A Differential Model Theory for Resource Lambda Calculi - Part II

Transcription

1 A Differential Model Theory for Resource Lambda Calculi - Part II Giulio Manzonetto (joint work with Bucciarelli, Ehrhard, Laird, McCusker) g.manzonetto@cs.ru.nl Intelligent Systems Radboud University Nijmegen FMCS Calgary - 11 th June 2011 Giulio Manzonetto (RU) Differential Model Theory 11/06/11 1 / 35

2 Introduction Outline 1 Cartesian (Closed) Differential Categories [BEM 10, Man 11] Typed and untyped models of differential/resource λ-calculus Soundness & Completeness Examples 2 Building Differential Categories [LMM 11] A general recipe: From SMC(C) Differential SMC(C) Differential C(C)C 3 Game Semantics [LMM 11] Instances: Categories of Games, The relational semantics Full abstraction for relational semantics w.r.t. Resource PCF 4 Conclusions Giulio Manzonetto (RU) Differential Model Theory 11/06/11 2 / 35

3 Cartesian Closed Differential Categories Differential Categories The differential λ-calculus inspired researchers working on category theory. Aim: Axiomatize a differential operator D( ) categorically. Differential categories Blute, Cockett and Seely proposed: BCS 06: (monoidal) differential categories point of view too fine BCS 09: Cartesian differential categories sound and complete for their term calculus, lack of higher order functions! Not enough for modeling the differential λ-calculus!!! Giulio Manzonetto (RU) Differential Model Theory 11/06/11 3 / 35

4 Cartesian Closed Differential Categories Differential Categories The differential λ-calculus inspired researchers working on category theory. Aim: Axiomatize a differential operator D( ) categorically. Differential categories Blute, Cockett and Seely proposed: BCS 06: (monoidal) differential categories point of view too fine BCS 09: Cartesian differential categories sound and complete for their term calculus, lack of higher order functions! Not enough for modeling the differential λ-calculus!!! Giulio Manzonetto (RU) Differential Model Theory 11/06/11 3 / 35

5 Cartesian Closed Differential Categories Left Additive Categories We have sums of λ-terms and the application is left-linear: ( i s i)t = i s it s( i t i) i st i We need a left-additive sum on morphisms! A category C is left-additive if: each homset has a structure of commutative monoid (C(A, B), + AB, 0 AB ), f ; (g + h) = (f ; g) + (f ; h) and f ; 0 = 0. When f satisfies also (g + h); f = (g; f ) + (h; f ) and 0; f = 0 it is called additive. (weak form of linearity) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 4 / 35

6 Cartesian Closed Differential Categories Left Additive Categories We have sums of λ-terms and the application is left-linear: ( i s i)t = i s it s( i t i) i st i We need a left-additive sum on morphisms! A category C is left-additive if: each homset has a structure of commutative monoid (C(A, B), + AB, 0 AB ), f ; (g + h) = (f ; g) + (f ; h) and f ; 0 = 0. When f satisfies also (g + h); f = (g; f ) + (h; f ) and 0; f = 0 it is called additive. (weak form of linearity) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 4 / 35

7 Cartesian Closed Differential Categories Cartesian (Closed) Left-additive Categories A category C is Cartesian left-additive if: C is a left-additive category, it is Cartesian (= it has products), all projections and pairings of additive maps are additive. A category C is Cartesian closed left-additive if: C is Cartesian left-additive, it is a ccc (Λ( ) = curry, ev =eval), it satisfies Λ(f + g) = Λ(f ) + Λ(g) and Λ(0) = 0. (implies f + g, h ; ev = f, h ; ev + g, h ; ev) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 5 / 35

8 Cartesian Closed Differential Categories Cartesian (Closed) Left-additive Categories A category C is Cartesian left-additive if: C is a left-additive category, it is Cartesian (= it has products), all projections and pairings of additive maps are additive. The Cartesian (closed) structure does not behaves automatically well with the left-additive enrichment! A category C is Cartesian closed left-additive if: C is Cartesian left-additive, it is a ccc (Λ( ) = curry, ev =eval), it satisfies Λ(f + g) = Λ(f ) + Λ(g) and Λ(0) = 0. (implies f + g, h ; ev = f, h ; ev + g, h ; ev) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 5 / 35

9 Cartesian Closed Differential Categories Cartesian Differential Categories Cartesian differential operator: D f : A B D (f ) : A A B Satisfying: D1. D (f + g) = D (f ) + D (g) and D (0) = 0 D2. h + k, v ; D (f ) = h, v ; D (f ) + k, v ; D (f ) and 0, v ; D (f ) = 0 D3. D (Id) = π 1, D (π 1 ) = π 1 ; π 1 and D (π 2 ) = π 1 ; π 2 D4. D ( f, g ) = D (f ), D (g) D5. D (g; f ) = D(g), π 2 ; g ; D (f ) D6. g, 0, h, k ; D (D (f )) = g, k ; D (f ) D7. 0, h, g, k ; D (D (f )) = 0, g, h, k ; D (D (f )) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 6 / 35

10 Cartesian Closed Differential Categories Subcategory of Linear Morphisms Linear morphisms A morphism f is linear if its differential is constant: D (f ) = π 1 ; f. f linear f additive f linear f additive Giulio Manzonetto (RU) Differential Model Theory 11/06/11 7 / 35

11 Cartesian Closed Differential Categories Partial differentiation Imagine we just want to differentiate f : C A B on C. D (f ) : (C A) (C A) B, we can obtain the partial derivative D C (f ) : C (C A) by zeroing out the A component, C (C A) Id C,0 C A Id C A (C A) (C A) B D (f ) D C (f ) = Id C, 0 C A Id C A; D (f ) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 8 / 35

12 Cartesian Closed Differential Categories Cartesian closed differential category Cartesian closed differential category [Bucciarelli-Ehrhard-Manzonetto 10] C is a Cartesian closed differential category if: C is a Cartesian differential category, it is Cartesian closed left-additive, it satisfies the following rule: For all f : C A B: D (Λ(f )) = Λ( π 1 0 A, π 2 Id A ; D (f )) Intuitively, the following methods for partial derivatives are equivalent: 1 Do Λ(f ) : C [A B] then apply D ( ), 2 Use the trick by zeroing out the A component as before. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 9 / 35

13 Cartesian Closed Differential Categories Categorical Interpretation (simply typed) Define f g = 0, π 1 ; g, Id ; D (f ): f : C A B g : C A f g : C A B Define Γ D s : σ = s σ Γ : Γ σ by: x σ Γ;x:σ = π 2, y τ Γ;x:σ = π 1 ; y τ Γ, (st ) τ Γ = s σ τ Γ, T σ Γ ; ev, (λx.s) σ τ Γ = Λ( s τ Γ;x:σ ), (D(s) t) σ τ Γ = Λ(Λ ( s σ τ Γ ) t σ Γ ), 0 σ Γ = 0, (s + S) σ Γ = s σ Γ + S σ Γ. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 10 / 35

14 Cartesian Closed Differential Categories Categorical Interpretation (simply typed) Define f g = 0, π 1 ; g, Id ; D (f ): f : C A B g : C A f g : C A B Define Γ D s : σ = s σ Γ : Γ σ by: x σ Γ;x:σ = π 2, y τ Γ;x:σ = π 1 ; y τ Γ, (st ) τ Γ = s σ τ Γ, T σ Γ ; ev, (λx.s) σ τ Γ = Λ( s τ Γ;x:σ ), (D(s) t) σ τ Γ = Λ(Λ ( s σ τ Γ ) t σ Γ ), 0 σ Γ = 0, (s + S) σ Γ = s σ Γ + S σ Γ. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 10 / 35

15 Cartesian Closed Differential Categories Categorical Interpretation (simply typed) Define f g = 0, π 1 ; g, Id ; D (f ): f : C A B g : C A f g : C A B Define Γ D s : σ = s σ Γ : Γ σ by: x σ Γ;x:σ = π 2, y τ Γ;x:σ = π 1 ; y τ Γ, (st ) τ Γ = s σ τ Γ, T σ Γ ; ev, (λx.s) σ τ Γ = Λ( s τ Γ;x:σ ), (D(s) t) σ τ Γ = Λ(Λ ( s σ τ Γ ) t σ Γ ), 0 σ Γ = 0, (s + S) σ Γ = s σ Γ + S σ Γ. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 10 / 35

16 Cartesian Closed Differential Categories Categorical Interpretation (simply typed) Define f g = 0, π 1 ; g, Id ; D (f ): f : C A B g : C A f g : C A B Define Γ D s : σ = s σ Γ : Γ σ by: x σ Γ;x:σ = π 2, y τ Γ;x:σ = π 1 ; y τ Γ, (st ) τ Γ = s σ τ Γ, T σ Γ ; ev, (λx.s) σ τ Γ = Λ( s τ Γ;x:σ ), (D(s) t) σ τ Γ = Λ(Λ ( s σ τ Γ ) t σ Γ ), 0 σ Γ = 0, (s + S) σ Γ = s σ Γ + S σ Γ. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 10 / 35

17 Cartesian Closed Differential Categories Soundness If C is a Cartesian closed differential category, then Th D (C) = {s = t Γ D s : σ Γ D t : σ s σ Γ = t σ Γ } is a differential λ-theory (i.e., it contains = D and it is contextual). Soundness Theorem [BEM 10] Cartesian closed differential categories are sound models for: Simply Typed Differential λ-calculus Giulio Manzonetto (RU) Differential Model Theory 11/06/11 11 / 35

18 Cartesian Closed Differential Categories Soundness If C is a Cartesian closed differential category, then Th D (C) = {s = t Γ D s : σ Γ D t : σ s σ Γ = t σ Γ } is a differential λ-theory (i.e., it contains = D and it is contextual). We can interpret the Resource Calculus by translation: Γ R M : σ = (M ) σ Γ we get that Th R (C) is a resource λ-theory. Soundness Theorem [BEM 10] Cartesian closed differential categories are sound models for: Simply Typed Differential λ-calculus Simply Typed Resource Calculus (by translation ( ) ) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 11 / 35

19 Soundness Cartesian Closed Differential Categories If C is a Cartesian closed differential category, then Th D (C) = {s = t Γ D s : σ Γ D t : σ s σ Γ = t σ Γ } is a differential λ-theory (i.e., it contains = D and it is contextual). Every model of the differential λ-calculus is also a model of the resource calculus We can interpret the Resource Calculus by translation: Γ R M : σ = (M ) σ Γ we get that Th R (C) is a resource λ-theory. Soundness Theorem [BEM 10] Cartesian closed differential categories are sound models for: Simply Typed Differential λ-calculus Simply Typed Resource Calculus (by translation ( ) ) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 11 / 35

20 Cartesian Closed Differential Categories Modelling the Untyped Differential λ-calculus As in the λ-calculus we need a reflexive object U (i.e., [U U] U) in a Cartesian closed differential category, but it is not enough! Linear reflexive object A reflexive object [U U] U is linear if A : U [U U], λ : [U U] U are both linear maps. We modify the interpretation in the obvious way: x i x = π i, st x = A s x, T x ; ev, λy.s x = Λ( s x,y ); λ, with y / x D(s) t x = Λ(Λ ( s x ; A) t x ); λ. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 12 / 35

21 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 13 / 35

22 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. Completeness Theorem for λ-calculus [Scott Koymans] Every λ-theories T is the theory of a reflexive object in a suitable CCC C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 13 / 35

23 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. Scott-Koymans completeness proof for regular λ-calculus (take T ): Λ/T Karoubi envelope C T = K(Λ/T ) I reflexive obj. Completeness Theorem for λ-calculus [Scott Koymans] Every λ-theories T is the theory of a reflexive object in a suitable CCC C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 13 / 35

24 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. Scott-Koymans completeness proof for regular λ-calculus (take T ): Λ/T Karoubi envelope C T = K(Λ/T ) I reflexive obj. M Λ o [λx.m] T Completeness Theorem for λ-calculus [Scott Koymans] Every λ-theories T is the theory of a reflexive object in a suitable CCC C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 13 / 35

25 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. Scott-Koymans completeness proof for regular λ-calculus (take T ): Λ/T Karoubi envelope C T = K(Λ/T ) Th(I) = T M Λ o [λx.m] T Completeness Theorem for λ-calculus [Scott Koymans] Every λ-theories T is the theory of a reflexive object in a suitable CCC C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 13 / 35

26 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. The category C T is described as follows (where M; N = λz.n(mz)): Objects: {A Λ/T A; A = A} Hom(A,B): {f Λ/T A; f ; B = f } Identity: Id A = A Composition: f ; g Completeness Theorem for λ-calculus [Scott Koymans] Every λ-theories T is the theory of a reflexive object in a suitable CCC C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 13 / 35

27 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. Idea: encode the categorical constructions via λ-terms: Products: f, g = λz.[fz, gz] π A 1,A 2 i = p i ; A i where [M, N] = λy.ymn, p 1 = λx.xk and p 2 = λx.xk (Church encoding). Exponents: [A B] = λz.a; z; B ev A,B = λz.b(p 1 z(a(p 2 z))) Λ(f ) = λxy.f [x, y] Completeness Theorem for λ-calculus [Scott Koymans] Every λ-theories T is the theory of a reflexive object in a suitable CCC C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 13 / 35

28 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. The category C T is described as follows (where M; N = λz.n(mz)): Objects: {A Λ/T A; A = A, A(x + y) = Ax + Ay} Hom(A,B): {f Λ/T A; f ; B = f } Identity: Id A = A Composition: f ; g Additive structure: the sum in the category is the sum of terms. Completeness Theorem for differential λ-calculus [Manzonetto 11] Every differential λ-theory T is the theory of a linear reflexive object in a suitable Cartesian closed differential category C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 14 / 35

29 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. Problem: Church s pairing is not additive! [M+M, N+N ] = λy.y(m+m )(N+N ) λy.ymn+λy.ym N = [M, N]+[M, N ] f + f, g + g f, g + f, g Completeness Theorem for differential λ-calculus [Manzonetto 11] Every differential λ-theory T is the theory of a linear reflexive object in a suitable Cartesian closed differential category C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 14 / 35

30 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. Solution: set-like encoding exploiting resource consciousness [M, N] = λy.m + λy.d(y) N p 1 = λx.x0 p 2 = λx.(d(x) I)00 We need to restrict to theories with idempotent sum! Completeness Theorem for differential λ-calculus [Manzonetto 11] Every differential λ-theory T with idempotent sum is the theory of a linear reflexive object in a suitable Cartesian closed differential category C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 14 / 35

31 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. The encoding of the differential operator is straightforward: D (f ) = λz.b((d(f ) (A(p 1 z)))(a(p 2 z))) : A A B Completeness Theorem for differential λ-calculus [Manzonetto 11] Every differential λ-theory T with idempotent sum is the theory of a linear reflexive object in a suitable Cartesian closed differential category C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 14 / 35

32 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. Completeness proof for differential λ-calculus (take T ): Λ d /T Karoubi envelope C T = K(Λ d /T ) I linear refl. obj. M Λ o d [λx. M] T ( ) : D(M) N λy.(d( M) Ñ)y, the identity otherwise. Completeness Theorem for differential λ-calculus [Manzonetto 11] Every differential λ-theory T with idempotent sum is the theory of a linear reflexive object in a suitable Cartesian closed differential category C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 14 / 35

33 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. Completeness proof for differential λ-calculus (take T ): Λ d /T Karoubi envelope C T = K(Λ d /T ) Th(I) T M Λ o d [λx. M] T We add equations Th(I) D(M) N = λy.(d(m) N)y (differentially extensional axiom) Completeness Theorem for differential λ-calculus [Manzonetto 11] Every differential λ-theory T with idempotent sum is the theory of a linear reflexive object in a suitable Cartesian closed differential category C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 14 / 35

34 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. Completeness proof for differential λ-calculus (take T ): Λ d /T Karoubi envelope C T = K(Λ d /T ) Th(I) = T M Λ o d [λx.m] T We restrict to differentially extensional T Completeness Theorem for differential λ-calculus [Manzonetto 11] Every differentionally extensional differential λ-theory T with idempotent sum is the theory of a linear reflexive object in a suitable Cartesian closed differential category C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 14 / 35

35 Cartesian Closed Differential Categories Soundness and Completeness Theorem [Manzonetto 11] Cartesian closed differential categories are sound and complete models of the differential λ-calculus. Completeness proof for differential λ-calculus (take T ): Λ d /T Karoubi envelope C T = K(Λ d /T ) Th(I) = T M Λ o d [λx.m] T All models arising naturally are differentionally extensional, also the non-extensional ones! Completeness Theorem for differential λ-calculus [Manzonetto 11] Every differentionally extensional differential λ-theory T with idempotent sum is the theory of a linear reflexive object in a suitable Cartesian closed differential category C T. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 14 / 35

36 Cartesian Closed Differential Categories Examples 1 MFin: Finiteness Spaces. [Ehrhard] [No reflexive objects in it!] 2 MRel: The Relational Semantics. The cokleisli of Rel + M f ( ). 3 Its variations with Infinite Multiplicities [Carraro-Ehrhard-Salibra] 4 Convenient differential category [Blute-Ehrhard-Tasson] 5 Categories of games... [Laird-Manzonetto-MCusker] (see later) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 15 / 35

37 Cartesian Closed Differential Categories Main Example: The Relational Semantics MRel Objects: sets, Morphisms: MRel(A, B) = P(M f (A) B) (relations between M f (A) and B). Given f : A B we can define: D(f ) = {(([a], m), b) (m [a], b) f } : A A B. Theorem [BEM 10] The category MRel is a Cartesian closed differential category. Corollary MRel is a model of the simply typed differential λ-calculus. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 16 / 35

38 Cartesian Closed Differential Categories Main Example: The Relational Semantics MRel Objects: sets, Morphisms: MRel(A, B) = P(M f (A) B) (relations between M f (A) and B). Given f : A B we can define: D(f ) = {(([a], m), b) (m [a], b) f } : A A B. Theorem [BEM 10] The category MRel is a Cartesian closed differential category. Corollary MRel is a model of the simply typed differential λ-calculus. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 16 / 35

39 Cartesian Closed Differential Categories Main Example: The Relational Semantics Untyped Models in MRel D: relational analogous of Scott s D (extensional), E: relational analogous of Engelers graph model (non extensional, but differentially extensional), Giulio Manzonetto (RU) Differential Model Theory 11/06/11 17 / 35

40 Cartesian Closed Differential Categories Main Example: The Relational Semantics Untyped Models in MRel D: relational analogous of Scott s D (extensional), Construction: D 0 = D n+1 = M f (D n ) (ω) D = n ω D n (m 1, m 2, m 3,...) D (m 1, (m 2, m 3,...)) M f (D) D = [D D] E: relational analogous of Engelers graph model (non extensional, but differentially extensional), Giulio Manzonetto (RU) Differential Model Theory 11/06/11 17 / 35

41 Cartesian Closed Differential Categories Main Example: The Relational Semantics Untyped Models in MRel D: relational analogous of Scott s D (extensional), E: relational analogous of Engelers graph model (non extensional, but differentially extensional), MRel models the Taylor Expansion In every linear reflexive object U of MRel we have S U x = S U x Giulio Manzonetto (RU) Differential Model Theory 11/06/11 17 / 35

42 Cartesian Closed Differential Categories Main Example: The Relational Semantics Untyped Models in MRel D: relational analogous of Scott s D (extensional), E: relational analogous of Engelers graph model (non extensional, but differentially extensional), MRel models the Taylor Expansion In every linear reflexive object U of MRel we have S U x = S U x Full Abstraction [Bucciarelli-Carraro-Ehrhard-Manzonetto 11] D is a fully abstract model of the resource calculus with tests. M N [ C( ) C(M) C(N) ] Giulio Manzonetto (RU) Differential Model Theory 11/06/11 17 / 35

43 Building Differential Categories Building Differential Categories SMC C sup-lattice enrichment countable biproducts Giulio Manzonetto (RU) Differential Model Theory 11/06/11 18 / 35

44 Building Differential Categories Building Differential Categories SMC C sup-lattice enrichment countable biproducts SMC K(C) Karoubi envelope symmetric tensor power coalgebra modality differential operator Giulio Manzonetto (RU) Differential Model Theory 11/06/11 18 / 35

45 Building Differential Categories Building Differential Categories SMC C sup-lattice enrichment countable biproducts Karoubi envelope Differential category K(C) sup-lattice enrichment Giulio Manzonetto (RU) Differential Model Theory 11/06/11 18 / 35

46 Building Differential Categories Building Differential Categories SMC C sup-lattice enrichment countable biproducts Karoubi envelope Differential category K(C) sup-lattice enrichment co-kleisli Cart. differential K! (C) cpo-enriched Giulio Manzonetto (RU) Differential Model Theory 11/06/11 18 / 35

47 Building Differential Categories Building Differential Categories monoidal closed SMC C sup-lattice enrichment countable biproducts Karoubi envelope monoidal closed Differential category K(C) sup-lattice enrichment co-kleisli Cartesian closed Cart. differential K! (C) cpo-enriched Giulio Manzonetto (RU) Differential Model Theory 11/06/11 18 / 35

48 Building Differential Categories Building Differential Categories SMC C sup-lattice enrichment countable biproducts all R-exponentials A R Karoubi envelope Differential category K(C) sup-lattice enrichment co-kleisli Cart. differential K! (C) cpo-enriched R-exponentials cat. Cartesian closed Giulio Manzonetto (RU) Differential Model Theory 11/06/11 18 / 35

49 Building Differential Categories General recipe for coalgebra modalities Take an SMC. In presence of the equalizer A n (as in many models of LL): A n equalizer A n. n! permutations A n A n gives the n-th layer of the free commutative monoid!a. Coalgebra modality!a = n ω A n This works for symmetric monoidal categories where the tensor distributes over the infinite product: X ( A n ) = (X A n ) n ω n ω Giulio Manzonetto (RU) Differential Model Theory 11/06/11 19 / 35

50 Building Differential Categories General recipe for coalgebra modalities Take an SMC. In presence of the equalizer A n (as in many models of LL): A n equalizer A n. n! permutations A n A n gives the n-th layer of the free commutative monoid!a. Coalgebra modality!a = n ω A n This works for symmetric monoidal categories where the tensor distributes over the infinite product: X ( A n ) = (X A n ) n ω n ω Giulio Manzonetto (RU) Differential Model Theory 11/06/11 19 / 35

51 Building Differential Categories Always works in the Karoubi Envelope K(C) (for sup-semilattice enriched C with infinite biproducts): Obj: Hom((A,f),(B,g)): (A, f ), A C, f : A A idempotent h C(A, B), such that f ; h = h g; h = h We can construct the symmetric tensor product A n using Θ A,n = σ S n σ (A n, f n ; Θ A,n ) Storage modality f n ; Θ A,n (A n, f n ) (A n, f n ). n! permutations!(a, f ) = n ω(a, f ) n (free commutative comonoid) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 20 / 35

52 Building Differential Categories Always works in the Karoubi Envelope K(C) (for sup-semilattice enriched C with infinite biproducts): Obj: Hom((A,f),(B,g)): (A, f ), A C, f : A A idempotent h C(A, B), such that f ; h = h g; h = h We can construct the symmetric tensor product A n using Θ A,n = σ S n σ (A n, f n ; Θ A,n ) Storage modality f n ; Θ A,n (A n, f n ) (A n, f n ). n! permutations!(a, f ) = n ω(a, f ) n (free commutative comonoid) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 20 / 35

53 Building Differential Categories Always works in the Karoubi Envelope Differential transformation d : (A, f )!(A, f )!(A, f ) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 21 / 35

54 Building Differential Categories Always works in the Karoubi Envelope Differential transformation d : (A, f ) (A, f ) n (A, f ) n n ω n ω 1 For each n we have: 2 Tupling all these gives a map f n+1 ; Θ A,n+1 (A, f ) (A, f ) n (A, f ) n+1 (A, f )!(A, f ) n ω (A, f ) n+1 3 Finally pairing with 0 : (A, f )!(A, f ) gives (A, f )!(A, f )!(A, f ) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 21 / 35

55 Building Differential Categories Always works in the Karoubi Envelope Differential transformation d : (A, f ) (A, f ) n (A, f ) n n ω n ω 1 For each n we have: 2 Tupling all these gives a map f n+1 ; Θ A,n+1 (A, f ) (A, f ) n (A, f ) n+1 (A, f )!(A, f ) n ω (A, f ) n+1 3 Finally pairing with 0 : (A, f )!(A, f ) gives (A, f )!(A, f )!(A, f ) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 21 / 35

56 Building Differential Categories Always works in the Karoubi Envelope Differential transformation d : (A, f ) (A, f ) n (A, f ) n n ω n ω 1 For each n we have: (A, f )!(A, f ) =; π n f n+1 ; Θ A,n+1 (A, f ) (A, f ) n (A, f ) n+1 2 Tupling all these gives a map (A, f )!(A, f ) n ω (A, f ) n+1 3 Finally pairing with 0 : (A, f )!(A, f ) gives (A, f )!(A, f )!(A, f ) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 21 / 35

57 Building Differential Categories Always works in the Karoubi Envelope Differential transformation d : (A, f ) (A, f ) n (A, f ) n n ω n ω 1 For each n we have: (A, f )!(A, f ) =; π n f n+1 ; Θ A,n+1 (A, f ) (A, f ) n (A, f ) n+1 2 Tupling all these gives a map (A, f )!(A, f ) n ω (A, f ) n+1 3 Finally pairing with 0 : (A, f )!(A, f ) gives (A, f )!(A, f )!(A, f ) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 21 / 35

58 Building Differential Categories Always works in the Karoubi Envelope Differential transformation d : (A, f ) (A, f ) n (A, f ) n n ω n ω 1 For each n we have: (A, f )!(A, f ) =; π n f n+1 ; Θ A,n+1 (A, f ) (A, f ) n (A, f ) n+1 2 Tupling all these gives a map (A, f )!(A, f ) n ω (A, f ) n+1 3 Finally pairing with 0 : (A, f )!(A, f ) gives (A, f )!(A, f )!(A, f ) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 21 / 35

59 Building Differential Categories Always works in the Karoubi Envelope What if I don t have sup-lattice enrichment or infinite biproducts? Differential transformation d : (A, f ) (A, f ) n (A, f ) n n ω n ω 1 For each n we have: (A, f )!(A, f ) =; π n f n+1 ; Θ A,n+1 (A, f ) (A, f ) n (A, f ) n+1 2 Tupling all these gives a map (A, f )!(A, f ) n ω (A, f ) n+1 3 Finally pairing with 0 : (A, f )!(A, f ) gives (A, f )!(A, f )!(A, f ) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 21 / 35

60 Free completions Building Differential Categories SMC C Giulio Manzonetto (RU) Differential Model Theory 11/06/11 22 / 35

61 Building Differential Categories Free completions SMC C + sup-lattice enrichment sup-lattice completion SMC C Giulio Manzonetto (RU) Differential Model Theory 11/06/11 22 / 35

62 Building Differential Categories Free completions SMC BP(C + ) sup-lattice enrichment countable biproducts biproduct completion SMC C + sup-lattice enrichment sup-lattice completion SMC C Giulio Manzonetto (RU) Differential Model Theory 11/06/11 22 / 35

63 Building Differential Categories Free completions SMC BP(C + ) sup-lattice enrichment countable biproducts biproduct completion Karoubi envelope SMC C + sup-lattice enrichment Differential category K(BP(C + )) sup-lattice enrichment sup-lattice completion co-kleisli SMC C Cartesian differential K! (BP(C + )) cpo-enriched Giulio Manzonetto (RU) Differential Model Theory 11/06/11 22 / 35

64 Building Differential Categories Easy way to Build Models of Resource PCF Resource PCF Simply Typed Resource Calculus + Constants for natural numbers (of ground type Nat) + Fixed Point Combinator Y, + If-zero? instruction. Operational semantics: Linear Head Reduction. Denotational Models Cartesian Closed Differential Categories, + Fixpoints, + (weak) Natural number object. Remark on the construction We recover MRel and build categories of games. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 23 / 35

65 Building Differential Categories Easy way to Build Models of Resource PCF Resource PCF Simply Typed Resource Calculus + Constants for natural numbers (of ground type Nat) + Fixed Point Combinator Y, + If-zero? instruction. Operational semantics: Linear Head Reduction. Denotational Models Cartesian Closed Differential Categories, + Fixpoints, + (weak) Natural number object. Remark on the construction We recover MRel and build categories of games. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 23 / 35

66 Building Differential Categories Example from SMCC 1 (1 object, 1 morphism) Terminal SMCC 1 Giulio Manzonetto (RU) Differential Model Theory 11/06/11 24 / 35

67 Building Differential Categories Example from SMCC 1 (1 object, 1 morphism) SMCC BP(1 + ) = Rel biproduct completion SMCC 1 + sup-lattice completion Terminal SMCC 1 Giulio Manzonetto (RU) Differential Model Theory 11/06/11 24 / 35

68 Building Differential Categories Example from SMCC 1 (1 object, 1 morphism) SMCC BP(1 + ) = Rel biproduct completion Karoubi envelope SMCC 1 + Differential category K(Rel) im(rel),! = M f sup-lattice completion co-kleisli Terminal SMCC 1 Cart. closed differential MRel K! (BP(C + )) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 24 / 35

69 Building Differential Categories Playing with games... Arena An arena A is given by: a finite bipartite forest over two sets of moves MA P and MO A, an edge relation (enabling), a labelling function (Questions and Answers). Strategies Given an arena A, a strategy over A is: a set of complete sequences (= every question is answered exactly once), the Opponent plays first (then they alternate), satisfying P-visibility and P-bracketing. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 25 / 35

70 Building Differential Categories Playing with games... Arena An arena A is given by: a finite bipartite forest over two sets of moves MA P and MO A, an edge relation (enabling), a labelling function (Questions and Answers). Strategies Given an arena A, a strategy over A is: a set of complete sequences (= every question is answered exactly once), the Opponent plays first (then they alternate), satisfying P-visibility and P-bracketing. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 25 / 35

71 Building Differential Categories A category G of games Category G of games [Harmer-McCusker] The category G: Objects: arenas whose roots are all O-moves, Hom(A,B): strategies on A B, Composition: usual parallel composition plus hiding construction, Identities: copycat strategies. = disjoint union, A B = Arena B with a copy of A attached below each initial move (to maintain the forest structure) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 26 / 35

72 Building Differential Categories A category G of games Category G of games [Harmer-McCusker] The monoidal closed category G: Objects: arenas whose roots are all O-moves, Hom(A,B): strategies on A B, Composition: usual parallel composition plus hiding construction, Identities: copycat strategies. = disjoint union, A B = Arena B with a copy of A attached below each initial move (to maintain the forest structure) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 26 / 35

73 Building Differential Categories The subcategory G Remark Every object of G can be endowed with a comonoid structure Comonoid homomorphisms = maps whose choice of move at any stage depends only on the current thread. Subcategory of comonoid homomorphisms Cartesian closed category G. Theorem [Harmer-McCusker 99] G is fully abstract for Erratic Idealized Algol. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 27 / 35

74 Building Differential Categories The subcategory G Remark Every object of G can be endowed with a comonoid structure Comonoid homomorphisms = maps whose choice of move at any stage depends only on the current thread. Subcategory of comonoid homomorphisms Cartesian closed category G. Theorem [Harmer-McCusker 99] G is fully abstract for Erratic Idealized Algol. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 27 / 35

75 Building Differential Categories G is a Cartesian Closed Differential Category Given a complete play (playing in A at least once): s : A B Its derivative s : A A B plays in the left A exactly once. Derivative combinator D (σ) = {s comp(a A B) s is a derivative of some s σ} The derivative is programmable in Erratic Idealized Algol. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 28 / 35

76 Building Differential Categories G is a Cartesian Closed Differential Category Given a complete play (playing in A at least once): s : A B Its derivative s : A A B plays in the left A exactly once. Derivative combinator D (σ) = {s comp(a A B) s is a derivative of some s σ} The derivative is programmable in Erratic Idealized Algol. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 28 / 35

77 Building Differential Categories Analysis of G Path: non-repeating enumeration of all moves, respects the order given by the edge relation in the arena, the first move is by O, moves alternate polarity thereafter. Exhausting strategy: set of even paths satisfying P-visibility. Category of Exhausting games The category EG: Objects: finite O-rooted arenas, Hom(A,B): exhausting strategy on A B, monoidal structure = disjoint union, has all R-exponentials A R, where R = arena with a single O-move. Let s apply (the second part of) our construction... Giulio Manzonetto (RU) Differential Model Theory 11/06/11 29 / 35

78 Building Differential Categories Analysis of G Path: non-repeating enumeration of all moves, respects the order given by the edge relation in the arena, the first move is by O, moves alternate polarity thereafter. Exhausting strategy: set of even paths satisfying P-visibility. Category of Exhausting games The category EG: Objects: finite O-rooted arenas, Hom(A,B): exhausting strategy on A B, monoidal structure = disjoint union, has all R-exponentials A R, where R = arena with a single O-move. Let s apply (the second part of) our construction... Giulio Manzonetto (RU) Differential Model Theory 11/06/11 29 / 35

79 Building Differential Categories Analysis of G Path: non-repeating enumeration of all moves, respects the order given by the edge relation in the arena, the first move is by O, moves alternate polarity thereafter. Exhausting strategy: set of even paths satisfying P-visibility. Category of Exhausting games The sup-lattice enriched category EG: Objects: finite O-rooted arenas, Hom(A,B): exhausting strategy on A B, monoidal structure = disjoint union, has all R-exponentials A R, where R = arena with a single O-move. Let s apply (the second part of) our construction... Giulio Manzonetto (RU) Differential Model Theory 11/06/11 29 / 35

80 Recovering G Building Differential Categories SMC EG Giulio Manzonetto (RU) Differential Model Theory 11/06/11 30 / 35

81 Recovering G Building Differential Categories SMC BP(EG) biproduct completion Karoubi envelope SMC EG Differential category K(BP(EG)) co-kleisli Cart. differential K! (BP(EG)) CCC R-exponentials Giulio Manzonetto (RU) Differential Model Theory 11/06/11 30 / 35

82 Recovering G Building Differential Categories SMC BP(EG) biproduct completion Karoubi envelope SMC EG Differential category K(BP(EG)) co-kleisli Category G full faithful functor Cart. differential K! (BP(EG)) CCC R-exponentials Giulio Manzonetto (RU) Differential Model Theory 11/06/11 30 / 35

83 Recovering G Building Differential Categories SMC BP(EG) biproduct completion Karoubi envelope SMC EG Differential category K(BP(EG)) co-kleisli Category G Model of Resource PCF full faithful functor Cart. differential K! (BP(EG)) CCC R-exponentials Giulio Manzonetto (RU) Differential Model Theory 11/06/11 30 / 35

84 Recovering G Building Differential Categories SMC BP(EG) biproduct completion Karoubi envelope SMC EG Differential category K(BP(EG)) co-kleisli Category G No full abstraction :( full faithful functor Cart. differential K! (BP(EG)) CCC R-exponentials Giulio Manzonetto (RU) Differential Model Theory 11/06/11 30 / 35

85 Building Differential Categories Refining We consider a switch equivalence relation: = least equivalence relation s o p o p t s o p o p t -closed strategies = causal independent strategies The category EG Objects: O-rooted arenas, Hom(A,B): deterministic -closed strategies. Remark Nondeterminism arise with the sup-lattice completion. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 31 / 35

86 Let s try again... Building Differential Categories SMC EG Giulio Manzonetto (RU) Differential Model Theory 11/06/11 32 / 35

87 Let s try again... Building Differential Categories SMC BP(EG ) biproduct completion Karoubi envelope SMC EG Differential category K(BP(EG )) co-kleisli Cart. differential K! (BP(EG )) CCC R-exponentials Giulio Manzonetto (RU) Differential Model Theory 11/06/11 32 / 35

88 Let s try again... Building Differential Categories SMC BP(EG ) biproduct completion Karoubi envelope SMC EG Differential category K(BP(EG )) co-kleisli Category G full faithful functor Cart. differential K! (BP(EG )) CCC R-exponentials Giulio Manzonetto (RU) Differential Model Theory 11/06/11 32 / 35

89 Let s try again... Building Differential Categories SMC BP(EG ) biproduct completion Karoubi envelope SMC EG Differential category K(BP(EG )) co-kleisli Category G Model of Resource PCF full faithful functor Cart. differential K! (BP(EG )) CCC R-exponentials Giulio Manzonetto (RU) Differential Model Theory 11/06/11 32 / 35

90 Let s try again... Building Differential Categories SMC BP(EG ) biproduct completion Karoubi envelope You re kidding, right? Why are you bothering us? SMC EG Differential category K(BP(EG )) co-kleisli Category G No full abstraction but... full faithful functor Cart. differential K! (BP(EG )) CCC R-exponentials Giulio Manzonetto (RU) Differential Model Theory 11/06/11 32 / 35

91 Let s try again... Building Differential Categories SMC BP(EG ) biproduct completion Karoubi envelope G has the finite definability property, moreover... SMC EG Differential category K(BP(EG )) co-kleisli Category G full faithful functor Cart. differential K! (BP(EG )) CCC R-exponentials Giulio Manzonetto (RU) Differential Model Theory 11/06/11 32 / 35

92 Building Differential Categories Full Abstraction for Resource PCF EG QA : Arenas where each question enables a unique answer (Unique) Full Functor : EG QA 1 Giulio Manzonetto (RU) Differential Model Theory 11/06/11 33 / 35

93 Building Differential Categories Full Abstraction for Resource PCF G embeds in K! (BP((EG QA ) + )) by construction! (Unique) Full Functor Full Functor : EG QA 1 F : K! (BP((EG QA ) + )) K! (Rel) (lifted) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 33 / 35

94 Building Differential Categories Full Abstraction for Resource PCF (Unique) Full Functor Full Functor Full Functor : EG QA 1 F : K! (BP((EG QA ) + )) K! (Rel) F : G MRel (lifted) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 33 / 35

95 Building Differential Categories Full Abstraction for Resource PCF (Unique) Full Functor Full Functor Full Functor : EG QA 1 F : K! (BP((EG QA ) + )) K! (Rel) F : G MRel (lifted) Finite Definability Property Both G and MRel have the finite definability property. (def. G def. MRel ) Corollary: MRel is Fully Abstract for Resource PCF Let M, N of type A. M N [ C( ) C(M) C(N) ] Giulio Manzonetto (RU) Differential Model Theory 11/06/11 33 / 35

96 Building Differential Categories Full Abstraction for Resource PCF Finite Definability Property Both G and MRel have the finite definability property. (def. G def. MRel ) Corollary: MRel is Fully Abstract for Resource PCF Let M, N of type A. M N [ C( ) C(M) C(N) ] Proof (sketch) Suppose a M N, By finite definability {(a, 0)} : A Nat denotes some term x : A C(x), Hence C(M) = zero and C(N) =, Thus C(M) but C(N) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 33 / 35

97 Conclusion Conclusions We have: Defined Cartesian closed differential categories, Shown they are sound and complete models of untyped differential λ-calculus, Provided a construction for building differential categories, Applied it to categories of games and categories of sets and relations, shown that MRel is fully abstract for Resource PCF. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 34 / 35

98 Conclusion Thanks for your attention! Questions? Giulio Manzonetto (RU) Differential Model Theory 11/06/11 35 / 35