Proof-Nets www.irif.fr/˜kesner

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Delia KESNER kesner@irif.fr www.irif.fr/˜kesner Multiplicative Linear Logic (MLL) MLL Proof-Nets Pre Proof-Nets Correctness Criteria Multiplicative Exponential Linear Logic (MELL) MELL Proof-Nets

Proof-Nets

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Agenda for Today

1

Multiplicative Linear Logic (MLL)

2

MLL Proof-Nets

3

Pre Proof-Nets

4

Correctness Criteria

5

Multiplicative Exponential Linear Logic (MELL)

6

MELL Proof-Nets

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Delia KESNER kesner@irif.fr www.irif.fr/˜kesner

Multiplicative Linear Logic (MLL)

MLL Proof-Nets Pre Proof-Nets Correctness Criteria Multiplicative Exponential Linear Logic (MELL) MELL Proof-Nets

Agenda

1

Multiplicative Linear Logic (MLL)

2

MLL Proof-Nets

3

Pre Proof-Nets

4

Correctness Criteria

5

Multiplicative Exponential Linear Logic (MELL)

6

MELL Proof-Nets

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MLL Formulae

Atomic Formulae: p and p .

Formulae: A :: = p | p | A O B | A ⊗ B

(Involutory) Negation:

p

^⊥

:= p p

^⊥

:= p

(A O B)

^⊥

:= A

^⊥

⊗ B

^⊥

(A ⊗ B)

^⊥

:= A

^⊥

O B

^⊥

Remark (A

^⊥

)

^⊥

= A

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MLL (Unilateral) Sequent Presentation

(Axiom)

` A

^⊥

, A

` A, Γ ` A

^⊥

, ∆ (Cut)

` Γ, ∆

` Γ, A, B, ∆ (Perm)

` Γ, B, A, ∆

` A, B, Γ (Par)

` A O B, Γ

` A, Γ ` B, ∆

(T ensor)

` A ⊗ B, Γ, ∆

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Agenda

1

Multiplicative Linear Logic (MLL)

2

MLL Proof-Nets

3

Pre Proof-Nets

4

Correctness Criteria

5

Multiplicative Exponential Linear Logic (MELL)

6

MELL Proof-Nets

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Towards Proof-Nets

How many proofs ending in ` (A

1

O A

2

), . . . , (A

2n−1

O A

2n

) if we start from a derivation of

` A

1

, . . . , A

2n

?

If n = 2 , then

` A

1

, A

2

, A

3

, A

4

(Par)

` A

1

O A

2

, A

3

, A

4

(Par)

` A

1

O A

2

, A

3

O A

4

` A

1

, A

2

, A

3

, A

4

(Par)

` A

1

, A

2

, A

3

O A

4

(Par)

` A

1

O A

2

, A

3

O A

4

In general n! : any sequent derivation must sequentialize the order of the rules to be applied, one by one.

Is there any better representation which makes abstraction of such bureaucracy?

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MLL Proof-Nets

An MLL Proof-Net (PN) with conclusions A

1

, . . . , A

n

is a graph defined by induction as follows:

For every MLL formula A , we have a PN with conclusions A, A

^⊥

having the form:

ax

A

^⊥

A

Given a PN with conclusions Γ, A, B on the left, we can construct the PN with conclusions Γ, A O B on the right.

B Γ A

O

A O B

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MLL Proof-Nets

Given a PN with conclusions Γ, A and a PN with conclusions ∆, B on the left, we can construct the PN with conclusions Γ, A ⊗ B, ∆ on the right.

Γ A B ∆

⊗

A ⊗ B

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MLL Proof-Nets

Given a PN with conclusions Γ, A and a PN with conclusions ∆, A

^⊥

on the left, we can construct the PN with conclusions Γ, ∆ on the right.

Γ A A

^⊥

∆

Γ A A

^⊥

∆

cut

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Digression

In proof-nets there is no trace of bureaucracy (the order of application of independent rules is abstracted away).

In graph-like structures there is no trace of the constructor history.

Closing the gap: how do we check that a given (graph-like) structure is in fact a

proof-net?

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Agenda

1

Multiplicative Linear Logic (MLL)

2

MLL Proof-Nets

3

Pre Proof-Nets

4

Correctness Criteria

5

Multiplicative Exponential Linear Logic (MELL)

6

MELL Proof-Nets

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Graph-Like Structures: Pre Proof-Nets

Pre Proof-Nets are generated by the following links:

(Axiom) (Cut) (Par) (T ensor)

ax

A

^⊥

A ^cut

A A

^⊥

O

A B

A O B

⊗

A B

A ⊗ B and satisfy the following conditions:

Every formula is the conclusion of exactly one link.

Every formula is the premise of at most one link.

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A Pre Proof-Net which is not a Proof-Net

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A Pre Proof-Net which is a Proof-Net

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Agenda

1

Multiplicative Linear Logic (MLL)

2

MLL Proof-Nets

3

Pre Proof-Nets

4

Correctness Criteria

5

Multiplicative Exponential Linear Logic (MELL)

6

MELL Proof-Nets

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Goal

We need some correctness criteria on graph-like structures to check if they are in

fact proof-nets

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Some References

Long-trip (Girard 87) Contractibility (Danos 90) Acyclic-Connected (Danos 90)

Graph Parsing (Guerrini, Martini, Masini 97)

. . .

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Long-Trip Criteria - Definitions

Each link is a router:

Ports: formulas

Rules: enter port(s) and exit port(s)

Axiom and Cut: one possible configuration, giving two paths

Tensor and Par: two possible configurations (L) and (R), each one giving three paths Leaves: connect the enter port with the exit port

Rules

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How to Read the Rules?

• If a path arrives to A from the bottom, then the path goes on to the top of A

^⊥

.

• If a path arrives to A

^⊥

from the bottom, then the path goes on to the top of A .

• If a path arrives to B from the top, then the path goes on to the bottom of A .

• If a path arrives to A from the top, then the path goes on to the top of A ⊗ B .

• If a path arrives to A ⊗ B from the bottom, then the path goes on to the bottom of B .

• If a path arrives to B from the top, then the path goes on to the bottom of B .

• If a path arrives to A from the top, then the path goes on to the top of A O B .

• If a path arrives to A O B from the bottom,

then the path goes on to the bottom of A .

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Long-Trip Criteria

Theorem

A Pre PN is a PN iff for every possible configuration one obtains a long trip (i.e.starting from any node there is a path visiting each node exactly once in each direction).

Exponential Complexity

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A Pre PN which is not a PN

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A Pre PN which is a PN

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Acyclic Connected Criteria (ACC) - Definitions

A Pre PN is abstracted by a paired graph S equipped with A set V of vertices

A set E of edges

A set C(S ) of pairwise disjoint pairs of co-incident (i.e.one node in common) edges.

They are marked with an arc in red.

For every Pre PN P we define a paired graph P

⁻

by using the following constructions:

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Examples

Paired Graph P1 Paired Graph P2

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Removing Edges

Definition

Let S be a paired graph. Then R(S ) is the set of graphs obtained from S by removing exactly one edge for each pair in C(S ) .

Coming back to the paired graph P2:

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Acyclic Connected Criteria (ACC)

Remark: Every Pre PN P can be seen (i.e. abstracted) as a paired graph P

⁻

.

Theorem

A Pre PN P is a PN iff every graph in R(P

⁻

) is connected and acyclic (i.e. a tree).

Still Exponential: 2 ^p , where p is the number of par links in the graph

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Contractibility - Definitions

We consider the following rewriting system with two rules:

Given a paired graph S :

The first rule can only be applied to a pairwise disjoint pair of edges of C(S ) connecting the same two nodes.

The second rule can only be applied to an edge (not in C(S) ) connecting two

different nodes.

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Examples

reduces to

Theorem

A Pre PN P is a PN iff P

⁻

is contractible (i.e. reduces to a single node).

Quadratic Time

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Agenda

1

Multiplicative Linear Logic (MLL)

2

MLL Proof-Nets

3

Pre Proof-Nets

4

Correctness Criteria

5

Multiplicative Exponential Linear Logic (MELL)

6

MELL Proof-Nets

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MELL Formulae

Atomic Formulae: p and p .

Formulae: A ::= p | p | A O B | A ⊗ B) | ?A | !A

(Involutory) Negation:

p

^⊥

:= p p

^⊥

:= p

(A O B)

^⊥

: = A

^⊥

⊗ B

^⊥

(A ⊗ B)

^⊥

: = A

^⊥

O B

^⊥

(?A)

^⊥

:= !A

^⊥

(!A)

^⊥

:= ?A

^⊥

Remark (A

^⊥

)

^⊥

= A

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MELL (Unilateral) Sequent Presentation

(Axiom)

` A

^⊥

, A

` A, Γ ` A

^⊥

, ∆ (Cut)

` Γ, ∆

` Γ, A, B, ∆ (Perm)

` Γ, B, A, ∆

` A, B, Γ (Par)

` A O B, Γ

` A, Γ ` B, ∆

(T ensor)

` A ⊗ B, Γ, ∆

` Γ

(Weak)

` ?A, Γ

`?A, ?A, Γ (Cont)

` ?A, Γ

` A, Γ (Der)

` ?A, Γ

` A, ?Γ (Box)

` !A, ?Γ

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Agenda

1

Multiplicative Linear Logic (MLL)

2

MLL Proof-Nets

3

Pre Proof-Nets

4

Correctness Criteria

5

Multiplicative Exponential Linear Logic (MELL)

6

MELL Proof-Nets

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MELL Pre Proof-Nets

(Axiom) (Cut) (Weakening) (Contraction)

ax

A

^⊥

A cut

A A

^⊥

w

?A

c

?A ?A

?A

(Par) (T ensor) (Dereliction) (Box)

O

A B

A O B

⊗

A B

A ⊗ B

d A

?A

! A

!A ?B

1

?B

n

· · ·

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MELL Proof-Nets

A MELL Proof-Net (PN) with conclusions A

1

, . . . , A

n

is defined by induction as follows:

Every MLL PN is a MELL PN.

Given a PN with conclusions Γ on the left, we can construct the PN with conclusions Γ, ?A on the right.

Γ Γ

w

?A

Given a PN with conclusions Γ, ?A, ?A on the left, we can construct the PN with conclusions Γ, ?A on the right.

Γ ?A ?A Γ ?A ?A

c

?A

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MELL Proof-Nets

Given a PN with conclusions Γ, A on the left, we can construct the PN with conclusions Γ, ?A on the right.

Γ

A Γ

A d

?A

Given a PN with conclusions ?Γ, A , we can construct the following PN with conclusions ? Γ, !A

A ?Γ

!

!A ?Γ

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Example

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Transforming MELL Proof-Nets

Some intuitions:

Structural transformations:

Equivalences Additional Rules

Cut Elimination transformations:

Multiplicative

Exponential

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Equivalence Relation for MELL Proof-Nets

` Γ, ?A

¹

, ?A

²

, ?A

³

` Γ, ?A

^1,2

, ?A

³

` Γ, ?A

^1,2,3

` Γ, ?A

¹

, ?A

²

, ?A

³

` Γ, ?A

¹

, ?A

^2,3

` Γ, ?A

^1,2,3

· · ·

?A ?A c ?A

c

?A

≡

_A(c)

· · ·

?A

?A ?A c c

?A

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Equivalence Relation for MELL Proof-Nets

` ?Γ, ?B, ?B, A

` ?Γ, ?B, A

`? Γ, ?B, !A

` ?Γ, ?B, ?B, A

` ?Γ, ?B, ?B, !A

`? Γ, ?B, !A

A

!

!A

?B ?B c

?B · · ·

≡

_IO(c)

A

!

!A

?B ?B c

?B · · ·

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Additional Rewriting Rules for MELL Proof-Nets

` Γ, ?A

` Γ, ?A, ?A

` Γ, ?A

w

?A ?A

c

?A · · ·

7→

_w

_-

_c

?A · · ·

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Additional Rewriting Rules for MELL Proof-Nets

`? Γ, A

`?Γ, A, ?B

`?Γ, !A, ?B

`? Γ, A

`?Γ, !A

`?Γ, !A, ?B

A

!

!A · · ·

w

?B

7→

_w

_-

_b

A

!

!A · · ·

w

?B

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Multiplicative Cut Elimination Rules for MELL Proof-Nets

` Γ, A ` A

^⊥

, A

` Γ, A

Γ A A

^⊥

A

cut ax

7→

_C(a)

Γ A

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Multiplicative Cut Elimination Rules for MELL Proof-Nets

` Γ, A, B

` Γ, A O B

` ∆, A

^⊥

` Π, B

^⊥

` ∆, A

^⊥

⊗ B

^⊥

, Π

` Γ, ∆, Π

` Γ, A, B ` Π, B

^⊥

` Γ, Π, A ` ∆, A

^⊥

` Γ, ∆, Π

B A

O

AOB

Γ ∆ A^⊥ B^⊥ Π

⊗

A^⊥⊗B^⊥ cut

7→

_C(O,⊗)

B

Γ A B^⊥ Π ∆ A^⊥

cut

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Exponential Cut Elimination Rules for MELL Proof-Nets

` ∆, A

` ∆, ?A

`?Γ, A

^⊥

`?Γ, !A

^⊥

` ∆, ?Γ

` ∆

=======

` ∆, ?Γ

A

^⊥

!

!A

^⊥

?Γ

w

?A

cut 7→

_C(w,b)

w

?Γ

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Exponential Cut Elimination Rules for MELL Proof-Nets

` ∆, A

` ∆, ?A

`? Γ, A

^⊥

`?Γ, !A

^⊥

` ∆, ?Γ

` ∆, A `?Γ, A

^⊥

` ∆, ?Γ

! A

^⊥

!A

^⊥

?Γ

d

?A cut A

∆

7→

C(d,b)

A

^⊥

?Γ A

∆

cut

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Exponential Cut Elimination Rules for MELL Proof-Nets

` ∆, ?A, ?A

` ∆, ?A

`?Γ, A

^⊥

` ?Γ, !A

^⊥

` ∆, ? Γ

` ∆, ?A, ?A

`?Γ, A

^⊥

`?Γ, !A

^⊥

` ∆, ?Γ, ?A

`?Γ, A

^⊥

` ?Γ, !A

^⊥

` ∆, ? Γ, ? Γ

===========================================

` ∆, ? Γ

! A^⊥

!A^⊥ ?Γ

c

?A cut

?A

∆

7→

_C(c,b)

! A^⊥

!A^⊥ ?Γ

cut

?A

∆

! A^⊥

!A^⊥ ?Γ

cut

c

?Γ

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Exponential Cut Elimination Rules for MELL Proof-Nets

`? Γ, B

^⊥

, ?A

`? Γ, !B

^⊥

, ?A

`? ∆, A

^⊥

`? ∆, !A

^⊥

`?Γ, ?∆, !B

^⊥

`? Γ, B

^⊥

, ?A

`? ∆, A

^⊥

`? ∆, !A

^⊥

`?Γ, ?∆, B

^⊥

`?Γ, ?∆, !B

^⊥

! B^⊥

!B^⊥ ?Γ ?A

! A^⊥

!A^⊥ ?∆

cut

7→

_C(b,b)

B^⊥

?Γ ?A !

A^⊥

!A^⊥

?∆

cut

!

!B^⊥ ?Γ

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The Reduction Relation for MELL Proof-Nets

Let consider the following relations:

R : = {C(a), C( O , ⊗), C(w, b), C(d, b), C(c, b), C(b, b), w - b, w - c } E := {A(c), IO(c)}

The reduction relation →

R

is the closure by all Proof-Net contexts of the rewriting rules in R .

The congruence '

E

is the reflexive, symmetric, transitive, closed by proof-net contexts relation on proof-nets generated by equations E .

We shall write →

_R/E

for the reduction relation on MELL proof-nets generated by the rules in R modulo the equations in E , i.e.

p →

_R/E

p

⁰

iff ∃ p

1

, p

2

such that p '

E

p

1

→

R

p

2

'

E

p

⁰

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MELL Proof-Nets Properties

Definition

A reduction relation S is said to be confluent if and only if for every t, u, v such that t →

^∗_S

u and t →

^∗_S

v there is t

⁰

such that u →

^∗_S

t

⁰

and v →

^∗_S

t

⁰

.

Theorem (Confluence)

The reduction →

R/E

is confluent on MELL Proof-Nets.

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MELL Proof-Nets Properties

Definition

A reduction relation S is said to be terminating if and only if for every t there is no infinite →

S

-sequence starting at t (i.e. every →

S

-reduction sequence starting at any term is terminating).

A reduction relation S is said to be strongly normalizing if and only if every typed object t is terminating.