A Very Rough Introduction to Linear Logic

Transcription

1 A Very Rough Introduction to Multicore Group Seminar January 7, 2014

2 Introduced by Jean-Yves Girard in 1987

3 Introduced by Jean-Yves Girard in 1987 Classical logic: is my formula true?

4 Introduced by Jean-Yves Girard in 1987 Classical logic: is my formula true? Intuitionistic logic: is my formula provable?

5 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

6 Vending machines Let: D := One dollar M := A pack of Marlboros C := A pack of Camels

7 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

8 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)

9 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

10 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

11 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

12 More vending machines Let R be a vending machine that accepts rubles and dispenses packs of rolos, and let S be a vending machine that accepts shillings and dispenses packs of softmints.

13 More vending machines Let R be a vending machine that accepts rubles and dispenses packs of rolos, and let S be a vending machine that accepts shillings and dispenses packs of softmints. Let P 1, P 2, P 3 and P 4 be four purchase orders.

14 More vending machines Let R be a vending machine that accepts rubles and dispenses packs of rolos, and let S be a vending machine that accepts shillings and dispenses packs of softmints. Let P 1, P 2, P 3 and P 4 be four purchase orders. P 1 (R S): If Ally hands in purchase order P 1, the college will install both machines side-by-side.

15 More vending machines Let R be a vending machine that accepts rubles and dispenses packs of rolos, and let S be a vending machine that accepts shillings and dispenses packs of softmints. Let P 1, P 2, P 3 and P 4 be four purchase orders. P 1 (R S): If Ally hands in purchase order P 1, the college will install both machines side-by-side. P 2 (R & S): If Ally hands in purchase order P 2, the college will install one machine of Ally s choice.

16 More vending machines Let R be a vending machine that accepts rubles and dispenses packs of rolos, and let S be a vending machine that accepts shillings and dispenses packs of softmints. Let P 1, P 2, P 3 and P 4 be four purchase orders. P 1 (R S): If Ally hands in purchase order P 1, the college will install both machines side-by-side. P 2 (R & S): If Ally hands in purchase order P 2, the college will install one machine of Ally s choice. P 3 (R S): If Ally hands in purchase order P 3, the college will install one machine of their choice.

17 More vending machines Let R be a vending machine that accepts rubles and dispenses packs of rolos, and let S be a vending machine that accepts shillings and dispenses packs of softmints. Let P 1, P 2, P 3 and P 4 be four purchase orders. P 1 (R S): If Ally hands in purchase order P 1, the college will install both machines side-by-side. P 2 (R & S): If Ally hands in purchase order P 2, the college will install one machine of Ally s choice. P 3 (R S): If Ally hands in purchase order P 3, the college will install one machine of their choice. P 4 (R ` S): If Ally hands in purchase order P 4, the college will install both machines side-by-side

18 More vending machines Let R be a vending machine that accepts rubles and dispenses packs of rolos, and let S be a vending machine that accepts shillings and dispenses packs of softmints. Let P 1, P 2, P 3 and P 4 be four purchase orders. P 1 (R S): If Ally hands in purchase order P 1, the college will install both machines side-by-side. P 2 (R & S): If Ally hands in purchase order P 2, the college will install one machine of Ally s choice. P 3 (R S): If Ally hands in purchase order P 3, the college will install one machine of their choice. P 4 (R ` S): If Ally hands in purchase order P 4, the college will install both machines side-by-side

19 More vending machines Let R be a vending machine that accepts rubles and dispenses packs of rolos, and let S be a vending machine that accepts shillings and dispenses packs of softmints. Let P 1, P 2, P 3 and P 4 be four purchase orders. P 1 (R S): If Ally hands in purchase order P 1, the college will install both machines side-by-side. P 2 (R & S): If Ally hands in purchase order P 2, the college will install one machine of Ally s choice. P 3 (R S): If Ally hands in purchase order P 3, the college will install one machine of their choice. P 4 (R ` S): If Ally hands in purchase order P 4, the college will install both machines side-by-side, and pre-load one of them with a coin.

20 More vending machines Let R be a vending machine that accepts rubles and dispenses packs of rolos, and let S be a vending machine that accepts shillings and dispenses packs of softmints. Let P 1, P 2, P 3 and P 4 be four purchase orders. P 1 (R S): If Ally hands in purchase order P 1, the college will install both machines side-by-side. P 2 (R & S): If Ally hands in purchase order P 2, the college will install one machine of Ally s choice. P 3 (R S): If Ally hands in purchase order P 3, the college will install one machine of their choice. P 4 (R ` S): If Ally hands in purchase order P 4, the college will install both machines side-by-side, and pre-load one of them with a coin. If a customer puts a coin in the other slot,

21 More vending machines Let R be a vending machine that accepts rubles and dispenses packs of rolos, and let S be a vending machine that accepts shillings and dispenses packs of softmints. Let P 1, P 2, P 3 and P 4 be four purchase orders. P 1 (R S): If Ally hands in purchase order P 1, the college will install both machines side-by-side. P 2 (R & S): If Ally hands in purchase order P 2, the college will install one machine of Ally s choice. P 3 (R S): If Ally hands in purchase order P 3, the college will install one machine of their choice. P 4 (R ` S): If Ally hands in purchase order P 4, the college will install both machines side-by-side, and pre-load one of them with a coin. If a customer puts a coin in the other slot, then one of the machines will consume its coin and dispense its confectionery!

22 Exponentials!A: Produces an unlimited number of A s?a: Consumes an unlimited number of A s Connected via linear negation: (?A) =!(A )

23 Jeroen s Burger Shack Set menu $5 per person Hamburger Fries or Wedges Unlimited Pepsi Ice-cream or sorbet (subject to availability)

24 Jeroen s Burger Shack Set menu $5 per person Hamburger Fries or Wedges Unlimited Pepsi Ice-cream or sorbet (subject to availability) D D D D D H (F & W )!P (I S)

25 Gentzen s LK The sequent A 1,..., A m B 1,..., B n means (A 1 A m ) (B 1 B n )

26 Gentzen s LK vs. A A Axiom Γ, A, A Γ, A Cont L Γ A, A, Γ A, Cont R Γ, A A, Γ Γ, Γ, Cut Γ, A Γ, B L Γ, A B Γ Γ, A W L Γ A, Γ A B, R1 Γ Γ A, W R Γ B, Γ A B, R2 Γ, A Γ, A B L1 Γ, B Γ, A B L2 Γ A, Γ B, R Γ A B,

27 Gentzen s LK vs. A A Axiom Γ, A, A Γ, A Cont L Γ A, A, Γ A, Cont R Γ, A A, Γ Γ, Γ, Cut Γ, A Γ, B L Γ, A B Γ Γ, A W L Γ A, Γ A B, R1 Γ Γ A, W R Γ B, Γ A B, R2 Γ, A Γ, A B L1 Γ, B Γ, A B L2 Γ A, Γ B, R Γ A B, Γ, A Γ, B Γ, Γ, A B, L Γ, A, B Γ, A B L Γ A, B, Γ A B, R Γ A, Γ B, Γ, Γ A B,, R

28 Gentzen s LK vs. A A Axiom Γ, A A, Γ Γ, Γ, Cut Γ, A Γ, B L Γ, A B 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, R Γ A B, Γ, A Γ, B Γ, Γ, A B, L Γ, A, B Γ, A B L Γ A, B, Γ A B, R Γ A, Γ B, Γ, Γ A B,, R

29 Gentzen s LK vs. A A Axiom Γ, A A, Γ Γ, Γ, Cut Γ, A Γ, B L Γ, A B 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, & R Γ A & B, Γ, A Γ, B Γ, Γ, A ` B, `L Γ, A, B Γ, A B L Γ A, B, Γ A ` B, `R Γ A, Γ B, Γ, Γ A B,, R

30 Gentzen s LK vs. A A Axiom Γ,!A,!A Γ,!A Cont L Γ?A,?A, Γ?A, Cont R Γ, A A, Γ Γ, Γ, Cut Γ, A Γ, B L Γ, A B Γ Γ,!A W L Γ A, Γ A B, R1 Γ Γ?A, W R Γ B, Γ A B, R2 Γ, A Γ, A & B & L1 Γ, B Γ, A & B & L2 Γ A, Γ B, & R Γ A & B, Γ, A Γ, B Γ, Γ, A ` B, `L Γ, A, B Γ, A B L Γ A, B, Γ A ` B, `R Γ A, Γ B, Γ, Γ A B,, R

31 Application: Petri nets A B C D

32 Application: Petri nets A B C D!((A C) B)!((B D D) (C D)) A C D D

33 Application: Petri nets A B C D!((A C) B)!((B D D) (C D)) (C D) A C D D

34 Application: Petri nets A B C D!((A C) B)!((B D D) (C D)) (C D) A C D D

35 Application: Petri nets A B C D!((A C) B)!((B D D) (C D)) (C D) A C D D

36 Application: Petri nets A B C D!((A C) B)!((B D D) (C D)) (C D) A C D D Other applications include: finding optimisations in Haskell, logic programming, quantum physics, linguistics,...

37 Connection to separation logic Linear logic models the number of times a resource is used. To reason about computer memory, we want to control how resources are shared. The logic of bunched implications (BI) does this. (Separation logic is based on BI.)

38 The idea of BI Sequents in classical/linear logic have this form: where Γ ::= A Γ, Γ Γ Γ

39 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

40 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

41 Some proof rules for BI Weakening and Contraction are allowed for ; but not for, B A B; B A W B; B A B A Cont

42 Some proof rules for BI Weakening and Contraction are allowed for ; but not for, B A B; B A W B; B A B A Cont The ; corresponds to (ordinary conjunction) B A B A B; B A A I

43 Some proof rules for BI Weakening and Contraction are allowed for ; but not for, B A B; B A W B; B A B A Cont 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

44 Conclusion Linear logic... is a logic behind logics

45 Conclusion Linear logic... is a logic behind logics models production and consumption of resources (but not sharing)

46 Conclusion Linear logic... is a logic behind logics models production and consumption of resources (but not sharing) has two types of conjunction, and two types of disjunction

47 Conclusion Linear logic... is a logic behind logics models production and consumption of resources (but not sharing) has two types of conjunction, and two types of disjunction has exponentials, which relax the strict rules on resource usage

48 Conclusion Linear logic... is a logic behind logics models production and consumption of resources (but not sharing) has two types of conjunction, and two types of disjunction has exponentials, which relax the strict rules on resource usage is obtained by removing Weakening and Contraction from classical logic

49 Conclusion Linear logic... is a logic behind logics models production and consumption of resources (but not sharing) has two types of conjunction, and two types of disjunction has exponentials, which relax the strict rules on resource usage is obtained by removing Weakening and Contraction from classical logic has various applications, e.g. modelling Petri nets

50 Bibliography [1] J.-Y. Girard. Linear logic. Theoretical Computer Science, 50:1 102, [2] P. Lincoln. Linear logic. SIGACT, [3] P. W. O Hearn and D. J. Pym. The logic of bunched implications. Bulletin of Symbolic Logic, 5(2): , [4] A. Saurin. Introduction to proof theory