A Very Rough Introduction to Linear Logic Very Rough Introduction to Linear Logic John Wickerson, Imperial College London Multicore Group Seminar January 7, 2014 John Wickerson, Imperial

A Very Rough Introduction to Linear Logic

John Wickerson, Imperial College London

Multicore Group Seminar

January 7, 2014

John Wickerson, Imperial College London Linear Logic

Linear Logic

Introduced by Jean-Yves Girard in 1987

Classical logic: is my formula true?

Intuitionistic logic: is my formula provable?

Linear logic is a bit different


Linear Logic






Linear Logic






Linear Logic






Vending machines

Let:

D := “One dollar”

M := “A pack of Marlboros”

C := “A pack of Camels”

Some axioms for a vending machine:

D ⇒M

D ⇒ C

In classical or intuitionistic logic, we can deduce:

D ⇒ (M ∧ C)


Vending machines

Let:





D ⇒M

D ⇒ C


D ⇒ (M ∧ C)


Vending machines

Let:





D ⇒M

D ⇒ C


D ⇒ (M ∧ C)


Linear logic operators

Two conjunctions:

Multiplicative conjunction (A⊗B):I can have both A and B at the same time

Additive conjunction (A & B):I can have A or B (not both), and I choose which

Two disjunctions:

Multiplicative disjunction (A`B):[we’ll come back to this one]

Additive disjunction (A⊕B):I can have A or B (not both), but I don’t choose which

De Morgan rules:

(A⊗B)⊥ ≡ A⊥ `B⊥

(A & B)⊥ ≡ A⊥ ⊕B⊥



Two conjunctions:



Two disjunctions:



De Morgan rules:

(A⊗B)⊥ ≡ A⊥ `B⊥

(A & B)⊥ ≡ A⊥ ⊕B⊥



Two conjunctions:



Two disjunctions:



De Morgan rules:

(A⊗B)⊥ ≡ A⊥ `B⊥

(A & B)⊥ ≡ A⊥ ⊕B⊥


More vending machines

Let R be a vending machine that accepts rubles and dispensespacks of rolos, and let S be a vending machine that acceptsshillings and dispenses packs of softmints.

Let P1, P2, P3 andP4 be four purchase orders.

P1 ( (R⊗ S): If Ally hands in purchase order P1, thecollege will install both machines side-by-side.

P2 ( (R & S): If Ally hands in purchase order P2, thecollege will install one machine of Ally’s choice.

P3 ( (R⊕ S): If Ally hands in purchase order P3, thecollege will install one machine of their choice.

P4 ( (R` S): If Ally hands in purchase order P4, thecollege will install both machines side-by-side, and pre-loadone of them with a coin. If a customer puts a coin in theother slot, then one of the machines will consume its coinand dispense its confectionery!



Let R be a vending machine that accepts rubles and dispensespacks of rolos, and let S be a vending machine that acceptsshillings and dispenses packs of softmints. Let P1, P2, P3 andP4 be four purchase orders.











P4 ( (R` S): If Ally hands in purchase order P4, thecollege will install both machines side-by-side

, and pre-loadone of them with a coin. If a customer puts a coin in theother slot, then one of the machines will consume its coinand dispense its confectionery!







































P4 ( (R` S): If Ally hands in purchase order P4, thecollege will install both machines side-by-side, and pre-loadone of them with a coin.

If a customer puts a coin in theother slot, then one of the machines will consume its coinand dispense its confectionery!







P4 ( (R` S): If Ally hands in purchase order P4, thecollege will install both machines side-by-side, and pre-loadone of them with a coin. If a customer puts a coin in theother slot,

then one of the machines will consume its coinand dispense its confectionery!









Exponentials

!A: Produces an unlimited number of A’s

?A: Consumes an unlimited number of A’s

Connected via linear negation: (?A)⊥ = !(A⊥)


Jeroen’s Burger Shack

Set menu – $5 per person

HamburgerFries or WedgesUnlimited Pepsi

Ice-cream or sorbet (subject to availability)

D ⊗D ⊗D ⊗D ⊗D(

H ⊗ (F & W )⊗ !P ⊗ (I ⊕ S)


Jeroen’s Burger Shack

Set menu – $5 per person

HamburgerFries or WedgesUnlimited Pepsi

Ice-cream or sorbet (subject to availability)

D ⊗D ⊗D ⊗D ⊗D(

H ⊗ (F & W )⊗ !P ⊗ (I ⊕ S)


Gentzen’s LK

The sequentA1, . . . , Am ` B1, . . . , Bn

means(A1 ∧ · · · ∧Am)⇒ (B1 ∨ · · · ∨Bn)


Gentzen’s LK vs. Linear Logic

A ` AAxiom

Γ,

!

A,

!

A ` ∆

Γ,

!

A ` ∆ContL

Γ `

?

A,

?

A,∆

Γ `

?

A,∆ContR

Γ ` ∆, A A,Γ′ ` ∆′

Γ,Γ′ ` ∆,∆′ CutΓ ` ∆

Γ,

!

A ` ∆WL

Γ ` ∆

Γ `

?

A,∆WR

Γ, A ` ∆ Γ, B ` ∆

Γ, A ∨B ` ∆∨L

Γ ` A,∆

Γ ` A ∨B,∆∨R1

Γ ` B,∆

Γ ` A ∨B,∆∨R2

Γ, A ` ∆

Γ, A ∧B ` ∆∧L1

Γ, B ` ∆

Γ, A ∧B ` ∆∧L2

Γ ` A,∆ Γ ` B,∆

Γ ` A ∧B,∆∧R

Γ, A ` ∆ Γ′, B ` ∆′

Γ,Γ′, A ∨B ` ∆,∆′ ∨′LΓ ` A,B,∆

Γ ` A ∨B,∆∨′R

Γ, A,B ` ∆

Γ, A ∧B ` ∆∧′L

Γ ` A,∆ Γ′ ` B,∆′

Γ,Γ′ ` A ∧B,∆,∆′ ∧′R



A ` AAxiom

Γ,

!

A,

!

A ` ∆

Γ,

!

A ` ∆ContL

Γ `

?

A,

?

A,∆

Γ `

?

A,∆ContR

Γ ` ∆, A A,Γ′ ` ∆′

Γ,Γ′ ` ∆,∆′ CutΓ ` ∆

Γ,

!

A ` ∆WL

Γ ` ∆

Γ `

?

A,∆WR

Γ, A ` ∆ Γ, B ` ∆

Γ, A ∨B ` ∆∨L

Γ ` A,∆

Γ ` A ∨B,∆∨R1

Γ ` B,∆

Γ ` A ∨B,∆∨R2

Γ, A ` ∆

Γ, A ∧B ` ∆∧L1

Γ, B ` ∆

Γ, A ∧B ` ∆∧L2

Γ ` A,∆ Γ ` B,∆

Γ ` A ∧B,∆∧R

Γ, A ` ∆ Γ′, B ` ∆′

Γ,Γ′, A ∨B ` ∆,∆′ ∨′LΓ ` A,B,∆

Γ ` A ∨B,∆∨′R

Γ, A,B ` ∆

Γ, A ∧B ` ∆∧′L

Γ ` A,∆ Γ′ ` B,∆′

Γ,Γ′ ` A ∧B,∆,∆′ ∧′R



A ` AAxiom

Γ,

!

A,

!

A ` ∆

Γ,

!

A ` ∆ContL

Γ `

?

A,

?

A,∆

Γ `

?

A,∆ContR

Γ ` ∆, A A,Γ′ ` ∆′

Γ,Γ′ ` ∆,∆′ Cut

Γ ` ∆

Γ,

!

A ` ∆WL

Γ ` ∆

Γ `

?

A,∆WR

Γ, A ` ∆ Γ, B ` ∆

Γ, A ∨B ` ∆∨L

Γ ` A,∆

Γ ` A ∨B,∆∨R1

Γ ` B,∆

Γ ` A ∨B,∆∨R2

Γ, A ` ∆

Γ, A ∧B ` ∆∧L1

Γ, B ` ∆

Γ, A ∧B ` ∆∧L2

Γ ` A,∆ Γ ` B,∆

Γ ` A ∧B,∆∧R

Γ, A ` ∆ Γ′, B ` ∆′

Γ,Γ′, A ∨B ` ∆,∆′ ∨′LΓ ` A,B,∆

Γ ` A ∨B,∆∨′R

Γ, A,B ` ∆

Γ, A ∧B ` ∆∧′L

Γ ` A,∆ Γ′ ` B,∆′

Γ,Γ′ ` A ∧B,∆,∆′ ∧′R



A ` AAxiom

Γ,

!

A,

!

A ` ∆

Γ,

!

A ` ∆ContL

Γ `

?

A,

?

A,∆

Γ `

?

A,∆ContR

Γ ` ∆, A A,Γ′ ` ∆′

Γ,Γ′ ` ∆,∆′ Cut

Γ ` ∆

Γ,

!

A ` ∆WL

Γ ` ∆

Γ `

?

A,∆WR

Γ, A ` ∆ Γ, B ` ∆

Γ, A⊕B ` ∆⊕L

Γ ` A,∆

Γ ` A⊕B,∆⊕R1

Γ ` B,∆

Γ ` A⊕B,∆⊕R2

Γ, A ` ∆

Γ, A & B ` ∆&L1

Γ, B ` ∆

Γ, A & B ` ∆&L2

Γ ` A,∆ Γ ` B,∆

Γ ` A & B,∆&R

Γ, A ` ∆ Γ′, B ` ∆′

Γ,Γ′, A`B ` ∆,∆′ `LΓ ` A,B,∆

Γ ` A`B,∆`R

Γ, A,B ` ∆

Γ, A⊗B ` ∆⊗L

Γ ` A,∆ Γ′ ` B,∆′

Γ,Γ′ ` A⊗B,∆,∆′ ⊗R



A ` AAxiom

Γ, !A, !A ` ∆

Γ, !A ` ∆ContL

Γ ` ?A, ?A,∆

Γ ` ?A,∆ContR

Γ ` ∆, A A,Γ′ ` ∆′

Γ,Γ′ ` ∆,∆′ CutΓ ` ∆

Γ, !A ` ∆WL

Γ ` ∆

Γ ` ?A,∆WR

Γ, A ` ∆ Γ, B ` ∆

Γ, A⊕B ` ∆⊕L

Γ ` A,∆

Γ ` A⊕B,∆⊕R1

Γ ` B,∆

Γ ` A⊕B,∆⊕R2

Γ, A ` ∆

Γ, A & B ` ∆&L1

Γ, B ` ∆

Γ, A & B ` ∆&L2

Γ ` A,∆ Γ ` B,∆

Γ ` A & B,∆&R

Γ, A ` ∆ Γ′, B ` ∆′

Γ,Γ′, A`B ` ∆,∆′ `LΓ ` A,B,∆

Γ ` A`B,∆`R

Γ, A,B ` ∆

Γ, A⊗B ` ∆⊗L

Γ ` A,∆ Γ′ ` B,∆′

Γ,Γ′ ` A⊗B,∆,∆′ ⊗R


Application: Petri nets

A B

C D

!((A⊗ C) ( B)⊗!((B ⊗D ⊗D) ( (C ⊗D))⊗A⊗ C ⊗D ⊗D

( (C ⊗D)

Other applications include: finding optimisations inHaskell, logic programming, quantum physics, linguistics,. . .



A B

C D

!((A⊗ C) ( B)⊗!((B ⊗D ⊗D) ( (C ⊗D))⊗A⊗ C ⊗D ⊗D

( (C ⊗D)




A B

C D

!((A⊗ C) ( B)⊗!((B ⊗D ⊗D) ( (C ⊗D))⊗A⊗ C ⊗D ⊗D

( (C ⊗D)




A B

C D

!((A⊗ C) ( B)⊗!((B ⊗D ⊗D) ( (C ⊗D))⊗A⊗ C ⊗D ⊗D

( (C ⊗D)




A B

C D

!((A⊗ C) ( B)⊗!((B ⊗D ⊗D) ( (C ⊗D))⊗A⊗ C ⊗D ⊗D

( (C ⊗D)




A B

C D

!((A⊗ C) ( B)⊗!((B ⊗D ⊗D) ( (C ⊗D))⊗A⊗ C ⊗D ⊗D

( (C ⊗D)



Connection to separation logic

Linear logic models the number of times a resource is used.

To reason about computer memory, we want to control howresources are shared.

The logic of bunched implications (BI) does this.(Separation logic is based on BI.)


The idea of BI

Sequents in classical/linear logic have this form:

Γ ` Γ

where Γ ::= A | Γ,Γ

Sequents in intuitionistic logic have this form:

Γ ` A

Sequents in BI have this form:

B ` A

where B ::= A | B,B | B;B


The idea of BI


Γ ` Γ

where Γ ::= A | Γ,ΓSequents in intuitionistic logic have this form:

Γ ` A


B ` A



The idea of BI


Γ ` Γ

where Γ ::= A | Γ,ΓSequents in intuitionistic logic have this form:

Γ ` A


B ` A



Some proof rules for BI

Weakening and Contraction are allowed for ‘;’ but not for ‘,’

B ` A

B;B′ ` AW

B;B ` A

B ` ACont

The ‘;’ corresponds to ‘∧’ (ordinary conjunction)

B ` A B′ ` A′

B;B′ ` A ∧A′ ∧I

The ‘,’ corresponds to ‘∗’ (separating conjunction)

B ` A B′ ` A′

B,B′ ` A ∗A′ ∗I




B ` A

B;B′ ` AW

B;B ` A

B ` ACont


B ` A B′ ` A′

B;B′ ` A ∧A′ ∧I


B ` A B′ ` A′

B,B′ ` A ∗A′ ∗I




B ` A

B;B′ ` AW

B;B ` A

B ` ACont


B ` A B′ ` A′

B;B′ ` A ∧A′ ∧I


B ` A B′ ` A′

B,B′ ` A ∗A′ ∗I


Conclusion

Linear logic . . .

is ‘a logic behind logics’

models production and consumption of resources (but notsharing)

has two types of conjunction, and two types of disjunction

has exponentials, which relax the strict rules on resourceusage

is obtained by removing Weakening and Contractionfrom classical logic

has various applications, e.g. modelling Petri nets


Conclusion

Linear logic . . .








Conclusion

Linear logic . . .








Conclusion

Linear logic . . .








Conclusion

Linear logic . . .








Conclusion

Linear logic . . .








Bibliography

[1] J.-Y. Girard.Linear logic.Theoretical Computer Science, 50:1–102, 1987.

[2] P. Lincoln.Linear logic.SIGACT, 1992.http://tinyurl.com/q7fewbb.

[3] P. W. O’Hearn and D. J. Pym.The logic of bunched implications.Bulletin of Symbolic Logic, 5(2):215–244, 1999.

[4] A. Saurin.Introduction to proof theory.http://tinyurl.com/nka48ul, 2004.


Documents

A Very Rough Introduction to Linear Logic Very Rough Introduction to Linear Logic John Wickerson, Imperial College London Multicore Group Seminar January 7, 2014 John Wickerson, Imperial