Compressing Polarized Boxes Beniamino Accattoli Carnegie Mellon University

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Compressing Polarized Boxes

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linear logic and boxes

Proof nets: the graphical syntax for linear logic. Brought new deep perspectives about normalization: 1

Optimal reductions;

2

Implicit computational complexity;

3

Explicit substitutions;

4

Strong normalization.

Key tool: boxes for the promotion rule, the heart of the system. This work: a new understanding of boxes, via polarity.

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Compressing Polarized Boxes

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Multiplicative Linear Logic (MLL)

Identity rules:

Multiplicative rules:

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` A⊥ , A

ax

` Γ, A ` ∆, B ⊗ ` Γ, ∆, A ⊗ B

Compressing Polarized Boxes

` Γ, A

` A⊥ , ∆ cut ` Γ, ∆

` Γ, A, B ` ` Γ, A ` B

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Proof nets for MLL

` A⊥ , A

ax

ax A⊥

π :

σ :

` Γ, A

` ∆, A⊥ ` Γ, ∆

π? cut

σ? A cut A⊥

Γ

∆

π?

π :

A

Γ

` Γ, A, B

`

` Γ, A ` B

`

B

A`B

π :

σ :

` Γ, A

` ∆, B

` Γ, ∆, A ⊗ B

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A

π? Γ ⊗

A

B

⊗

σ? ∆

A⊗B

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Cut-elimination for MLL

ax

A A⊥ cut

A

A⊥ B ⊥

A

`

B

⊗

A

→ax

→`

cut

P⊥

Q⊥ P cut

Q

cut

No duplication/erasure of subnets ⇒ Everything works fine

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Multiplicative Exponential Linear Logic (MELL)

MLL + Exponential rules:

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` Γ, A d ` Γ, ?A

`?Γ, A ! `?Γ, !A

` Γ, ?A, ?A c ` Γ, ?A

`Γ w ` Γ, ?A

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Exponential Cut-elimination Consider the following cut with contraction: π

ρ

:

:

` ?A⊥ , ?A⊥ , Γ

`?∆, A c ! `?∆, !A ` ?A⊥ , Γ cut `?∆, Γ

Its elimination requires to duplicate ρ: ρ ρ :

`?∆, A ! `?∆, !A

:

`?∆, A ! `?∆, !A

π : ⊥

` ?A , ?A⊥ , Γ

`?∆, ?A⊥ , Γ `?∆, ?∆, Γ c .. . c `?∆, Γ

cut

cut

Similarly, weakening induces erasure of sub-proofs. B. Accattoli (CMU)

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Na¨ıve proof nets for MELL π :

`Γ w ` Γ, ?A

π?

w

Γ

π

?A

π?

:

A

` Γ, A d ` Γ, ?A

d ?A

π

π?

:

` Γ, ?A, ?A c ` Γ, ?A

Γ

?A ?A

π

π?

:

` ?Γ, A ! ` ?Γ, !A

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?A

c

?Γ

Compressing Polarized Boxes

A

! !A

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How to eliminate cuts? Na¨ıve translation of promotion: π

π?

:

` ?Γ, A ! ` ?Γ, !A

A

!

?Γ

!A

Given this cut in a generic net: ?A⊥ ?A⊥

A

c

!

?A⊥

cut

!A

There is no way of recovering the sub-proof to duplicate. Then !-rules are represented as boxes: π π?

:

` ?Γ, A ! ` ?Γ, !A B. Accattoli (CMU)

A

! ?Γ

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!A 9 / 39

Exponential cut elimination implemented using boxes w ?A

⊥

A⊥

P

d

cut!A ?A⊥

?A⊥

cut

A

?∆

A⊥

A

P

cut Γ

!A

!

... ?B1 ?Bk

!B cut

! ?Γ

→c

! ... ?A⊥ ?A⊥ !A cut cut

! !A . . .

c

c

... ?B1 ?Bk

→

P ! A

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w

Γ

P !

...

?B1 . . . ?Bk

→d

!

?A⊥ ?A⊥

c

w

→w

! ... !A cut ?B ?Bk 1

?∆

Compressing Polarized Boxes

!B cut

!

?Γ 10 / 39

Boxes

Boxes solve the problem of defining cut-elimination. However, the solution is drastic, equivalent to give up. Some fragments seem to have an inherent notion of box. Where does the problem lie? Is there a logic feature that internalizes boxes?

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Last rule 1 Main problem: in proof nets there is no last rule. Re-consider: π :

ρ : `?∆, !A

!

` ?A⊥ , Γ `?∆, Γ

c

→

`?∆, !A

cut

π :

`?∆, A !

`?∆, !A

`?∆, A

` ?A⊥ , ?A⊥ , Γ

`?∆, A

ρ :

ρ : !

` ?A⊥ , ?A⊥ , Γ

`?∆, ?A⊥ , Γ `?∆, ?∆, Γ . . . `?∆, Γ

cut

cut

c c

In sequent calculus: rule occurrence r 7→ sub-proof ending on r . No such thing in proof nets! B. Accattoli (CMU)

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Last rule 2

Intuition: Internalizing a notion of last rule will internalize boxes Partially internalized boxes: Olivier Laurent’s polarized MELL. Abstract last rule = last positive rule. This work: totally internalized boxes for MELLP. Expressiveness: MELLP codes classical logic/λµ-calculus.

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Compressing Polarized Boxes

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Outline

1

Polarized MELL

2

Compressing polarized boxes

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Polarization Formulas: P, Q ::= X N, M ::= X ⊥

| 1 | P ⊗Q | ⊥ | N `M

| !N | ?P

Sequents: `Γ;P

or

`Γ;

Multiplicative rules: ` Γ; P

` ∆, P ⊥ ; [Q] cut ` Γ, ∆; [Q] ` Γ; [P] ⊥ ` Γ, ⊥; [P]

` Γ, N, M; [P] ` ` Γ, N ` M; [P] B. Accattoli (CMU)

ax

` P ⊥; P

`; 1

1

` Γ; P ` ∆; Q ⊗ ` Γ, ∆; P ⊗ Q

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Laurent’s MELLP: adding exponentials Exponential rules: ` Γ; [P] w ` Γ, N; [P]

` Γ; P ` Γ, ?P;

d

` Γ, N, N; [P] c ` Γ, N; [P]

` Γ, N; ` Γ; !N

!

Difference with linear logic: Promotion, contraction, and weakening do not need the ? modality. Important: Only positives are duplicated/erased. Positives are last rules and every positive will have a box. B. Accattoli (CMU)

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`; 1

1

1

` P ⊥; P

1 ax

ax

π?

` Γ; P ` ∆; Q ⊗ ` Γ, ∆; P ⊗ Q

P⊥

P

P

Q

Γ

θ? ⊗

∆

P⊗Q

` Γ; P

` ∆, P ⊥ ; [Q] ` Γ, ∆; [Q]

π? cut

θ? ⊥ cut P

P Γ

∆

[Q]

π? ` Γ, N, M; [P] ` Γ, N ` M; [P]

`

N Γ

`

M [P]

N`M

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w

` Γ, N; [P]

` Γ; P ` Γ, ?P;

π?

w

` Γ; [P]

N

Γ

[P]

π? d

P

d

Γ

?P

π? ` Γ, N, N; [P] ` Γ, N; [P]

c

N Γ

N

c

[P]

N

` Γ, N; ` Γ; !N

π? !

! Γ

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N

Compressing Polarized Boxes

!N

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Positive Trees ` Γ; P

` ∆, P ⊥ ; [Q] cut ` Γ, ∆; [Q] ` Γ; [P] ⊥ ` Γ, ⊥; [P]

` Γ, N, M; [P] ` ` Γ, N ` M; [P]

ax

` P ⊥; P

`; 1

1

` Γ; P ` ∆; Q ⊗ ` Γ, ∆; P ⊗ Q

` Γ; [P] w ` Γ, N; [P]

` Γ; P ` Γ, ?P;

d

` Γ, N, N; [P] c ` Γ, N; [P]

` Γ, N; ` Γ; !N

!

Note: positives have a forest structure. B. Accattoli (CMU)

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Positive Tree Positive connectives: 1, ⊗, !. Explicit boxes for ! ⇒ induced box for every positive: H

H

→

! N1

!

N ... k

N1

Q

P

⊗ P⊗Q

P⊥

→

ax P

P⊥

N ... k

Q

P

→

ax P

1 ⊗

→

1

P⊗Q

My contribution: explicit boxes for ! can be made implicit.

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Generalized rewriting rule

Laurent uses the positive tree to generalize box rules: T (+) w

+

→w

...

cut N1

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w

...

w

N1

...

Nk

Nk

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Polarized cut-elimination 1

ax P⊥ cut

P

ax

N

N⊥ cut

N

P⊥ Q

⊥

`

P

→ax+

Q

P

⊗

P

→ax−

→`

N

P⊥

Q⊥ P cut

Q

cut

cut

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Compressing Polarized Boxes

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Polarized cut-elimination 2 N⊥

N⊥

E d

!N cut

w P⊥

cut

!

P

→d

... N1

+

w

T (+)

→w

...

c P

P cut

w

N1 . . . Nk

P⊥ P⊥

+

...

T (+)

→c

... N1

P cut

T (+) ...

+

+

Q

... M1 Mh

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cut

P

+

T (+) ... N1

Nk

→

E !

cut ... M1 Mh

Compressing Polarized Boxes

c

c

Nk

E

T (+) ...

P cut

N1

!

Nk

Nk

P⊥ P⊥

⊥

... N1

Nk

N1

E

N cut

P

...

+

Nk

T (+) ... N1

Nk

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Outline

1

Polarized MELL

2

Compressing polarized boxes

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Matching property ` Γ; P

` ∆, P ⊥ ; [Q] cut ` Γ, ∆; [Q] ` Γ; [P] ⊥ ` Γ, ⊥; [P]

` Γ, N, M; [P] ` ` Γ, N ` M; [P]

ax

` P ⊥; P

`; 1

1

` Γ; P ` ∆; Q ⊗ ` Γ, ∆; P ⊗ Q

` Γ; [P] w ` Γ, N; [P]

` Γ; P ` Γ, ?P;

d

` Γ, N, N; [P] c ` Γ, N; [P]

` Γ, N; ` Γ; !N

!

Matching property: every !-rule is enabled by a d-rule. B. Accattoli (CMU)

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Materializing the matching property 1 Consider: w !

1 d

w !

w

1 d

w

Problem: without box the content is disconnected. Idea: let’s materialize the matching property with an additional edge. w !

i

1 d

w

The content and the positive sub-graphs are now connected. The induced box: the positive tree plus the negative trees on it.

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Quotient and weakenings Let’s do it again: w !

1 d

w !

w

1 d

w

We do not recover the original box: w !

i

1 d

w

w !

1 d

w

Interpretation: we are quotienting proof nets with explicit boxes. Remark: weakenings are not attached! ⇒ improvement over Francois Lamarche’s essential nets. B. Accattoli (CMU)

Compressing Polarized Boxes

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Box borders

Let’s do it again: w !

1 d

w

w !

1 d

i

1

1

d

d

!

!

w

i

Remark: we are not attaching the border of the box. ⇒ improvement over Ian Mackie’s interaction nets technique.

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Compressing Polarized Boxes

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Implicit boxes Recipe: Take a cut-free proof net. Matching: every !-box has a unique dereliction at level 0. Remove the explicit box and add the matching edge. Then: The induced boxes define a net with explicit boxes. Induced boxes are locally reconstructable. There is a simple correctness criterion (i.e. not ad-hoc). It is a canonical representation (i.e. no choice). B. Accattoli (CMU)

Compressing Polarized Boxes

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Summing up

In a cut-free proof net the explicit box of a ! can be replaced by a single edge in a canonical and more parallel way.

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Cuts Cuts introduce a problem: ax

ax cut

!

d

The positive sub-graph is no longer connected. Let’s iterate the same idea: ax

!

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ax cut

i i

d

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Example ax ax

w w c N M ` ! d N`M ?!P

P

⊗ d

P

?(P ⊗ P)

⊥

w !

w cut

w

1

c ! d

?1

cut

ax ax

w w c N M ` ! d N`M

i

⊗ d i

w !

w cut

w

P

⊗ d

P

?(P ⊗ P)

?!P ⊥

⊥

ax ax

w w `

1 d

i

1 d

w w

1

c ! d

1 d

1

?1

cut

w cut ⊥

ax ax

w w `

!

i

⊗ d i

w !

w

i

1 d

cut

w

1 cut

cut

Implicit box: one dereliction plus the cuts at level 0. Induced box: positive tree plus negative sub-trees. Novelty: ` commutes with box borders! B. Accattoli (CMU)

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Cut elimination ax

X⊥ cut

X

!

P⊥ Q

X

Q

P

`

→ax−

⊗ !

→`

i !0

→ax+ !

X⊥

N

N⊥

d

!

c P⊥

!

P cut

P⊥ P⊥

+

+

i

P

cut

!

i

P

+

N1

N1

w →w Nk

!0

i i

!00

+ c

i

(+) ...

N cut

(+) ...

P

i

⊥

→d

i

(+) ... cut

!

Nk

w

B. Accattoli (CMU)

P cut

→c

... N1

i

cut

(+)

cut

i

!00

P⊥ P⊥

Q

i

i

X⊥

i

cut

!

ax X cut

Q⊥ P

P⊥

cut

!

i

N⊥

X⊥

!

⊥

X

...

c ...

Nk

w

N1 . . . Nk

!

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Side effects 1 Cut elimination has ’side effects’. Consider the weakening rule: w P

⊥

cut

!

P

+

w

(+) ... N1

→w Nk

...

w

N1 . . . Nk

!

i

It automatically pushes the created weakenings out of boxes: w !

+ cut i

(+)

...

... N1

→w

!

Nk

...

w

w

... N1 Nk

Similarly for contraction.

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No commutative cuts

There is no commutative rule. It is included inside the axiom and dereliction rules. The axiom rule: ax

X⊥ cut

X

!

X

X

→ax− !

i

and its action through box borders: P

ax

... P cut

!

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+

(+) ... N1

Nk

→ax−

i

Compressing Polarized Boxes

+ P

(+)

N1

... Nk

!

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No commutative cuts 2 The dereliction rule: N⊥ i !0

N

d

!

N⊥ i

→d

cut

!00

N cut

!0

i

i i

!00

and an example of its action:

i !0

d0

... cut

!00

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w !

i

1 d

w cut

→d

i i !0

1 d

...

!00

i

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Cut-elimination On η-expanded nets: Cut elimination is strongly normalizing (SN). Confluence requires assoc., comm., and neutrality for contractions. Then: Church-Rosser modulo and SN modulo hold. The general case: More and wilder ’side effects’. Difficult critical pairs and known techniques do not work. By-product: new simpler proof of SN for linear logic (RTA 2013).

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Conclusions An alternative representation of boxes: Simple; Canonical; More parallel; Provided of a correctness criterion; A local reconstruction of boxes; Results on the dynamics.

New perspective on polarity. It already lead to new understanding of SN for linear logic.

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THANKS!

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