ

*

.

Remarks

.

•

If a language is generable by a Post system, it is generable by a normal system.

•

A language is generable by a Post system iff it is generable by a semi-Thue system. In this sense, Post systems and semi-Thue systems are “

equivalent

”.

•

Instead of allowing for one antecedent and one consequent in any production, one can have a more generalized system where one production involves a finite number of antecedents as well as a finite number of consequents:

{

a

11

⁢

S

11

⁢

a

12

⁢

S

12

⁢

⋯

⁢

a

1

⁢

n

1

⁢

S

1

⁢

n

1

⁢

a

1

,

n

1

+

1

,

⋮

a

p

⁢

1

⁢

S

p

⁢

1

⁢

a

p

⁢

2

⁢

S

p

⁢

2

⁢

⋯

⁢

a

p

⁢

n

p

⁢

S

p

⁢

n

p

⁢

a

p

,

n

p

+

1

}

→

{

b

11

⁢

S

ϕ

1

⁢

(

1

)

⁢

b

12

⁢

S

ϕ

1

⁢

(

2

)

⁢

⋯

⁢

b

1

⁢

m

⁢

S

ϕ

1

⁢

(

m

1

)

⁢

b

1

,

m

1

+

1

,

⋮

b

q

⁢

1

⁢

S

ϕ

q

⁢

(

1

)

⁢

b

q

⁢

2

⁢

S

ϕ

q

⁢

(

2

)

⁢

⋯

⁢

b

q

⁢

m

⁢

S

ϕ

q

⁢

(

m

q

)

⁢

b

q

,

m

+

1

}

where each

ϕ

i

is a function from

{

1

,

…

,

m

i

}

to

{

(

1

,

1

)

,

…

,

(

1

,

n

1

)

,

…

,

(

p

,

1

)

,

…

,

(

p

,

n

p

)

}

. We may define

b

to be immediately derivable from

a

if

a

can be expressed using

each

of the antecedents by substituting the variables

S

i

⁢

j

by words

c

i

⁢

j

∈

Σ

*

, and

b

can be expressed in

at least one

of the consequents by the corresponding

substitutions

(of

S

ϕ

i

⁢

(

j

)

into

c

ϕ

i

⁢

(

j

)

). It can be shown that any language generated by this more general system is in fact Post generable!

•

It can be shown that a language is Post-generable iff it is

recursively enumerable

.

Post Computability

For any positive integer

m

, and an

m

-tuple

n

¯

:=

(

n

1

,

…

,

n

m

)

of

natural numbers

, we may associate a word

E

⁢

(

n

¯

)

:=

a

⁢

b

n

1

⁢

a

⁢

b

n

2

⁢

a

⁢

⋯

⁢

a

⁢

b

n

m

⁢

a

.

Let

f

:

ℕ

m

→

ℕ

be a

partial function

. Define

L

⁢

(

f

)

:=

{

E

⁢

(

n

¯

)

⁢

c

⁢

E

⁢

(

f

⁢

(

n

¯

)

)

∣

n

¯

∈

dom

⁡

(

f

)

}

.

We say that

f

is

Post-computable

if

L

⁢

(

f

)

is Post-generable. As expected from the last remark in the previous

section

, a partial function is

Turing-computable

iff it is Post-computable.

References

1

M. Davis,

Computability and Unsolvability

. Dover Publications, New York (1982).

2

N. Cutland,

Computability: An

Introduction

to

Recursive Function

Theory

. Cambridge University Press, (1980).

3

M. Minsky,

Computation

: Finite and

Infinite

Machines

. Prentice Hall, (1967).

Title

Post system

Canonical name

PostSystem

Date of creation

2013-03-22 17:33:28

Last modified on

2013-03-22 17:33:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

19

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q42

Classification

msc 03D03

Synonym

normal system

Synonym

Post generable

Synonym

Post computable

Related topic

FormalGrammar

Defines

Post canonical system

Defines

Post normal system

Defines

normal production

Defines

generable by a Post system

Defines

Post-computable

Generated on Sat Feb 10 12:44:39 2018 by

LaTeXML