[PDF] - Linear Logi vs Ane Logi Linear Logi Examples A A; !( A PDF Document

SLIDE 1 Linear Logi vs AÆne Logi Linear Logi Examples A

A;

!(A

Æ

B ); !(A

Æ

C ) ` B

C

A

A
A;

!(A

Æ

B ); !(A

Æ

C ) 6` B

C

A

A
A;

!(A

Æ

B ); !(A

Æ

C ) ` A

B
C

Theorem (Lin oln, Mit hel, S edro v, Shank ar) LL is unde idable. Linear AÆne Logi =Linear Logi + W eak ening (LL W) The w eak ening rules:

`
;

A `

`
`

; B All three sequen ts from the ab

v

e example are deriv able in LL W. Theorem. LL W is de idable.

SLIDE 2 P etri Nets Def. A P etri net is a pair (P ; T ) where P is a nite set

f

pla es T

(!

P

!

P ) is a nite set

f

transa tions Def. A marking

f

a P etri net is a v e tor M 2 ! P Def. A transa tion t = (u; v ) is reable at M if 8p : P :M p

u

p Def. A transa tion t = (u; v ) res form M to M if M = M

u

+ v Def. A state M is rea hable from M if there is a sequen e M = M ; M 1 ; : : : ; M n = M su h that M i+1 is

btained

from M i after the ring

f

some transa tion. Theorem (Ma y er 1981, Kosara ju 1982). The rea habilit y problem is de idable.

SLIDE 3 Horn fragmen t

f

LL A simple pr

du t

is a tensor pro du t

f

literals (e.g. p

p
q

). A Horn impli ation is an impli ation

f

the form: A

Æ

B , where A and B are simple pro d- u ts. A Horn se quent is a sequen t

f

the form W ; ! ` Z where W and Z are simple pro du ts and

is

a set

f

Horn impli ations.

SLIDE 4 En o ding P etri nets in the Horn fragmen t Ea h pla e

rresp
nds

to a literal. V e tors (markings)

rresp
nds

to simple pro d- u ts. T ransa tion

rresp
nds

to Horn impli ations. P etri net R

rresp
nds

to the set

f

Horn impli ations

R

Theorem. M is rea hable from M in a P etri net R i the sequen t M ; ! R ` M is deriv able in LL. Theorem. The sequen t M ; ! R ` M is deriv- able in LL W i there is M 00

M

, su h that M 00 is rea hable from M in a P etri net R .

SLIDE 5 Normal fragmen t A

Horn

impli ation is an impli ation

f

the form: A

Æ

(B

C

), where A, B and C are simple pro du ts. A simple disjun tion is a disjun tion

f

the form: B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C , where B and C are simple pro d- u ts. A normal se quent is a sequen t

f

the form W ; ! ` ? ; where W is a simple pro du t,

is

a m ultiset

f

Horn impli ations,

Horn

impli ations and simple disjun tions, and

is

a m ultiset

f

simple pro du ts.

SLIDE 6 Example L 1 = hal k bl a k boar dpaper

Æ

pr esentation L 2 = sl idespr

j

e tor paper

Æ

pr esentation L 3 = bl a k boar d

pr
j

e tor paper sl ides hal k ; !L 1 ; !L 2 ; !L 3 ` pr esentation L 3 = bl a k boar d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pr

j

e tor paper

paper
sl

ides

hal

k ; !L 1 ; !L 2 ; !L 3 ` pr esentation; pr esentation paper

paper
sl

ides

hal

k ; !L 1 ; !L 2 ; !L 3 ` ?pr esentation

SLIDE 7 Let

=

(W ; ! ` ?). Game A

1.

Initially , all v e tors from ~

are

written

n

the bla kb

ard.

2. W e ma y write new v e tors with natur al

r

dinates b y the follo wing rules: (a) If X

Æ

Y 2

and

a v e tor a + ~ Y has b een already written, then w e ma y write a + ~ X . (b) If X

Æ

(Y 1

Y

2 ) 2

and

v e tors a + ~ Y 1 and a + ~ Y 2 ha v e b een already written, then w e ma y write a + ~ X . ( ) If Y 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y 2 2

and

v e tors a 1 + ~ Y 1 and a 2 + ~ Y 2 ha v e b een already written, then w e ma y write a 1 + a 2 . 3. The aim

f

the game is to

btain

~ W . Game B

4.

If a v e tor a has b een written and a

then

w e ma y write .

SLIDE 8 Theorem. (Computational in terpretation) 1. The normal sequen t

is

deriv able in LL i it is p

ssible

to rea h the aim in the game A

2.

The normal sequen t

is

deriv able in LL W i it is p

ssible

to rea h the aim in the game B

Pro
f.

It is easy to pro v e it b y indu tion

n

deriv ation and

n

the n um b er

f

steps that w e needed to a hiev e the aim in the games.

SLIDE 9 Redu tion to the normal fragmen t Theorem. F

r

an y sequen t

ne

an ee tiv ely

nstru t

a normal sequen t

su

h that LLW `

(

) LLW ` ; LL `

(

) LL ` : Lemma. Let

`
b

e a sequen t. Let x b e an atom in the sequen t. Let A b e an arbitrary form ula. Let

=

[x := A℄ and

=

[x := A℄. Then

`
is

deriv able i !(x

Æ

A); !(A

Æ

x);

`
is

deriv able.

SLIDE 10 The de idabilit y

f

LL W Theorem. The problem whether w e an rea h the aim in the game B

is

de idable. Lemma. An y set

f

pairwise in omparable v e - tors from ! n is nite. Def. Let

b

e a set

f

normal sequen ts. Let A

!

n . W e sa y that A is

losed

when 1. If X

Æ

Y 2

then

8a 2 ! n a + ~ Y 2 A ) a + ~ X 2 A: 2. If X

Æ

(Y 1

Y

2 ) 2

then

8a 2 ! n a+ ~ Y 1 2 A; a+ ~ Y 2 2 A ) a+ ~ X 2 A: 3. If Y 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y 2 2

then

8a 1 ; a 2 2 ! n a 1 + ~ Y 1 2 A; a 2 + ~ Y 2 2 A ) a 1 +a 2 2 A: W e sa y that A is w

losed

when 8a 2 A 8

a

2 A:

SLIDE 11 The set

f

all rea hable v e tors in the game B

is
losed

and w

losed.

Lemma. It is p

ssible

to rea h the aim in the game B

if

and

nly

if the follo wing holds: F

r

an y A

!

n if A is

losed

and w

losed

and ~

A,

then ~ W 2 A. Lemma. If A is losed under the w eak ening then for some nite B A = [ z 2B K z ; where K z = fx j x

z

g. Lemma. The prop ert y

f

the set A = [ z 2B K z to b e

lose

is de idable. Corollary . The set

f

deriv able normal sequen ts is

en

umerable.