philippe de groote, bruno guillaume, sylvain salvati- vector addition tree automata & multiplicative...

8/3/2019 Philippe de Groote, Bruno Guillaume, Sylvain Salvati- Vector Addition Tree Automata & Multiplicative Exponential Linear Logic

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VATA & MELL GDR-ALP September 24, 2004

Vector Addition Tree Automata&Multiplicative Exponential Linear Logic

Philippe de Groote, Bruno Guillaume, Sylvain Salvati

LORIA & INRIA Lorraine

September 24, 2004
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Overview

Introduction

An example

The automata side

The linear logic side

Bridging the two sides
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Introduction
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Introduction

There is an encoding of a fragment (!-Horn) of MELL in Petri Nets or Vector

Addition System with States (Kanovich).
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Introduction


Addition System with States (Kanovich). Our goal is to extend this encoding to full [I]MELL.
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Introduction



This yields a reformulation of the (open) problem of the decidability of[I]MELL in an equivalent problem in automata theory.
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Introduction



This yields a reformulation of the (open) problem of the decidability of[I]MELL in an equivalent problem in automata theory.

!-Horn fragmentVector Addition

System with States Petri Nets

Finite State Automata

IMELLVector AdditionTree Automata

Tree Automata
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An example

Find a proof in IMELL of

!((A B S) S), !(S S S), !(A S), !(B S) S
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An example


!((A B S) S), !(S S S), !(A S), !(B S) S

is equivalent to find a closed linear -term of type S on the signature:

f : (A B S) S g : S S S a : A S b : B S
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An example


!((A B S) S), !(S S S), !(A S), !(B S) S



Lets go...
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An example


!((A B S) S), !(S S S), !(A S), !(B S) S



Lets go...

TS ::= f (xy.TS[x : A, y : B])| g TS TS
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An example


!((A B S) S), !(S S S), !(A S), !(B S) S



Lets go...TS ::= f (xy.TS[x : A, y : B])

| g TS TS

TS[x : A, y : B] ::= f (xy.TS[x : A, y : B, x : A, y : B])

| g TS TS[x : A, y : B]

| g TS[x : A] TS[y : B]

| g TS[y : B] TS[x : A]

| g TS[x : A, y : B] TS
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An example


!((A B S) S), !(S S S), !(A S), !(B S) S



Lets go...TS ::= f (xy.TS[x : A, y : B])

| g TS TS

TS[x : A, y : B] ::= f (xy.TS[x : A, y : B, x : A, y : B])

| g TS TS[x : A, y : B]

| g TS[x : A] TS[y : B]

| g TS[y : B] TS[x : A]

| g TS[x : A, y : B] TS

TS[x : A] ::= a x

| . . .

G A S b
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An example

The signature:


In fact, only the number of free variables matters.

VATA & MELL GDR ALP S t b 24 2004
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An example

The signature:

f : (A

B

S)

S g : S

S

S a : A

S b : B

S

In fact, only the number of free variables matters.Let x = (n1, n2) N2, TS[x] stands for the set of terms of type S with n1 free variables of type A and n2 freevariables of type B.

VATA & MELL GDR ALP S t b 24 2004
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An example

The signature:

f : (A

B

S)

S g : S

S

S a : A

S b : B

S


TS[x] ::= f ( : A : B TS[x + (1, 1)])

| g TS[x1] TS[x2] x = x1 + x2| a if x = (1, 0)

| b if x = (0, 1)

VATA & MELL GDR ALP September 24 2004
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An example

The signature:

f : (A

B

S)

S g : S

S

S a : A

S b : B

S


TS[x] ::= f ( : A : B TS[x + (1, 1)])

| g TS[x1] TS[x2] x = x1 + x2| a if x = (1, 0)

| b if x = (0, 1)

Putting it as a (tree automata like) rewriting system:

a q[(1, 0)]

b q[(0, 1)]

f(q[x]) q[x (1, 1)]

g(q[x1], q[x2]) q[x1 + x2]

VATA & MELL GDR ALP September 24 2004
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An example

= (A B S) S, S S S, A S, B S

f

g

dddd

f a

g

dddd

g a

d

ddd

bb

VATA & MELL GDR-ALP September 24 2004
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An example

= (A B S) S, S S S, A S, B S

f

g

dddd

f a

g

dddd

g a

d

ddd

bbq(0,1)

(b)!, B S

VATA & MELL GDR-ALP September 24 2004
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VATA & MELL GDR ALP September 24, 2004

An example

= (A B S) S, S S S, A S, B S

f

g

dddd

f a

g

dddd

g a

d

ddd

bb q(0,1)q(0,1)

(b)!, B S

(b)!, B S

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VATA & MELL GDR ALP September 24, 2004

An example

= (A B S) S, S S S, A S, B S

f

g

dddd

f a

g

dddd

g aq(0,2)

(b)!, B S

(b)!, B S

(g)!, B, B S

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& p ,

An example

= (A B S) S, S S S, A S, B S

f

g

dddd

f a

g

dddd

g aq(0,2) q(1,0)

(b)!, B S

(b)!, B S

(g)!, B, B S

(a)!, A S

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p ,

An example

= (A B S) S, S S S, A S, B S

f

g

dddd

f a

gq(1,2)

(b)!, B S

(b)!, B S

(g)!, B, B S

(a)!, A S

(g)!, A,B,B S

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An example

= (A B S) S, S S S, A S, B S

f

g

dddd

f aq(0,1) (b)!, B S

(b)!, B S

(g)!, B, B S

(a)!, A S

(g)!, A,B,B S

(f)!, B S

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An example

= (A B S) S, S S S, A S, B S

f

g

dddd

f aq(0,1) q(1,0) (b)!, B S

(b)!, B S

(g)!, B, B S

(a)!, A S

(g)!, A,B,B S

(f)!, B S

(a)!, A S

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An example

= (A B S) S, S S S, A S, B S

f

gq(1,1)

(b)!, B S

(b)!, B S

(g)!, B, B S

(a)!, A S

(g)!, A,B,B S

(f)!, B S

(a)!, A S

(g)!, A, B S

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An example

= (A B S) S, S S S, A S, B S

fq(0,0)

(b)!, B S

(b)!, B S

(g)!, B, B S

(a)!, A S

(g)!, A,B,B S

(f)!, B S

(a)!, A S

(g)!, A, B S

(f)! S

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Vector Addition Tree Automata

A k-VATA is a quadruple F, Q, Cf, where:

F is a ranked alphabet;

Q is a finite set of states;

Cf is a finite set of accepting configurations (i.e. elements of Q Nk);

is a finite set of transition rules of the form:

f(q0[x0], . . . , qn1[xn1]) q in (xi ci) + c

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f(q0[x0], . . . , qn1[xn1]) q in (xi ci) + cwhere:

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f(q0[x0], . . . , qn1[xn1]) q in (xi ci) + cwhere:

f F is a functional symbol of arity n;

q0 . . . qn1 Q;

c, c0, . . . cn1 Nk are given vectors, proper to the transition rule;

x0, . . . xn1 are variables in Nk.

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f(q0[x0], . . . , qn1[

xn1]) q in (xi ci) + c

where:

f F is a functional symbol of arity n;

q0 . . . qn1 Q;

c, c0, . . . cn1 Nk are given vectors, proper to the transition rule;

x0, . . . xn1 are variables in Nk.

The rewriting relation is induced by the transition rules with the constraint that xici Nk (this correspondsto the positivity condition in VASS).

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Vector Addition Tree Automata under Normal Form

A k-VATA is a normal form if

it is deterministic;

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Cf = (qf, 0);

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Cf = (qf, 0);

each transition rule is in one of the three forms:

f q[ei] for some i k

f(q0[x0]) q[x0 ei] for some i k

f(q0[x0], q1[x1]) q[x0 + x1]

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Cf = (qf, 0);

each transition rule is in one of the three forms:

f q[ei] for some i k

f(q0[x0]) q[x0 ei] for some i k

f(q0[x0], q1[x1]) q[x0 + x1]

Proposition 1 For any k-VATA A there exists a k-VATA A in normal form such that LA = iff LA = .

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Linear Logic

IMELL

F ::= 1 | A | F F | F F | !F

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Linear Logic

IMELL

F ::= 1 | A | F F | F F | !F

IMELL0

F0 ::= M | !MM ::= 1 | A | M M | MM

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Linear Logic

IMELL

F ::= 1 | A | F F | F F | !F

IMELL0

F0 ::= M | !MM ::= 1 | A | M M | MM

IMELL0F0 ::= M | !M

M ::= A | MM


Li L i
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Linear Logic

IMELL

F ::= 1 | A | F F | F F | !F

IMELL0

F0 ::= M | !MM ::= 1 | A | M M | MM

IMELL0F0 ::= M | !M

M ::= A | MM

s-IMELL0

s-F0 ::= A | !(A A) | !(A A A) | !((A A) A)


F IMELL t IMELL
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From IMELL to IMELL0

Proposition 2 IMELL is decidable iff IMELL0 is decidable.


F IMELL t IMELL
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Proof

Let A an IMELL sequent.


F IMELL t IMELL
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Proof


Define A to be the sequent obtained by replacing each exponential subformula !F by a freshatomic proposition pF.


From IMELL to IMELL
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Proof



Add , the following set of formulas to the antecedent of the sequent:

!(pF F),

!(pF pF pF),

!(pF 1),

!(pF1 pFn pF) whenever pF1 pFn F is provable.


From IMELL to IMELL
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Proof




!(pF F),

!(pF pF pF),

!(pF 1),


If IMELL0 is decidable then the construction is effective.


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Proof




!(pF F),

!(pF pF pF),

!(pF 1),


If IMELL0 is decidable then the construction is effective.

A is provable if and only if , A is provable.


From IMELL0 to IMELL
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From IMELL0 to IMELL0

Proposition 3 IMELL0 is decidable iff IMELL

0 is decidable.


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0 is decidable.

Proof Let b a fresh atomic proposition, we define two translation on multiplicative formulas:

A+ = A b, for any formula A

1 = b

a = a b

(A B) = A+ (B+ b)

(A B) = (A+ B+) b

and extend it to IMELL0 with (!A)+ = !(A+).


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0 is decidable.

Proof Let b a fresh atomic proposition, we define two translation on multiplicative formulas:

A+ = A b, for any formula A

1 = b

a = a b

(A B) = A+ (B+ b)

(A B) = (A+ B+) b

and extend it to IMELL0 with (!A)+ = !(A+).

A an IMELL0 sequent is provable iff + A+ is provable.


From IMELL0 to s-IMELL0
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From IMELL0 to s IMELL0

Proposition 4 IMELL0 is decidable iff s-IMELL

0 is decidable.


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From IMELL0 to s IMELL0


0 is decidable.

Proof

Lemma 1 Let A an IMELL sequent.

. . . F . . . F . . .

. . . F . . . A

!(p F), !(F p), . . . p . . . p . . .

. . . p . . . A


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0 0


0 is decidable.

Proof

Lemma 1 Let A an IMELL sequent.

. . . F . . . F . . .

. . . F . . . A

!(p F), !(F p), . . . p . . . p . . .

. . . p . . . A

!, A is a provable IMELL0 sequent

!, atomic

a is a provable IMELL0 sequent

!

2,

atomic a is a provable IMELL0 sequent


From VATA to IMELL
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Let A = F, Q, {(qf, 0)}, a k-VATA in normal form.


From VATA to IMELL
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Let A = F, Q, {(qf, 0)}, a k-VATA in normal form.We define the set of atomic types A = Q {a0, . . . , ak1} and by:

f q[ei] ai q

f(q0[x0]) q[x0 ei] (ai q0) q

f(q0[x0], q1[x1]) q[x0 + x1] q0 q1 q


From VATA to IMELL
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Let A = F, Q, {(qf, 0)}, a k-VATA in normal form.We define the set of atomic types A = Q {a0, . . . , ak1} and by:

f q[ei] ai q

f(q0[x0]) q[x0 ei] (ai q0) q

f(q0[x0], q1[x1]) q[x0 + x1] q0 q1 q

Proposition 5 L(A) = iff ! qf.


From IMELL to VATA
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Let !, a0 an s-IMELL0 sequent and {a0, . . . , ak1} an enumeration of the atomic formulas of the sequent.


From IMELL to VATA
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Let !, a0 an s-IMELL0 sequent and {a0, . . . , ak1} an enumeration of the atomic formulas of the sequent.We define the set of state Q = {q0, . . . , qk1}, the only final configuration (q0, ||) and the set of transitions:

ci qi[ei]

aj al f(qj[x]) ql[x]

(aj al) am g(ql[x]) qm[x ej ]

aj al am h(qj [x0], ql[x1]) qm[x0 + x1]


From IMELL to VATA
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Let !, a0 an s-IMELL0 sequent and {a0, . . . , ak1} an enumeration of the atomic formulas of the sequent.We define the set of state Q = {q0, . . . , qk1}, the only final configuration (q0, ||) and the set of transitions:

ci qi[ei]

aj al f(qj[x]) ql[x]

(aj al) am g(ql[x]) qm[x ej ]

aj al am h(qj [x0], ql[x1]) qm[x0 + x1]

Proposition 6 !, a0 is provable in s-IMELL0 iff L(A) = .


Conclusion and future work
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We have so far:

Decidability of MELL is equivalent to decidability of reachability in VATA


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We have so far:

Decidability of MELL is equivalent to decidability of reachability in VATA

Future work:

Prove the decidability of MELL
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philippe de groote, bruno guillaume, sylvain salvati- vector addition tree automata & multiplicative...

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