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Linear Logic vs AÆne Logic
Linear Logic
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 (Lincoln, Mitchel, Scedrov, Shankar)
LL is undecidable.
Linear AÆne Logic
=Linear Logic + Weakening (LLW)
The weakening rules:
`
`
;A ` ` ; B
All three sequents from the above example are
derivable in LLW.
Theorem. LLW is decidable.
Petri Nets
Def. A Petri net is a pair (P; T ) where
P is a nite set of places
T (!P !P ) is a nite set of transactions
Def. A marking of a Petri net is a vector M
!P
2
Def. A transaction t = (u; v ) is reable at M if
8p : P:Mp up
Def. A transaction t = (u; v ) res form M to
M if
M =M u+v
0
0
Def. A state M is reachable from M if there is
a sequence M = M0; M1; : : : ; Mn = M such
that Mi+1 is obtained from Mi after the ring
of some transaction.
0
0
Theorem (Mayer 1981, Kosaraju 1982).
The reachability problem is decidable.
Horn fragment of LL
A simple product
(e.g. p p q ).
is a tensor product of literals
is an implication of the
form: A Æ B , where A and B are simple products.
A Horn implication
A Horn sequent
is a sequent of the form
W; !
`Z
where W and Z are simple products and is a
set of Horn implications.
Encoding Petri nets in the Horn fragment
Each place corresponds to a literal.
Vectors (markings) corresponds to simple products.
Transaction corresponds to Horn implications.
Petri net R corresponds to the set of Horn implications R
Theorem. M is reachable from M in a Petri
net R i the sequent M; ! R ` M is derivable
in LL.
0
0
Theorem. The sequent M; ! R ` M is derivable in LLW i there is M M , such that
M is reachable from M in a Petri net R.
0
00
00
0
Normal fragment
-Horn implication is an implication of the
form: A Æ (B C ), where A, B and C are
A
simple products.
is a disjunction of the
form: B C , where B and C are simple products.
A simple disjunction
......................
.. .
..................
... ....
...
A normal sequent
is a sequent of the form
W; !
` ?;
where W is a simple product, is a multiset
of Horn implications, -Horn implications and
simple disjunctions, and is a multiset of simple products.
Example
L1 = chalk
blackboard
paper Æpresentation
L2 = slides
projector
paper Æpresentation
L3 = blackboard projector
paper
slides
chalk; !L1; !L2; !L3 ` presentation
L3 = blackboard projector
0
.......
.............
. .. ..
......
... ..........
.... ...
...
paper paper slides chalk; !L1; !L2; !L3 `
presentation; presentation
paper paper slides chalk; !L1; !L2; !L3 `
?presentation
0
0
Let = (W; !
Game A
` ?).
~ are written on
1. Initially, all vectors from the blackboard.
2. We may write new vectors with natural coordinates by the following rules:
(a) If X Æ Y 2 and a vector a + Y~ has
been already written, then we may write
a + X~ .
(b) If X Æ (Y1 Y2) 2 and vectors a + Y~1
and a + Y~2 have been already written,
~.
then we may write a + X
(c) If Y1 Y2 2 and vectors a1 + Y~1 and
a2 + Y~2 have been already written, then
we may write a1 + a2.
~.
3. The aim of the game is to obtain W
......................
.. .
...................
.........
Game B
4. If a vector a has been written and a
then we may write c.
c
Theorem. (Computational interpretation)
1. The normal sequent is derivable in LL i
it is possible to reach the aim in the game 2. The normal sequent is derivable in LLW i
it is possible to reach the aim in the game A
B
Proof.
It is easy to prove it by induction on derivation
and on the number of steps that we needed to
achieve the aim in the games.
Reduction to the normal fragment
Theorem. For any sequent one can eectively
construct a normal sequent such that
LLW ` () LLW ` ;
LL ` () LL ` :
Lemma. Let ` be a sequent.
Let x be an atom in the sequent.
Let A be an arbitrary formula.
Let = [x := A] and = [x := A].
Then ` is derivable i
0
0
0
0
!(x
is derivable.
Æ A); !(A Æ x); ` The decidability of LLW
Theorem. The problem whether we can reach
the aim in the game is decidable.
B
Lemma. Any set of pairwise incomparable vectors from ! n is nite.
Def. Let be a set of normal sequents. Let
A !n. We say that A is -closed when
1. If X Æ Y 2 then
8a 2 !n a + Y~ 2 A ) a + X~ 2 A:
2. If X Æ (Y1 Y2) 2 then
8a 2 !n
a+Y~1 2 A; a+Y~2 2 A ) a+X~ 2 A:
3. If Y1 Y2 2 then
8a1; a2 2 !n a1+Y~1 2 A; a2+Y~2 2 A ) a1+a2 2 A:
.........................
.. .
..................
... ....
...
We say that A is w-closed when
8a 2 A 8c a
c 2 A:
B
The set of all reachable vectors in the game is -closed and w-closed.
Lemma. It is possible to reach the aim in the
game if and only if the following holds:
B
For any A ! n if A is -closed and
~ A, then W
~ 2 A.
w-closed and Lemma. If A is closed under the weakening
then for some nite B
A = z [ Kz ;
where Kz = fx j x z g.
2B
Lemma. The property of the set A = [z Kz
to be -close is decidable.
2B
Corollary. The set of derivable normal sequents
is coenumerable.