Denotational semantics of linear logic

Transcription

1 Denotational semantics of linear logic Lionel Vaux I2M, Université d Aix-Marseille, France LL2016, Lyon school: 7 and 8 November 2016 L. Vaux (I2M) Denotational semantics of linear logic LL / 31

2 Denotational semantics Not about formulas (truth) but about the meaning of proofs Originated in the study of programming (λ-calculus) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

3 Denotational semantics Not about formulas (truth) but about the meaning of proofs Originated in the study of programming (λ-calculus) A A Logic Computation Denotational semantics Formula Type Object (mathematical space) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

4 Denotational semantics Not about formulas (truth) but about the meaning of proofs Originated in the study of programming (λ-calculus) π. A π A Logic Computation Denotational semantics Formula Type Object (mathematical space) Proof Program Point L. Vaux (I2M) Denotational semantics of linear logic LL / 31

5 Denotational semantics Not about formulas (truth) but about the meaning of proofs Originated in the study of programming (λ-calculus) π. Γ A π : Γ A Logic Computation Denotational semantics Formula Type Object (mathematical space) Proof Program Point/Morphism (function) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

6 Denotational semantics Not about formulas (truth) but about the meaning of proofs Originated in the study of programming (λ-calculus) π. Γ A B π : Γ A B Logic Computation Denotational semantics Formula Type Object (mathematical space) Proof Program Point/Morphism (function) Connective Type constructor Operation on spaces L. Vaux (I2M) Denotational semantics of linear logic LL / 31

7 Denotational semantics Not about formulas (truth) but about the meaning of proofs Originated in the study of programming (λ-calculus) π π.. Γ A Γ B Γ A B π, π : Γ A B Logic Computation Denotational semantics Formula Type Object (mathematical space) Proof Program Point/Morphism (function) Connective Type constructor Operation on spaces Deduction rule Term constructor Operation on morphisms L. Vaux (I2M) Denotational semantics of linear logic LL / 31

8 Denotational semantics Not about formulas (truth) but about the meaning of proofs Originated in the study of programming (λ-calculus) π π.. Γ A A B cut Γ B π π : Γ B Logic Computation Denotational semantics Formula Type Object (mathematical space) Proof Program Point/Morphism (function) Connective Type constructor Operation on spaces Deduction rule Term constructor Operation on morphisms Cut Application Composition L. Vaux (I2M) Denotational semantics of linear logic LL / 31

9 Denotational semantics Not about formulas (truth) but about the meaning of proofs Originated in the study of programming (λ-calculus) π. Γ A c.e. π. Γ A π = π : Γ A Logic Computation Denotational semantics Formula Type Object (mathematical space) Proof Program Point/Morphism (function) Connective Type constructor Operation on spaces Deduction rule Term constructor Operation on morphisms Cut Application Composition Cut elimination Evaluation = L. Vaux (I2M) Denotational semantics of linear logic LL / 31

10 Denotational semantics Not about formulas (truth) but about the meaning of proofs Originated in the study of programming (λ-calculus) π. Γ A c.e. π. Γ A π = π : Γ A Logic Computation Denotational semantics Formula Type Object (mathematical space) Proof Program Point/Morphism (function) Connective Type constructor Operation on spaces Deduction rule Term constructor Operation on morphisms Cut Application Composition Cut elimination Evaluation = a notion of morphism = a category L. Vaux (I2M) Denotational semantics of linear logic LL / 31

11 Denotational semantics Not about formulas (truth) but about the meaning of proofs Originated in the study of programming (λ-calculus) π. Γ A c.e. π. Γ A π = π : Γ A Logic Computation Denotational semantics Formula Type Object (mathematical space) Proof Program Point/Morphism (function) Connective Type constructor Operation on spaces Deduction rule Term constructor Operation on morphisms Cut Application Composition Cut elimination Evaluation = a notion of morphism = a category a denotational model = a category with some additional structure L. Vaux (I2M) Denotational semantics of linear logic LL / 31

12 How linear logic got its name Models of typed λ-calculus are cartesian closed categories (, ). Quantitative semantics, in a nutshell (Girard, early 80 s, before LL) objects = sets morphisms QS(A, B) = analytic functions R A R B alternative to Scott domains, motivated by non-determinism Analytic functions are defined by power series: there is a family of coefficients (ϕ a,β ) a Mf (A),b B such that ϕ(α) b = a M f (A) ϕ a,bα a where α [a1,...,an] = α a1 α an L. Vaux (I2M) Denotational semantics of linear logic LL / 31

13 How linear logic got its name Models of typed λ-calculus are cartesian closed categories (, ). Quantitative semantics, in a nutshell (Girard, early 80 s, before LL) objects = sets morphisms QS(A, B) = analytic functions R A R B alternative to Scott domains, motivated by non-determinism Analytic functions are defined by power series: there is a family of coefficients (ϕ a,β ) a Mf (A),b B such that ϕ(α) b = a M f (A) ϕ a,bα a where α [a1,...,an] = α a1 α an Linear decomposition of the function space Setting α! a = αa we obtain α! R M f (A) and ϕ(α) = ϕ α! where denotes matrix application: An ( R A, R B) = Lin ( R M f (A), R B) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

14 How linear logic got its name Models of typed λ-calculus are cartesian closed categories (, ). Quantitative semantics, in a nutshell (Girard, early 80 s, before LL) objects = sets morphisms QS(A, B) = analytic functions R A R B alternative to Scott domains, motivated by non-determinism Analytic functions are defined by power series: there is a family of coefficients (ϕ a,β ) a Mf (A),b B such that ϕ(α) b = a M f (A) ϕ a,bα a where α [a1,...,an] = α a1 α an Linear decomposition of the function space Setting α! a = αa we obtain α! R M f (A) and ϕ(α) = ϕ α! where denotes matrix application: An ( R A, R B) = Lin ( R M f (A), R B) A B =!A B L. Vaux (I2M) Denotational semantics of linear logic LL / 31

15 The finite dimensional matricial model There is a catch with quantitative semantics: infinite sums might not converge L. Vaux (I2M) Denotational semantics of linear logic LL / 31

16 The finite dimensional matricial model There is a catch with quantitative semantics: infinite sums might not converge If we stick to the linear fragment, we can consider finite dimensional spaces. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

17 The finite dimensional matricial model There is a catch with quantitative semantics: infinite sums might not converge If we stick to the linear fragment, we can consider finite dimensional spaces. A category of finite dimensional vector spaces: Mat objects = finite sets morphisms Mat(A, B) = matrices R A B The finite set A is the canonical base of R A Matrices R A B represent linear maps from R A to R B L. Vaux (I2M) Denotational semantics of linear logic LL / 31

18 Proofs as matrices We interpret formulas A as finite sets A and proofs π : A B as matrices Mat(A, B): for all a A and b B, we have π a,b R. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

19 Proofs as matrices We interpret formulas A as finite sets A and proofs π : A B as matrices Mat(A, B): for all a A and b B, we have π a,b R. The axiom is the identity matrix ax A, A = δ a,a a,a L. Vaux (I2M) Denotational semantics of linear logic LL / 31

20 Proofs as matrices We interpret formulas A as finite sets A and proofs π : A B as matrices Mat(A, B): for all a A and b B, we have π a,b R. The axiom is the identity matrix ax A, A = δ a,a a,a Cut is composition, i.e. matrix product: π. A, B A, C π. B, C cut a,c = b B π a,b π b,c L. Vaux (I2M) Denotational semantics of linear logic LL / 31

21 The symmetric monoidal closed structure:, Tensor product Given matrices ϕ R A B and ϕ R A B, we can form their tensor product ϕ ϕ R (A A ) (B B ) setting (ϕ ϕ ) (a,a ),(b,b ) = ϕ a,b ϕ a,b Set A B = A B This is associative and commutative (up to isomorphism) There is a unit 1 = { } Space of linear maps A B is also the object of morphisms from A to B: Set A B = A B We have an adjunction: Mat(A B, C) = Mat(A, B C) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

22 The -autonomous structure: A = A Dual space The dual of R A is the set of linear forms on R A : Lin(R A, R) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

23 The -autonomous structure: A = A Dual space The dual of R A is the set of linear forms on R A : Lin(R A, R) Fix to be a singleton: = { } so that Lin(R A, R) = Mat(A, ) By monoidal closedness, we can derive a canonical morphism ϕ Mat(A, (A ) ) from the identity on A : ϕ a,((a, ), ) = δ a,a ϕ is an isomorphism: this is -autonomy L. Vaux (I2M) Denotational semantics of linear logic LL / 31

24 The -autonomous structure: A = A Dual space The dual of R A is the set of linear forms on R A : Lin(R A, R) Fix to be a singleton: = { } so that Lin(R A, R) = Mat(A, ) By monoidal closedness, we can derive a canonical morphism ϕ Mat(A, (A ) ) from the identity on A : ϕ a,((a, ), ) = δ a,a ϕ is an isomorphism: this is -autonomy This isomorphism reflects the identity A = A One recovers A ` B as ( A B ) In fact here R A { } = R A : we can set, more simply, A = A In particular A B = A ` B: the model is compact closed L. Vaux (I2M) Denotational semantics of linear logic LL / 31

25 Interpretation of MLL General guidelines Interpret a proof of A 1,..., A n B as a morphism: A 1 A n B By the -autonomous structure, this is equivalent to a morphism 1 A 1 ` ` A n ` B This reflects the equiprovability of Γ, Γ, and Γ, In Mat If π is a proof of A 1,..., A n then π R A1 An L. Vaux (I2M) Denotational semantics of linear logic LL / 31

26 Interpretation of MLL Rules π. Γ, A ax A, A = δ a,a a,a π. B, Γ, A B, π. Γ, A, B ` Γ, A ` B π. Γ, A π. s,(a,b) A, cut Γ, s,(a,b),t s,t = π s,a π b,t = π s,a,b = a A π s,a π a,t L. Vaux (I2M) Denotational semantics of linear logic LL / 31

27 Additives Cartesian product Vector spaces admit cartesian products: R A R B = R A+B where A + B denotes the disjoint union ({1} A) ({2} B) A & B = A + B This is the product type of programming languages: given two morphisms ϕ Mat(C, A) and ϕ Mat(C, B), one forms ϕ, ψ Mat(C, A + B), hence the rule Γ, A Γ, B & Γ, A & B L. Vaux (I2M) Denotational semantics of linear logic LL / 31

28 Additives Cartesian product Vector spaces admit cartesian products: R A R B = R A+B where A + B denotes the disjoint union ({1} A) ({2} B) A & B = A + B This is the product type of programming languages: given two morphisms ϕ Mat(C, A) and ϕ Mat(C, B), one forms ϕ, ψ Mat(C, A + B), hence the rule Γ, A Γ, B & Γ, A & B This is the additive conjunction L. Vaux (I2M) Denotational semantics of linear logic LL / 31

29 Additives Cartesian product Vector spaces admit cartesian products: R A R B = R A+B where A + B denotes the disjoint union ({1} A) ({2} B) A & B = A + B This is the product type of programming languages: given two morphisms ϕ Mat(C, A) and ϕ Mat(C, B), one forms ϕ, ψ Mat(C, A + B), hence the rule Γ, A Γ, B & Γ, A & B This is the additive conjunction (Note that A B is not the type of pairs) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

30 Additives Direct sums By duality we obtain A B = ( A & B ) Here A B = A & B (because of compact closedness) This is the sum type of programming languages: it comes with injections ι l Mat(A, A + B) and ι r Mat(B, A + B) hence the rules Γ, A Γ, B l and r Γ, A B Γ, A B The additive disjunction L. Vaux (I2M) Denotational semantics of linear logic LL / 31

31 Additives Direct sums By duality we obtain A B = ( A & B ) Here A B = A & B (because of compact closedness) This is the sum type of programming languages: it comes with injections ι l Mat(A, A + B) and ι r Mat(B, A + B) hence the rules Γ, A Γ, B l and r Γ, A B Γ, A B The additive disjunction In fact we gave only half of the picture and we miss: projections for the products case definitions for sums but they follow from pairs and injections by duality (see the cut elimination later) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

32 Additives Semantics π 1. Γ, A π 2. Γ, B & Γ, A & B π. Γ, A Γ, A B π. Γ, B Γ, A B l r s,(i,c) s,(i,c) s,(i,c) = = = { π 1 s,a if (i, c) = (1, a) π 2 s,b if (i, c) = (2, b) { π s,a if (i, c) = (1, a) 0 if i = 2 { 0 if 1 = 2 π s,b if (i, c) = (2, b) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

33 Additives Cut elimination π 1 π 2.. Γ, A Γ, B & Γ, A & B π 1 π 2.. Γ, A Γ, B & Γ, A & B Γ, Γ, π., A, A B π., B, A B l cut r cut π 1. Γ, A π 2. Γ, B Γ, Γ, π., A π., B cut cut L. Vaux (I2M) Denotational semantics of linear logic LL / 31

34 Additives Cut elimination π 1 π 2.. Γ, A Γ, B & Γ, A & B π 1 π 2.. Γ, A Γ, B & Γ, A & B Γ, Γ, π., A, A B π., B, A B l cut r cut π 1. Γ, A π 2. Γ, B Γ, Γ, π., A π., B cut cut non-local! a good notion of proof nets with additives is difficult to design but additives live very naturally in the semantics (like units) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

35 The relational model Rel Setting R = B (booleans), we recover the model presented yesterday: write s π for π s = 1 a matrix ϕ B A B is a relation ϕ A B we no longer care about finiteness, since the composition of relations ϕ A B and ψ B C is always defined: (a, c) ψϕ iff there exists b B such that (a, b) ϕ and (b, c) ψ write Rel for the category of sets and relations Recall from yesterday that the relational model can be described by a kind of type system: π. a 1 a n is derivable iff (a 1,..., a n ) π. A 1,..., An A 1,..., A n L. Vaux (I2M) Denotational semantics of linear logic LL / 31

36 The relational model Rules for MLL A B = A ` B = A B A a, A a ax s Γ a, A A a, t cut Γ s, t Γ s, A a B b, t Γ s (a,b), A B, t Γ s, A a, B b Γ s (a,b), A ` B ` L. Vaux (I2M) Denotational semantics of linear logic LL / 31

37 The relational model Rules for MLL MALL A B = A ` B = A B A a, A a ax s Γ a, A A a, t cut Γ s, t Γ s, A a B b, t Γ s (a,b), A B, t Γ s, A a, B b Γ s (a,b), A ` B ` A & B = A B = A + B Γ s, A a Γ, B & Γ s (1,a), A & B Γ, A Γ s, B b & Γ s (2,b), A & B Γ s, A a Γ s (1,a), A B l Γ s, B b Γ s (2,b), A B r L. Vaux (I2M) Denotational semantics of linear logic LL / 31

38 Exponentials in Rel Ingredients Recall from quantitative semantics:!a = M f (A). L. Vaux (I2M) Denotational semantics of linear logic LL / 31

39 Exponentials in Rel Ingredients Recall from quantitative semantics:!a = M f (A). This defines an functor in Rel: if ϕ Rel(A, B) then!ϕ = {([a 1,..., a n ], [b 1,..., b n ]) ; (a i, b i ) ϕ for all i} Rel(!A,!B) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

40 Exponentials in Rel Ingredients Recall from quantitative semantics:!a = M f (A). This defines an functor in Rel: if ϕ Rel(A, B) then!ϕ = {([a 1,..., a n ], [b 1,..., b n ]) ; (a i, b i ) ϕ for all i} Rel(!A,!B) This has the structure of a comonad with der = {([a], a); a A} Rel(!A, A) and digg = {(a a n, [a 1,..., a n ]); a 1,..., a n!a} Rel(!A,!!A) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

41 Exponentials in Rel Ingredients Recall from quantitative semantics:!a = M f (A). This defines an functor in Rel: if ϕ Rel(A, B) then!ϕ = {([a 1,..., a n ], [b 1,..., b n ]) ; (a i, b i ) ϕ for all i} Rel(!A,!B) This has the structure of a comonad with der = {([a], a); a A} Rel(!A, A) and digg = {(a a n, [a 1,..., a n ]); a 1,..., a n!a} Rel(!A,!!A) There is an isomorphism!(a & B) =!A!B: {([(1, a 1 ),..., (1, a n ), (2, b 1 ),..., (2, b n )], ([a 1,..., a n ], [b 1,..., b p ]))} L. Vaux (I2M) Denotational semantics of linear logic LL / 31

42 Exponentials in Rel Ingredients Recall from quantitative semantics:!a = M f (A). This defines an functor in Rel: if ϕ Rel(A, B) then!ϕ = {([a 1,..., a n ], [b 1,..., b n ]) ; (a i, b i ) ϕ for all i} Rel(!A,!B) This has the structure of a comonad with der = {([a], a); a A} Rel(!A, A) and digg = {(a a n, [a 1,..., a n ]); a 1,..., a n!a} Rel(!A,!!A) There is an isomorphism!(a & B) =!A!B: {([(1, a 1 ),..., (1, a n ), (2, b 1 ),..., (2, b n )], ([a 1,..., a n ], [b 1,..., b p ]))} This (plus a bunch of other conditions to make everything work together) makes Rel a model of LL. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

43 Exponentials in Rel Interpretation Structural rules: resource management Γ s, A a?d Γ s [a],?a Γ s?w Γ s [],?A Γ s,?a a,?a a?c Γ s a+a,?a L. Vaux (I2M) Denotational semantics of linear logic LL / 31

44 Exponentials in Rel Interpretation Structural rules: resource management Γ s, A a?d Γ s [a],?a Γ s?w Γ s [],?A Γ s,?a a,?a a?c Γ s a+a,?a Promotion: resource factory s i a i?γ, A (for 1 i n) n! i =1 s i [a 1,...,a n]?γ,!a (Quantitative semantics relies on quite complicated formulas for the coefficients.) L. Vaux (I2M) Denotational semantics of linear logic LL / 31

45 Example A, A ax l B, B ax r A B, A?d A B, B?d?(A B ), A?(A B ), B!?(A B ),!A?(A B ),!B?(A B ),?(A B ),!A!B?c?(A B ),!A!B! L. Vaux (I2M) Denotational semantics of linear logic LL / 31

46 Example a a A, A ax l A B, A?d b b B, B ax r A B, B?d?(A B ), A?(A B ), B!?(A B ),!A?(A B ),!B?(A B ),?(A B ),!A!B?c?(A B ),!A!B! L. Vaux (I2M) Denotational semantics of linear logic LL / 31

47 Example a a A, A ax (1,a ) a A B, A?(A B ), A?(A B ),!A l?d! b b B, B ax (2,b ) b A B, B?(A B ), B?(A B ),!B?(A B ),?(A B ),!A!B?(A B ),!A!B r?d?c! L. Vaux (I2M) Denotational semantics of linear logic LL / 31

48 Example a a A, A ax (1,a ) a A B, A [(1,a )] a?(a B ), A?(A B ),!A l?d! b b B, B ax (2,b ) b A B, B [(2,b )] b?(a B ), B?(A B ),!B?(A B ),?(A B ),!A!B?(A B ),!A!B r?d?c! L. Vaux (I2M) Denotational semantics of linear logic LL / 31

49 Example a i a i A, A ax l (1,a i ) a i A B, A?d [(1,a i )] a i?(a B ), A! [(1,a 1),...,(1,a n)] [a 1,...,a n]?(a B ),!A b i b i B, B ax r (2,b i ) b i A B, B?d [(2,b i )] b i?(a B ), B! [(2,b 2),...,(2,b p)] [b 2,...,b p]?(a B ),!B?(A B ),?(A B ),!A!B?(A B ),!A!B?c L. Vaux (I2M) Denotational semantics of linear logic LL / 31

50 Example a i a i A, A ax l (1,a i ) a i A B, A?d [(1,a i )] a i?(a B ), A! [(1,a 1),...,(1,a n)] [a 1,...,a n]?(a B ),!A [(1,a 1),...,(1,a n)]?(a B ) b i b i B, B ax r (2,b i ) b i A B, B?d [(2,b i )] b i?(a B ), B! [(2,b 2),...,(2,b p)] [b 2,...,b p]?(a B ),!B [(2,b 2),...,(2,b p)] ([a 1,...,a n],[b 2,...,b p]),?(a B ),!A!B?(A B ),!A!B?c L. Vaux (I2M) Denotational semantics of linear logic LL / 31

51 Example a i a i A, A ax l (1,a i ) a i A B, A?d [(1,a i )] a i?(a B ), A! [(1,a 1),...,(1,a n)] [a 1,...,a n]?(a B ),!A [(1,a 1),...,(1,a n)]?(a B ) b i b i B, B ax r (2,b i ) b i A B, B?d [(2,b i )] b i?(a B ), B! [(2,b 2),...,(2,b p)] [b 2,...,b p]?(a B ),!B [(2,b 2),...,(2,b p)] ([a 1,...,a n],[b 2,...,b p]),?(a B ),!A!B [(1,a 1),...,(1,a n),(2,b 2),...,(2,b p)] ([a 1,...,a n],[b 2,...,b p])?(a B ),!A!B?c L. Vaux (I2M) Denotational semantics of linear logic LL / 31

52 Uniformity The relational model is highly non uniform: the union of arbitrary relations is still a relation. This is good for capturing non-deterministic superpositions of proofs. But it comes with many degeneracies: most notably A = A; and it does not see vicious cycles in proof structures. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

53 Uniformity The relational model is highly non uniform: the union of arbitrary relations is still a relation. This is good for capturing non-deterministic superpositions of proofs. But it comes with many degeneracies: most notably A = A; and it does not see vicious cycles in proof structures. Coherence spaces A coherence space A is the data of a set A (the web of A) and of a reflexive and symmetric relation A on A. The intuitive meaning of a A a is that a and a can appear simultaneously in the semantics of the same proof of A: we will interpret proofs by cliques. C (A) = {α A ; a A a, for all a, a α} L. Vaux (I2M) Denotational semantics of linear logic LL / 31

54 Duality Negation reverses the coherence (but we still need a symmetric relation). Define three new relations on A : a ˇA iff a A a ; a A iff a = a or a ˇA a ; a A iff a A a and a a. Dual Define A by A = A and A = A. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

55 Multiplicatives It is natural to set: A B = A B (a, b) A B (a, b ) a A a and b B b We deduce ` by de Morgan rules: (a, b) A`B (a, b ) a A a or b B b Linear implication A B = (A B ) follows: (a, b) A B (a, b ) a A a entails b B b It should be clear that if ϕ C (A B) and ψ C (B C), then ψϕ C (A C) (where composition is that of relations). L. Vaux (I2M) Denotational semantics of linear logic LL / 31

56 A -autonomous category of coherence spaces Coherence spaces and coherent relations Write Coh for the category of coherence spaces, with morphisms Coh(A, B) = C (A B). Setting = { } equipped with the only possible coherence relation it is clear that A = A. We no longer have A = A. The morphisms giving the -autonomous structure are exactly the same as in Rel. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

57 A -autonomous category of coherence spaces Coherence spaces and coherent relations Write Coh for the category of coherence spaces, with morphisms Coh(A, B) = C (A B). Setting = { } equipped with the only possible coherence relation it is clear that A = A. We no longer have A = A. The morphisms giving the -autonomous structure are exactly the same as in Rel. Lemma If we fix an interpretation X Coh of atoms as coherence spaces and consider the relational model where X Rel = X Coh, then π Rel C (Γ Coh ) for all MLL proof π of Γ. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

58 Additives The previous result extends to additives: set A & B = A B = A + B and i = i = 1 and c A c (i, c) A&B (i, c ) or i = i = 2 and c B c or i i { (i, c) A B (i, c i = i = 1 and c A c ) or i = i = 2 and c B c Now A & B = A B and cliques correspond with our intuition: a clique in A & B corresponds with a pair of cliques in A and B respectively; a clique in A B corresponds with either a clique in A or a clique in B. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

59 Experiments in MLL proof nets We can play the same game as with Rel: A a ax A a A a (a,b) A B B b A a (a,b) A ` B ` B b A a cut A a L. Vaux (I2M) Denotational semantics of linear logic LL / 31

60 Experiments in MLL proof nets We can play the same game as with Rel: A a ax A a A a (a,b) A B B b A a (a,b) A ` B ` B b A a cut A a But now we can detect vicious cycles: A a (a,a) A A (a, a) ˇA A (a, a ) as soon as a a A ax A a L. Vaux (I2M) Denotational semantics of linear logic LL / 31

61 Experiments in MLL proof nets We can play the same game as with Rel: A a ax A a A a (a,b) A B B b A a (a,b) A ` B ` B b A a cut A a But now we can detect vicious cycles: A a (a,a) A A (a, a) ˇA A (a, a ) as soon as a a A In fact one can show that (for a well chosen interpretation of atoms) a cut-free MLL structure is correct iff its relational semantics is clique. ax A a L. Vaux (I2M) Denotational semantics of linear logic LL / 31

62 Exponentials Multiset based A natural choice is!a = M f ( A ). Set [a 1,..., a n ]!A [ a 1,..., a p] iff ai A a j for all i and j. But this is not reflexive! L. Vaux (I2M) Denotational semantics of linear logic LL / 31

63 Exponentials Multiset based A natural choice is!a = M f ( A ). Set [a 1,..., a n ]!A [ a 1,..., a p] iff ai A a j for all i and j. But this is not reflexive! Multicliques We set!a to be the set of all finite multicliques (finite multisets whose support is a clique). Then a a iff a + a!a. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

64 Exponentials Multiset based A natural choice is!a = M f ( A ). Set [a 1,..., a n ]!A [ a 1,..., a p] iff ai A a j for all i and j. But this is not reflexive! Multicliques We set!a to be the set of all finite multicliques (finite multisets whose support is a clique). Then a a iff a + a!a.?a is the set of multicliques in A?A is a strange beast. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

65 Exponentials Interpretation of sequent calculus The system of annotations now comes with side conditions: Γ s, A a?d Γ s [a],?a Γ s?w Γ s [],?A Γ s,?a a,?a a a + a?a?c Γ s a+a,?a s i a i?γ, A (for 1 i n) n i=1 s i?γ! n i =1 s i [a 1,...,a n]?γ,!a L. Vaux (I2M) Denotational semantics of linear logic LL / 31

66 Exponentials Set based Coherence spaces were first introduced as a refinement of qualitative domains: particular sets of subsets. Cliques We set!a to be the set of all finite cliques of A. Then ã ã iff ã ã!a.?a is a still strange beast. And we still need annotations. This model is less fine but it has a close relationship with the notion of stable functions which was crucial in the invention of coherence spaces. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

67 Orthogonality Coherence spaces (at least for MLL) are an instance of a general construction: The core of the interpretation is the -autonomous structure: composition is the interaction between a proof Γ, A and a counterproof A,. We can refine a preexisting model by selecting morphisms that interact nicely. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

68 Orthogonality Coherence spaces as spaces of cliques Observe that if α C (A) and α C ( A ) then α α has at most one element. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

69 Orthogonality Coherence spaces as spaces of cliques Observe that if α C (A) and α C ( A ) then α α has at most one element. Let A be a set: for all α, α A write α α iff #(α α ) 1. If A P (A), let A = {α A; α α, for all α A}. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

70 Orthogonality Coherence spaces as spaces of cliques Observe that if α C (A) and α C ( A ) then α α has at most one element. Let A be a set: for all α, α A write α α iff #(α α ) 1. If A P (A), let A = {α A; α α, for all α A}. Observe that α A iff for all a 1, a 2 α, {a 1, a 2 } A. In other words, the sets A are exactly the sets of cliques of coherence spaces with web A. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

71 Orthogonality Coherence spaces as spaces of cliques Observe that if α C (A) and α C ( A ) then α α has at most one element. Let A be a set: for all α, α A write α α iff #(α α ) 1. If A P (A), let A = {α A; α α, for all α A}. Observe that α A iff for all a 1, a 2 α, {a 1, a 2 } A. In other words, the sets A are exactly the sets of cliques of coherence spaces with web A. Characterization of morphisms A relation ϕ A B is coherent iff for all α C (A) and all β C (B), ϕ α β and ϕ β α (where ϕ is the reverse relation). L. Vaux (I2M) Denotational semantics of linear logic LL / 31

72 Orthogonality Coherence spaces as spaces of cliques Observe that if α C (A) and α C ( A ) then α α has at most one element. Let A be a set: for all α, α A write α α iff #(α α ) 1. If A P (A), let A = {α A; α α, for all α A}. Observe that α A iff for all a 1, a 2 α, {a 1, a 2 } A. In other words, the sets A are exactly the sets of cliques of coherence spaces with web A. Characterization of morphisms A relation ϕ A B is coherent iff for all α C (A) and all β C (B), ϕ α β and ϕ β α (where ϕ is the reverse relation). The coherence space model is thus derived from the relational model by orthogonality. This kind of construction generalizes to other notions of orthogonality, and to a generic categorical framework. L. Vaux (I2M) Denotational semantics of linear logic LL / 31

73 Perspectives Systematic categorical descriptions of models of LL exist I was just lazy (see, e.g., Melliès, Panoramas et Synthèses 2009) one always recover a model of λ-calculus (co-kleisli:!a B) Characterization of computational properties normalizability bounds on the number of cut elimination step connexion with interaction types rely on models of pure λ-calculus: solve o =!o o Quantitative semantics can be described explicitly in a standard setting (Ehrhard s finiteness spaces) differential linear logic a guide for introducing and studying new models (see Michele Pagani s talk during the workshop) Duality given by interaction at the core of geometry of interaction (operational-ish semantics) the basis of game semantics, a multi-purpose family of models... L. Vaux (I2M) Denotational semantics of linear logic LL / 31

74 Have a nice LL2016 Tutorial time! L. Vaux (I2M) Denotational semantics of linear logic LL / 31