Attack of the Exponentials

Transcription

1 Attack of the Exponentials Damiano Mazza LIPN, CNRS Université Paris 13, France LL2016, Lyon school: 7 and 8 November 2016 D. Mazza (LIPN) Attack of the Exponentials LL / 26

2 Something is missing... Consider the following tautologies: 1. X Y X ; 2. (X Y Z) (X Y ) X Z. D. Mazza (LIPN) Attack of the Exponentials LL / 26

3 Something is missing... Consider the following tautologies: 1. X Y X ; 2. (X Y Z) (X Y ) X Z. Are they provable in multiplicative linear logic? D. Mazza (LIPN) Attack of the Exponentials LL / 26

4 Something is missing... Consider the following tautologies: 1. X Y X ; 2. (X Y Z) (X Y ) X Z. Are they provable in multiplicative linear logic? X Y X D. Mazza (LIPN) Attack of the Exponentials LL / 26

5 Something is missing... Consider the following tautologies: 1. X Y X ; 2. (X Y Z) (X Y ) X Z. Are they provable in multiplicative linear logic? X Y X D. Mazza (LIPN) Attack of the Exponentials LL / 26

6 Something is missing... Consider the following tautologies: 1. X Y X ; 2. (X Y Z) (X Y ) X Z. Are they provable in multiplicative linear logic? X Y X D. Mazza (LIPN) Attack of the Exponentials LL / 26

7 Something is missing... Consider the following tautologies: 1. X Y X ; 2. (X Y Z) (X Y ) X Z. Are they provable in multiplicative linear logic? X Y X D. Mazza (LIPN) Attack of the Exponentials LL / 26

8 Something is missing... Consider the following tautologies: 1. X Y X ; 2. (X Y Z) (X Y ) X Z. Are they provable in multiplicative linear logic? (X Y Z ) (X Y ) X Z D. Mazza (LIPN) Attack of the Exponentials LL / 26

9 Something is missing... Consider the following tautologies: 1. X Y X ; 2. (X Y Z) (X Y ) X Z. Are they provable in multiplicative linear logic? X Y Z X Y X Z D. Mazza (LIPN) Attack of the Exponentials LL / 26

10 Something is missing... Consider the following tautologies: 1. X Y X ; 2. (X Y Z) (X Y ) X Z. Are they provable in multiplicative linear logic? X Y Z X Y X Z X Y Z X Y X Z D. Mazza (LIPN) Attack of the Exponentials LL / 26

11 Something is missing... Consider the following tautologies: 1. X Y X ; 2. (X Y Z) (X Y ) X Z. Are they provable in multiplicative linear logic? X Y Z Y X X Z X Y Z X Y X Z D. Mazza (LIPN) Attack of the Exponentials LL / 26

12 Something is missing... Consider the following tautologies: 1. X Y X ; 2. (X Y Z) (X Y ) X Z. Are they provable in multiplicative linear logic? X Y Z X Y X Z X Y Z X Y X Z D. Mazza (LIPN) Attack of the Exponentials LL / 26

13 Something is missing... Consider the following tautologies: 1. X Y X ; 2. (X Y Z) (X Y ) X Z. Are they provable in multiplicative linear logic? X Y Z X Y X Z X Y Z X Y X Z D. Mazza (LIPN) Attack of the Exponentials LL / 26

14 Structural rules I Mathematical truth is cumulative and inexhaustible: ΓC weakening Γ, A C D. Mazza (LIPN) Attack of the Exponentials Γ, A, A C Γ, A C contraction LL / 26

15 Structural rules I Mathematical truth is cumulative and inexhaustible: ΓC weakening Γ, A C I Γ, A, A C Γ, A C contraction Linear logic replaces truth with the notion of resource: I no structural rules on arbitrary formulas; I in other words, no arbitrary erasing and duplication: A A D. Mazza (LIPN) 6( 1 6 ( AA Attack of the Exponentials LL / 26

16 The why not modality We introduce a unary connective (modality) why not, denoted by?. Structural rules are allowed only on formulas of the form?a:?a?a?w weakening contraction?a?a We need to explicitly declare a formula as contractible : A dereliction?a D. Mazza (LIPN) Attack of the Exponentials LL / 26

17 Revisiting classical tautologies Instead of X Y X, we prove X?Y X :?w X?Y X D. Mazza (LIPN) Attack of the Exponentials LL / 26

18 Revisiting classical tautologies Instead of (X Y Z) (X Y ) X Z, we prove (X Y Z ) (X Y )?X Z: X X X X?X?X Y Z Y Z X Y Z X Y?X Z D. Mazza (LIPN) Attack of the Exponentials LL / 26

19 The meaning of linear sequents In classical logic, Γ means one of the formulas in Γ is necessarily true. This is because Γ is Γ. In linear logic, Γ is Γ, with A B := A B. What does that mean??? A B A B A B simultaneous availability of both A and B A is needed to yield B (loosing A in the process) A is needed to yield B and B is needed to yield A So, in linear logic, A 1,..., A n means for any 1 i n, A 1,..., A i 1, A i+1,..., A n are needed to obtain A i. The meaning of why not is best understood via duality:?a A is needed an unspecified number of times!a A is available at will D. Mazza (LIPN) Attack of the Exponentials LL / 26

20 The of course modality The dual of why not is of course, or bang, denoted!. Two alternative presentations in nets: inductive: nets are defined inductively on their exponential depth: depth 0: as for MLL, with nodes,,,,,?w and ; depth n > 0: as above but also with nodes of the form!ρ...?c 1?C n!a where ρ is a net of depth < n, of conclusions?c 1,...,?C n, A. global: as depth 0 above but also with nodes! and p; each p has an associated!, and each! has an associated subnet, called box: A!!A?A p?a?c 1 p... ρ?c n A p!?c 1...?C n!a Boxes are either disjoint or included one in the other. D. Mazza (LIPN) Attack of the Exponentials LL / 26

21 Cut-elimination: dereliction ρ?c 1...?C n A p p!!a?c 1...?C n A?A ρ?c 1...?C n A A Dereliction opens a box. D. Mazza (LIPN) Attack of the Exponentials LL / 26

22 Cut-elimination: weakening ρ?c 1 p...?c n p A!?w?w?w?C 1...?C n!a?a?c 1...?C n Weakening erases a box. D. Mazza (LIPN) Attack of the Exponentials LL / 26

23 Cut-elimination: contraction?c1 p... ρ?cn A p!?a?a?c1 p ρ?c1?cn?cn A p!!a?c1... p...?c1 ρ?cn A p!?cn!a?a?a?c1...?cn!a?a...?c1?cn Contraction duplicates a box. D. Mazza (LIPN) Attack of the Exponentials LL / 26

24 Cut-elimination: commutative step ρ?c 1...?C n A p p!!a?c 1...?C n ρ?a B... p!...?a!b?c 1 p p...?c 1... ρ?c n A p!?c n!a p?c 1...?C n... ρ?a B...!!B A box may enter inside another box. D. Mazza (LIPN) Attack of the Exponentials LL / 26

25 Correctness Contractions are treated like nodes: or Boxes are collapsed to a single node (that s already the case in the inductive formulation): p p!... A proof net is a net such that: every switching (graph obtained as above) is acyclic; the contents of every box is a proof net. Preserved by -elimination: ρ correct, ρ ρ implies ρ correct. Lack of connectedness causes a little technical problem... D. Mazza (LIPN) Attack of the Exponentials LL / 26

26 Exponentials in sequent calculus Sequent calculus rules for the exponential modalities: Γ, A Γ,?A Γ Γ,?A?w Γ,?A,?A Γ,?A?Γ, A!?Γ,!A Exponential ioms ( storage laws ): functoriality:!(a B)!A!B dereliction:!a A (retrieve) digging:!a!!a (indirection) weakening:!a 1 (discard) contraction:!a!a!a (copy) Categorically:!( ) is a monoidal comonad and free coalgebras are comonoids. The exponential isomorphism:!(a & B) =!A!B (just like 2 a+b = 2 a 2 b ) Lack of connectedness in proof nets corresponds to mix: Γ 1... Γ n mix Γ 1,..., Γ n D. Mazza (LIPN) Attack of the Exponentials LL / 26

27 Recovering intuitionistic and classical logic Intuitionistic logic is actually a fragment of linear logic (at any order): A, B ::= X!A B ξ.a ξ.!a 0 A & B!A!B X A B ξ.a ξ.a A B A B Theorem (Embedding of intuitionistic logic) Γ A is provable in LJ iff!γ A is provable in the above fragment of LL. Categorically: intuitionistic logic is the Kleisli category of the comonad!( ). Other translations of intuitionistic logic in linear logic exist (CbN vs. CbV). Classical logic may also be translated: X + :=?!X ( X ) + :=?!?X (A B) + := A + B + (A B) + :=?(!A +!B + ) ( x.a) + :=?! x.a + ( x.a) + :=?!? x.!a + The principle is the generalized Gödel translation A F = (A F ) F, with F =. D. Mazza (LIPN) Attack of the Exponentials LL / 26

28 Example: the drinker s formula In any bar, there s someone such that, if he drinks, then everyone drinks: F := x(d(x) y D(y)). Proof: either everyone s drinking, or there s someone who is not (MrSober). 1st case: D(z) y D(y) is true for any z, anybody is the existential witness; 2nd case: D(MrSober) y D(y) is true, so MrSober is our witness. We used excluded middle (indeed, F is not provable intuitionistically). In LK: Dy, Dy weak Dy, Dy, ydy Dy, Dy ydy Dy, F ydy, F weak Dz, ydy, F Dz ydy, F F, F contr F In LL: F +? x(?d(x)? yd(y)) Dy Dy?w ydy?dy? ydy?w?dy? ydy?dz? ydy?dz? ydy x(?dx? ydy) x(?dx? ydy)? x(?dx? ydy)? x(?dx? ydy)? x(?dx? ydy) D. Mazza (LIPN) Attack of the Exponentials LL / 26

29 Curry-Howard: computation in linear logic The Curry-Howard correspondence logic formula proof -elimination computer science type program exeion Moral of the story: proof nets are programs! In particular: derivations of propositional NJ are simply-typed λ-terms. NJ (via LJ) is a fragment of LL = the λ-calculus embeds in proof nets. The decompostion A B =!A B appears at the level of exeion: (λx.m)n M{N/x} vs. (λx.m)n M[N/x] M{N/x}[N/x] M{N/x}. D. Mazza (LIPN) Attack of the Exponentials LL / 26

30 Example: Booleans and if... then... else Let Bool := X X X =!X!X X =?X (?X X ). Let true := X, X?X, X?w?X,?X, X?X,?X X Bool X?w?X?X Bool X?X X D. Mazza (LIPN) Attack of the Exponentials LL / 26

31 Example: Booleans and if... then... else Let Bool := X X X =!X!X X =?X (?X X ). Let false := X, X?X, X?X X?w?X,?X X Bool X?w?X?X X?X X Bool D. Mazza (LIPN) Attack of the Exponentials LL / 26

32 Example: Booleans and if... then... else Let Bool := X X X =!X!X X =?X (?X X ). Let X X X X?w?w true :=?X?X?X X false :=?X?X?X X Bool Bool D. Mazza (LIPN) Attack of the Exponentials LL / 26

33 Example: Booleans and if... then... else We have Let ρ, ρ : A and let Bool =!X (!X X ) ρ ρ if x then ρ else ρ := A!!A A!!A A!A A A x : Bool [A/X ] D. Mazza (LIPN) Attack of the Exponentials LL / 26

34 Example: Booleans and if... then... else Let us compute if (true[a/x ]) then ρ else ρ : ρ?w!! ρ D. Mazza (LIPN) Attack of the Exponentials LL / 26

35 Example: Booleans and if... then... else Let us compute if (true[a/x ]) then ρ else ρ : ρ ρ?w!! D. Mazza (LIPN) Attack of the Exponentials LL / 26

36 Example: Booleans and if... then... else Let us compute if (true[a/x ]) then ρ else ρ : ρ ρ?w!! D. Mazza (LIPN) Attack of the Exponentials LL / 26

37 Example: Booleans and if... then... else Let us compute if (true[a/x ]) then ρ else ρ : ρ! D. Mazza (LIPN) Attack of the Exponentials LL / 26

38 Example: Booleans and if... then... else Let us compute if (true[a/x ]) then ρ else ρ : ρ D. Mazza (LIPN) Attack of the Exponentials LL / 26

39 Example: Booleans and if... then... else Let us compute if (true[a/x ]) then ρ else ρ : ρ D. Mazza (LIPN) Attack of the Exponentials LL / 26

40 Example: Booleans and if... then... else Let us compute if (true[a/x ]) then ρ else ρ : ρ D. Mazza (LIPN) Attack of the Exponentials LL / 26

41 Example: Booleans and if... then... else Let us compute if (true[a/x ]) then ρ else ρ : ρ Observation: apart from type-checking the, we never used types/formulas. D. Mazza (LIPN) Attack of the Exponentials LL / 26

42 Untyped nets In fact, -elimination in nets makes sense even without formulas!! D. Mazza (LIPN) Attack of the Exponentials LL / 26

43 Untyped nets In fact, -elimination in nets makes sense even without formulas!!! D. Mazza (LIPN) Attack of the Exponentials LL / 26

44 Untyped nets In fact, -elimination in nets makes sense even without formulas!! D. Mazza (LIPN) Attack of the Exponentials LL / 26

45 Untyped nets In fact, -elimination in nets makes sense even without formulas!! D. Mazza (LIPN) Attack of the Exponentials LL / 26

46 Untyped nets In fact, -elimination in nets makes sense even without formulas!! Untyped nets are a Turing-complete model of computation. D. Mazza (LIPN) Attack of the Exponentials LL / 26

47 Untyped nets In fact, -elimination in nets makes sense even without formulas!! Untyped nets are a Turing-complete model of computation. Actually, the above net may be typed in presence of!r = R (Russel s antinomy). D. Mazza (LIPN) Attack of the Exponentials LL / 26

48 Integers Voilà the n-th (Church) numeral, denoted by n:... n {}}{... D. Mazza (LIPN) Attack of the Exponentials LL / 26

49 Integers What does the following net compute? D. Mazza (LIPN) Attack of the Exponentials LL / 26

50 Integers Let s find out!... n {}}{... D. Mazza (LIPN) Attack of the Exponentials LL / 26

51 Integers Let s find out!... n {}}{... D. Mazza (LIPN) Attack of the Exponentials LL / 26

52 Integers Let s find out!... n {}}{... D. Mazza (LIPN) Attack of the Exponentials LL / 26

53 Integers Let s find out!... n {}}{... D. Mazza (LIPN) Attack of the Exponentials LL / 26

54 Integers Let s find out!... n {}}{... D. Mazza (LIPN) Attack of the Exponentials LL / 26

55 Integers Let s find out!... n {}}{... It is the successor function! D. Mazza (LIPN) Attack of the Exponentials LL / 26

56 Heterodox exponentials Every occurrence of formula of the form?a in a proof net may be assigned a polynomial with non-negative integer coefficients: p q p?a?w p x 0 1 p + q x p Let Poly := (N[x],, x) be the monoid of polynomials under composition. Any submonoid M Poly induces a subsystem of linear logic (closed under -elimination and proving?a,!a for all A), as follows: Definition. Call an occurrence of?a in a proof net final if it is not the premise of a or p node. A proof net ρ belongs to MELL M just if, whenever?a is a final occurrence of ρ whose associated polynomial is p, we have p M. D. Mazza (LIPN) Attack of the Exponentials LL / 26

57 Heterodox exponentials LL Poly 4LL a nx n + + a 1 x or 0 no dereliction TLL + b no digging ELL PLL + b, a 1!A A!A LLL!( ) not monoidal SLL x or b!a A A D. Mazza (LIPN) Attack of the Exponentials LL / 26

58 Heterodox exponentials LL Poly 4LL a nx n + + a 1 x or 0 no dereliction TLL + b no digging light logics ELL PLL + b, a 1!A A!A LLL!( ) not monoidal SLL x or b!a A A D. Mazza (LIPN) Attack of the Exponentials LL / 26

59 Heterodox exponentials LL Poly 4LL a nx n + + a 1 x or 0 no dereliction TLL + b no digging light logics ELL parsimonious logic LLL!( ) not monoidal PLL + b, a 1!A A!A SLL x or b!a A A D. Mazza (LIPN) Attack of the Exponentials LL / 26

60 Implicit computational complexity Runtime = number of -elimination steps to normal form. Light logics have untyped -elimination: ELL: characterizes elementary time SLL: characterizes polynomial time LLL: characterizes polynomial time Parsimonious logic (with!a = A!A) is Turing-complete when untyped; however: propositional: characterizes logarithmic space linear 2nd order: characterizes polynomial time Two different approaches: stratification: (light logics) complexity is controlled globally; parsimony: complexity is controlled locally. The parsimonious approach also opens the way to non-uniformity, via approximations. D. Mazza (LIPN) Attack of the Exponentials LL / 26

61 Perspectives Approximations (exponentials as a limit): differential linear logic (DiLL) and Taylor expansion; affine approximations and complexity. Quantitative analyses: in DiLL, exponentials have non-deterministic -elimination; bounded linear logic (modalities parametriezed by semi-ring). Geometry of interaction: an exeion model based on tokens moving along proof nets. D. Mazza (LIPN) Attack of the Exponentials LL / 26