A Differential Model Theory for Resource Lambda Calculi - Part I

Transcription

1 A Differential Model Theory for Resource Lambda Calculi - Part I Giulio Manzonetto 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 / 29

2 Introduction The resource calculus Λ r Non-lazy version of Boudol s resource calculus Λ r etends the notion of the λ-calculus application along two directions: MN 1 a term is applied to a multiset of resources, called bag of resources 2 the resources can be reusable (available at will) or linear (to be used once) Ancestors: the λ-calculus with multiplicities by Gérard Boudol (1993) introduced to study the observational semantics induced on the lazy λ-calculus by Milner s translation into the π-calculus; the differential λ-calculus by Thomas Ehrhard and Laurent Regnier (2003) designed starting from a denotational model of linear logic. Synta formalized by Paolo Tranquilli (2008) it corresponds to the minimal fragment of the differential etension of linear logic by Thomas Ehrhard and Laurent Regnier (2006) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 2 / 29

3 Introduction The resource calculus Λ r Non-lazy version of Boudol s resource calculus Λ r etends the notion of the λ-calculus application along two directions: M[N 1,..., N n ] 1 a term is applied to a multiset of resources, called bag of resources 2 the resources can be reusable (available at will) or linear (to be used once) Ancestors: the λ-calculus with multiplicities by Gérard Boudol (1993) introduced to study the observational semantics induced on the lazy λ-calculus by Milner s translation into the π-calculus; the differential λ-calculus by Thomas Ehrhard and Laurent Regnier (2003) designed starting from a denotational model of linear logic. Synta formalized by Paolo Tranquilli (2008) it corresponds to the minimal fragment of the differential etension of linear logic by Thomas Ehrhard and Laurent Regnier (2006) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 2 / 29

4 Introduction The resource calculus Λ r Non-lazy version of Boudol s resource calculus Λ r etends the notion of the λ-calculus application along two directions: M[L 1,..., L l, N! 1,..., N! n] 1 a term is applied to a multiset of resources, called bag of resources 2 the resources can be reusable (available at will) or linear (to be used once) Ancestors: the λ-calculus with multiplicities by Gérard Boudol (1993) introduced to study the observational semantics induced on the lazy λ-calculus by Milner s translation into the π-calculus; the differential λ-calculus by Thomas Ehrhard and Laurent Regnier (2003) designed starting from a denotational model of linear logic. Synta formalized by Paolo Tranquilli (2008) it corresponds to the minimal fragment of the differential etension of linear logic by Thomas Ehrhard and Laurent Regnier (2006) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 2 / 29

5 Introduction The resource calculus Λ r Non-lazy version of Boudol s resource calculus Λ r etends the notion of the λ-calculus application along two directions: M[L 1,..., L l, N! 1,..., N! n] 1 a term is applied to a multiset of resources, called bag of resources 2 the resources can be reusable (available at will) or linear (to be used once) Ancestors: the λ-calculus with multiplicities by Gérard Boudol (1993) introduced to study the observational semantics induced on the lazy λ-calculus by Milner s translation into the π-calculus; the differential λ-calculus by Thomas Ehrhard and Laurent Regnier (2003) designed starting from a denotational model of linear logic. Synta formalized by Paolo Tranquilli (2008) it corresponds to the minimal fragment of the differential etension of linear logic by Thomas Ehrhard and Laurent Regnier (2006) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 2 / 29

6 Introduction The resource calculus Λ r Non-lazy version of Boudol s resource calculus Λ r etends the notion of the λ-calculus application along two directions: M[L 1,..., L l, N! 1,..., N! n] 1 a term is applied to a multiset of resources, called bag of resources 2 the resources can be reusable (available at will) or linear (to be used once) Ancestors: the λ-calculus with multiplicities by Gérard Boudol (1993) introduced to study the observational semantics induced on the lazy λ-calculus by Milner s translation into the π-calculus; the differential λ-calculus by Thomas Ehrhard and Laurent Regnier (2003) designed starting from a denotational model of linear logic. Synta formalized by Paolo Tranquilli (2008) it corresponds to the minimal fragment of the differential etension of linear logic by Thomas Ehrhard and Laurent Regnier (2006) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 2 / 29

7 Introduction Just another synta for Differential Lambda Calculus Precise link between: Resource Calculus Differential λ-calculus linear resources syntactic differentiation M[L 1,..., L l, N! 1,..., N! n] D l (M) (L 1,..., L l )( i N i) M[L 1,..., L l ] D l (M) (L 1,..., L l )(0) Taylor Epansion Formula (MN) 1 = n! Dn (M) (N,..., N)(0) }{{} n=0 n times Giulio Manzonetto (RU) Differential Model Theory 11/06/11 3 / 29

8 Introduction Just another synta for Differential Lambda Calculus Precise link between: Resource Calculus Differential λ-calculus linear resources syntactic differentiation M[L 1,..., L l, N! 1,..., N! n] D l (M) (L 1,..., L l )( i N i) M[L 1,..., L l ] D l (M) (L 1,..., L l )(0) Taylor Epansion Formula (MN) 1 = n! Dn (M) (N,..., N)(0) }{{} n=0 n times Giulio Manzonetto (RU) Differential Model Theory 11/06/11 3 / 29

9 The synta of Λ r There are three syntactic categories: terms are in functional positions, bags of resources are in argument position and represent multisets of linear and reusable terms, sums of terms are results of the computation. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 4 / 29

10 The synta of Λ r There are three syntactic categories: terms are in functional positions, bags of resources are in argument position and represent multisets of linear and reusable terms, sums of terms are results of the computation. Formally: M, N, L := λ.m MP terms P, Q, R := [] [M] [M! ] P Q bags M, N := 0 M M + N sums Giulio Manzonetto (RU) Differential Model Theory 11/06/11 4 / 29

11 The synta of Λ r It contains λ-calculus There are three syntactic categories: terms are in functional positions, bags of resources are in argument position and represent multisets of linear and reusable terms, sums of terms are results of the computation. Λ r λ-calculus Formally: M, N, L := λ.m MP terms P, Q, R := [] [M] [M! ] P Q bags M, N := 0 M M + N sums Giulio Manzonetto (RU) Differential Model Theory 11/06/11 4 / 29

12 The synta of Λ r It contains λ-calculus and a nondeterministic etension of λ-calculus There are three syntactic categories: terms are in functional positions, bags of resources are in argument position and represent multisets of linear and reusable terms, sums of terms are results of the computation. Λ r λ-calculus eponential fragment Formally: M, N, L := λ.m MP terms P, Q, R := [] [M] [M! ] P Q bags M, N := 0 M M + N sums Giulio Manzonetto (RU) Differential Model Theory 11/06/11 4 / 29

13 The synta of Λ r It contains λ-calculus and a nondeterministic etension of λ-calculus and a finite resource calculus There are three syntactic categories: terms are in functional positions, bags of resources are in argument position and represent multisets of linear and reusable terms, sums of terms are results of the computation. Λ r λ-calculus eponential fragment Formally: M, N, L := λ.m MP terms P, Q, R := [] [M] [M! ] P Q bags M, N := 0 M M + N sums Λ r f finite fragment Giulio Manzonetto (RU) Differential Model Theory 11/06/11 4 / 29

14 Will we have sums everywhere? Nope! All operators are linear... ecept the ( )! : λ.( i M i) := i λ.m i ( i M i)p := i M ip M( i P i) := i MP i M([ i N i] P) := i M([N i] P) M([( i N i)! ] P) := M([N! 1,..., N! n] P) 0 annihilates everything (ecept under ( )! ) λ.0 = 0 M([0] P) = 0 0P = 0 M([0! ] P) = MP Giulio Manzonetto (RU) Differential Model Theory 11/06/11 5 / 29

15 Two kind of substitutions Usual and Linear Substitution Two kinds of resources two kinds of substitution: M {N/} : usual capture free substitution, M N/ : linear substitution, N is substituted for eactly one linear occurrence of in M. M N! Usual Substitution substitute all occurrences! Giulio Manzonetto (RU) Differential Model Theory 11/06/11 6 / 29

16 Two kind of substitutions Usual and Linear Substitution Two kinds of resources two kinds of substitution: M {N/} : usual capture free substitution, M N/ : linear substitution, N is substituted for eactly one linear occurrence of in M. M N! Usual Substitution substitute all occurrences! Giulio Manzonetto (RU) Differential Model Theory 11/06/11 6 / 29

17 Two kind of substitutions Usual and Linear Substitution Two kinds of resources two kinds of substitution: M {N/} : usual capture free substitution, M N/ : linear substitution, N is substituted for eactly one linear occurrence of in M. M N M{N/}! =! Usual Substitution substitute all occurrences! Giulio Manzonetto (RU) Differential Model Theory 11/06/11 6 / 29

18 Two kind of substitutions Usual and Linear Substitution Two kinds of resources two kinds of substitution: M {N/} : usual capture free substitution, M N/ : linear substitution, N is substituted for eactly one linear occurrence of in M. M N M N/! = normal Linear Substitution (idea) choose non-deterministically one linear occurrence at time ( = differentiation) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 6 / 29

19 Two kind of substitutions Usual and Linear Substitution Two kinds of resources two kinds of substitution: M {N/} : usual capture free substitution, M N/ : linear substitution, N is substituted for eactly one linear occurrence of in M. M N M N/! =! normal Linear Substitution (idea) choose non-deterministically one linear occurrence at time ( = differentiation) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 6 / 29

20 Two kind of substitutions Usual and Linear Substitution Two kinds of resources two kinds of substitution: M {N/} : usual capture free substitution, M N/ : linear substitution, N is substituted for eactly one linear occurrence of in M. M N M N/! =! normal Linear Substitution (idea) choose non-deterministically one linear occurrence at time ( = differentiation) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 6 / 29

21 Two kind of substitutions Usual and Linear Substitution Two kinds of resources two kinds of substitution: M {N/} : usual capture free substitution, M N/ : linear substitution, N is substituted for eactly one linear occurrence of in M. M N M N/! =! normal Linear Substitution (idea) choose non-deterministically one linear occurrence at time ( = differentiation) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 6 / 29

22 Two kind of substitutions Usual and Linear Substitution Two kinds of resources two kinds of substitution: M {N/} : usual capture free substitution, M N/ : linear substitution, N is substituted for eactly one linear occurrence of in M. M N M N/! =! normal Linear Substitution (idea) choose non-deterministically one linear occurrence at time ( = differentiation) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 6 / 29

23 Two kind of substitutions Usual and Linear Substitution Two kinds of resources two kinds of substitution: M {N/} : usual capture free substitution, M N/ : linear substitution, N is substituted for eactly one linear occurrence of in M. M N M N/! =! normal Linear Substitution (idea) choose non-deterministically one linear occurrence at time ( = differentiation) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 6 / 29

24 Two kind of substitutions Usual and Linear Substitution Two kinds of resources two kinds of substitution: M {N/} : usual capture free substitution, M N/ : linear substitution, N is substituted for eactly one linear occurrence of in M. M N M N/! =! normal Linear Substitution (idea) choose non-deterministically one linear occurrence at time ( = differentiation) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 6 / 29

25 Linear Substitution (formally) M N/ : linear substitution On terms: On Bags: y N/ = { N if = y 0 otherwise (λy.m) N/ = λy.m N/ (MP) N/ = M N/ P + M(P N/ ) [] N/ = 0 [M] N/ = [M N/ ] [M! ] N/ = [M N/, M! ] (P R) N/ = P N/ R + P (R N/ ) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 7 / 29

26 The operational semantics of Λ r β- and η- reductions The β-reduction: (λ.m)[l 1,..., L l, N! 1,..., N! n] β M L 1 / L l / {Σ i N i /} The η-reduction: λ.m[! ] η M, where / FV(M) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 8 / 29

27 The operational semantics of Λ r β- and η- reductions The β-reduction: (λ.m)[l 1,..., L l, N! 1,..., N! n] β M L 1 / L l / {Σ i N i /} The η-reduction: λ.m[! ] η M, where / FV(M) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 8 / 29

28 The operational semantics of Λ r β- and η- reductions The β-reduction: (λ.m)[l 1,..., L l, N! 1,..., N! n] β M L 1 / L l / {Σ i N i /} The η-reduction: λ.m[! ] η M, where / FV(M) Eample Let I := λ.: (λ.)[i] I/ {0/} I {0/} I (λ.)[] {0/} 0 (λ.)[i, I] I/ I/ {0/} I I/ {0/} 0 (λ.y[][][])[i, z! ] β y[i][z][z] + y[z][i][z] + y[z][z][i] nice term starvation surfeit nondeterminism Giulio Manzonetto (RU) Differential Model Theory 11/06/11 8 / 29

29 The operational semantics of Λ r β- and η- reductions The β-reduction: (λ.m)[l 1,..., L l, N! 1,..., N! n] β M L 1 / L l / {Σ i N i /} The η-reduction: λ.m[! ] η M, where / FV(M) Eample Let I := λ.: (λ.)[i] I/ {0/} I {0/} I (λ.)[] {0/} 0 (λ.)[i, I] I/ I/ {0/} I I/ {0/} 0 (λ.y[][][])[i, z! ] β y[i][z][z] + y[z][i][z] + y[z][z][i] nice term starvation surfeit nondeterminism Giulio Manzonetto (RU) Differential Model Theory 11/06/11 8 / 29

30 The operational semantics of Λ r β- and η- reductions The β-reduction: (λ.m)[l 1,..., L l, N! 1,..., N! n] β M L 1 / L l / {Σ i N i /} The η-reduction: λ.m[! ] η M, where / FV(M) Eample Let I := λ.: (λ.)[i] I/ {0/} I {0/} I (λ.)[] {0/} 0 (λ.)[i, I] I/ I/ {0/} I I/ {0/} 0 (λ.y[][][])[i, z! ] β y[i][z][z] + y[z][i][z] + y[z][z][i] nice term starvation surfeit nondeterminism Giulio Manzonetto (RU) Differential Model Theory 11/06/11 8 / 29

31 The operational semantics of Λ r β- and η- reductions The β-reduction: (λ.m)[l 1,..., L l, N! 1,..., N! n] β M L 1 / L l / {Σ i N i /} The η-reduction: λ.m[! ] η M, where / FV(M) Eample Let I := λ.: (λ.)[i] I/ {0/} I {0/} I (λ.)[] {0/} 0 (λ.)[i, I] I/ I/ {0/} I I/ {0/} 0 (λ.y[][][])[i, z! ] β y[i][z][z] + y[z][i][z] + y[z][z][i] nice term starvation surfeit nondeterminism Giulio Manzonetto (RU) Differential Model Theory 11/06/11 8 / 29

32 The operational semantics of Λ r β- and η- reductions The β-reduction: (λ.m)[l 1,..., L l, N! 1,..., N! n] β M L 1 / L l / {Σ i N i /} The η-reduction: λ.m[! ] η M, where / FV(M) Theorem [Pagani-Tranquilli APLAS 09] β is confluent. β enjoys a standardization property. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 8 / 29

33 Simple Type System Resource Calculus Types: σ, τ ::= α σ τ (R) Γ() = σ Γ R : σ (Rλ) Γ, : σ R M : τ Γ R λ.m : σ τ (R@) Γ R M : σ τ Γ R P : σ Γ R MP : τ (Rb) Γ R N : σ Γ R P : σ Γ R [N (!) ] P : σ (R[]) Γ R [] : σ (R+) Γ R A i : σ for all i Γ R i A i : σ Remark Sums and bags are typed uniformly... Giulio Manzonetto (RU) Differential Model Theory 11/06/11 9 / 29

34 The differential λ-calculus: Synta Differential Lambda Terms: Reduction Rules ( D = β βd ): s, t, u ::= λ.s st D(s) t S, T, U ::= s s + T 0 (β) (λ.s)t β s{t/} (β D ) D(λ.s) t βd λ. s t Ideas: st = usual application of λ-calculus ( = s[t! ]) D( (D(s) t 1 ) ) t k = linear application ( = s[t 1,..., t k ]) s t = differential substitution ( = s t/ ) (su) t = ( s U t)u + (D(s) ( t))u ( = (s[u! ]) t/ = s t/ [U! ] + s[u t/, U! ]) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 10 / 29

35 The differential λ-calculus: Synta Differential Lambda Terms: Reduction Rules ( D = β βd ): s, t, u ::= λ.s st D(s) t S, T, U ::= s s + T 0 (β) (λ.s)t β s{t/} (β D ) D(λ.s) t βd λ. s t Ideas: st = usual application of λ-calculus ( = s[t! ]) D( (D(s) t 1 ) ) t k = linear application ( = s[t 1,..., t k ]) s t = differential substitution ( = s t/ ) (su) t = ( s U t)u + (D(s) ( t))u ( = (s[u! ]) t/ = s t/ [U! ] + s[u t/, U! ]) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 10 / 29

36 The differential λ-calculus: Synta Differential Lambda Terms: Reduction Rules ( D = β βd ): s, t, u ::= λ.s st D(s) t S, T, U ::= s s + T 0 (β) (λ.s)t β s{t/} (β D ) D(λ.s) t βd λ. s t Ideas: st = usual application of λ-calculus ( = s[t! ]) (D( (D(s) t 1 ) ) t k )0 = linear application ( = s[t 1,..., t k ]) s t = differential substitution ( = s t/ ) (su) t = ( s U t)u + (D(s) ( t))u ( = (s[u! ]) t/ = s t/ [U! ] + s[u t/, U! ]) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 10 / 29

37 The differential λ-calculus: Synta Differential Lambda Terms: Reduction Rules ( D = β βd ): s, t, u ::= λ.s st D(s) t S, T, U ::= s s + T 0 (β) (λ.s)t β s{t/} (β D ) D(λ.s) t βd λ. s t Ideas: st = usual application of λ-calculus ( = s[t! ]) D( (D(s) t 1 ) ) t k = linear application ( = s[t 1,..., t k ]) s t = differential substitution ( = s t/ ) (su) t = ( s U t)u + (D(s) ( t))u ( = (s[u! ]) t/ = s t/ [U! ] + s[u t/, U! ]) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 10 / 29

38 Translation between the two calculi We can define a translation map =, (λ.m) = λ.m, ( ) : Resource calculus Differential λ-calculus (M[ L, N! ]) = (D k (M ) L 1 L k )( i N i ), 0 = 0, ( i M i) = i M i. The translation is faithful For M, N resource terms: M β N implies M D N Giulio Manzonetto (RU) Differential Model Theory 11/06/11 11 / 29

39 Simple Types in Differential Calculus Γ() = σ Γ D : σ λ Γ; : σ D s : τ Γ D λ.s : σ Γ D s : σ τ Γ D t : σ Γ D st : τ D Γ D s : σ τ Γ D t : σ Γ D D(s) t : σ τ 0 Γ D 0 : σ sum Γ D s i : σ for all i Γ D i s i : σ Remark: Linear application does not decrease types. The translation remains faithful Let M be a resource term. If Γ R M : σ then Γ D M : σ Corollary Every model of the (typed/untyped) differential λ-calculus will also be a model of the resource calculus. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 12 / 29

40 Simple Types in Differential Calculus Γ() = σ Γ D : σ λ Γ; : σ D s : τ Γ D λ.s : σ Γ D s : σ τ Γ D t : σ Γ D st : τ D Γ D s : σ τ Γ D t : σ Γ D D(s) t : σ τ 0 Γ D 0 : σ sum Γ D s i : σ for all i Γ D i s i : σ Remark: Linear application does not decrease types. The translation remains faithful Let M be a resource term. If Γ R M : σ then Γ D M : σ Corollary Every model of the (typed/untyped) differential λ-calculus will also be a model of the resource calculus. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 12 / 29

41 Taylor Epansion: intuition Lambda Calculus: Taylor Epansion Formula For λ-terms M, N we have (MN) 1 = n! (Dn (M) (N,..., N))(0) }{{} n=0 n times Giulio Manzonetto (RU) Differential Model Theory 11/06/11 13 / 29

42 Taylor Epansion: intuition Lambda Calculus: Taylor Epansion Formula For λ-terms M, N we have (MN) 1 = n! M[N, }. {{.., N ] } n=0 n times Etension: From λ-calculus to resource calculus... Giulio Manzonetto (RU) Differential Model Theory 11/06/11 13 / 29

43 Taylor Epansion: intuition Lambda Calculus: Taylor Epansion Formula For λ-terms M, N we have (MN) = {M[N,..., N]} }{{} n times n=0 Etension: From λ-calculus to resource calculus... For the sake of simplicity we consider an idempotent sum Giulio Manzonetto (RU) Differential Model Theory 11/06/11 13 / 29

44 Resource Calculus: Full Taylor Epansion ( ) Λ r P(Λ r f ) The (support of the full) Taylor Epansion M of a term M: = {}, (λ.m) = λ.m (= {λ.m : M M }), (M[L 1,..., L l, N! 1,..., N! n]) = P Mf ( i N i )M ([L 1,..., L l ] P), It is etended to sums M by setting: ( i M i) = i M i. Eample M := (λz.[z! ]) = {λz.[z n ] : n Nat}, N := (λz.[] + λz.[z, z! ]) = {λz.[]} {λz.[z n+1 ] : n Nat}. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 14 / 29

45 Resource Calculus: Full Taylor Epansion ( ) Λ r P(Λ r f ) The (support of the full) Taylor Epansion M of a term M: = {}, (λ.m) = λ.m (= {λ.m : M M }), (M[L 1,..., L l, N! 1,..., N! n]) = P Mf ( i N i )M ([L 1,..., L l ] P), It is etended to sums M by setting: ( i M i) = i M i. Eample M := (λz.[z! ]) = {λz.[z n ] : n Nat}, N := (λz.[] + λz.[z, z! ]) = {λz.[]} {λz.[z n+1 ] : n Nat}. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 14 / 29

46 Resource Calculus: Full Taylor Epansion ( ) Λ r P(Λ r f ) The (support of the full) Taylor Epansion M of a term M: = {}, (λ.m) = λ.m (= {λ.m : M M }), (M[L 1,..., L l, N! 1,..., N! n]) = P Mf ( i N i )M ([L 1,..., L l ] P), It is etended to sums M by setting: ( i M i) = i M i. Eample M := (λz.[z! ]) = {λz.[z n ] : n Nat}, N := (λz.[] + λz.[z, z! ]) = {λz.[]} {λz.[z n+1 ] : n Nat}. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 14 / 29

47 What about semantics? Semantics The differential/resource calculi are born from the analysis of the semantics of LL... however their semantical investigations are only at the beginning! Categorical description: Giulio Manzonetto (RU) Differential Model Theory 11/06/11 15 / 29

48 What about semantics? Semantics The differential/resource calculi are born from the analysis of the semantics of LL... however their semantical investigations are only at the beginning! Categorical description: Giulio Manzonetto (RU) Differential Model Theory 11/06/11 15 / 29

49 What about semantics? Semantics The differential/resource calculi are born from the analysis of the semantics of LL... however their semantical investigations are only at the beginning! Categorical description: λ-calculus CCC Giulio Manzonetto (RU) Differential Model Theory 11/06/11 15 / 29

50 What about semantics? Semantics The differential/resource calculi are born from the analysis of the semantics of LL... however their semantical investigations are only at the beginning! Categorical description: λ-calculus CCC cokleisli SMCC Giulio Manzonetto (RU) Differential Model Theory 11/06/11 15 / 29

51 What about semantics? Semantics The differential/resource calculi are born from the analysis of the semantics of LL... however their semantical investigations are only at the beginning! Categorical description: differential λ-calculus CCC + differential structure cokleisli SMCC + differential structure Giulio Manzonetto (RU) Differential Model Theory 11/06/11 15 / 29

52 What about semantics? Semantics The differential/resource calculi are born from the analysis of the semantics of LL... however their semantical investigations are only at the beginning! Categorical description: differential λ-calculus CCC + differential structure cokleisli Differential Categories (Blute, Cockett & Seely 06) SMCC + differential structure Aiomatic characterization of a derivative operator in (possibly non-closed) Symmetric Monoidal Categories + the! is not necessarily monoidal. SMCCs + monoidal! constitute interesting instances. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 15 / 29

53 (Monoidal) Differential Categories (Monoidal) Differential Categories Additive Symmetric Monoidal Categories Sum on terms sum on morphisms A symmetric monoidal category is additive if all homsets are enriched with commutative monoids: (f + g); h = f ; h + g; h h; (f + g) = h; f + h; g 0; f = 0 = f ; 0 The tensor product preserves the commutative monoid structure: (f + g) h = f h + g h 0 f = 0 Remark Additive becomes left additive in the cokleisli. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 16 / 29

54 (Monoidal) Differential Categories (Monoidal) Differential Categories Coalgebra Modalities A comonad (!, δ, ε) is a coalgebra modality if each!a comes equipped with a natural coalgebra structure :!A!A!A, 1 (!,, e) is a comonoid e :!A, such that: Id e Id!A!A Id!A Id e!a!a!a!a!a!a!a Id Id!A!A!A 2 δ is a morphism of comonoids:!a δ!!a!a δ!!a e e!a!a δ δ!!a!!a Giulio Manzonetto (RU) Differential Model Theory 11/06/11 17 / 29

55 (Monoidal) Differential Categories (Monoidal) Differential Categories Coalgebra Modalities A comonad (!, δ, ε) is a coalgebra modality if each!a comes equipped with a natural coalgebra structure :!A!A!A, 1 (!,, e) is a comonoid e :!A, such that: Id e Id!A!A Id!A Id e!a!a!a!a!a!a!a Id Id!A!A!A 2 δ is a morphism of comonoids:!a δ!!a!a δ!!a e e!a!a δ δ!!a!!a Giulio Manzonetto (RU) Differential Model Theory 11/06/11 17 / 29

56 (Monoidal) Differential Categories (Monoidal) Differential Categories Coalgebra Modalities A comonad (!, δ, ε) is a coalgebra modality if each!a comes equipped with a natural coalgebra structure :!A!A!A, 1 (!,, e) is a comonoid e :!A, such that: Id e Id!A!A Id!A Id e!a!a!a!a!a!a!a Id Id!A!A!A 2 δ is a morphism of comonoids:!a δ!!a!a δ!!a e e!a!a δ δ!!a!!a Giulio Manzonetto (RU) Differential Model Theory 11/06/11 17 / 29

57 (Monoidal) Differential Categories (Monoidal) Differential Categories Coalgebra Modalities A comonad (!, δ, ε) is a coalgebra modality if each!a comes equipped with a The! is not (necessarily) monoidal! natural coalgebra structure :!A!A!A, e :!A, such that: 1 (!,, e) is a comonoid Id e Id!A!A Id!A Id e!a!a!a!a!a!a!a Id Id!A!A!A 2 δ is a morphism of comonoids:!a δ!!a!a δ!!a e e!a!a δ δ!!a!!a Giulio Manzonetto (RU) Differential Model Theory 11/06/11 17 / 29

58 (Monoidal) Differential Categories (Monoidal) Differential Categories Differential Combinator Intuitions: A map f : A B = linear map, A cokleisli map f :!A B = abstract differentiable map from A to B, cokleisli C! = category of abstract differentiable maps. Differential Combinator D : C(A, B) C(A!A, B) f :!A B D (f ) : A!A B D that must satisfy suitable equations... (see later) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 18 / 29

59 (Monoidal) Differential Categories (Monoidal) Differential Categories Differential Combinator - Additivity & functoriality Additivity: D (0) = 0, D (f + g) = D (f ) + D (g). Functoriality:!A f B A!A D (f ) B!u v u!u v!c g D C!C D (g) D Giulio Manzonetto (RU) Differential Model Theory 11/06/11 19 / 29

60 (Monoidal) Differential Categories (Monoidal) Differential Categories Differential Combinator - Aiom D1 [D1] Constant maps: D (e A ) = 0 Giulio Manzonetto (RU) Differential Model Theory 11/06/11 20 / 29

61 (Monoidal) Differential Categories (Monoidal) Differential Categories Differential Combinator - Aiom D1 [D1] Constant maps: D (e A ) = 0 Constant functions have derivative 0. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 20 / 29

62 (Monoidal) Differential Categories (Monoidal) Differential Categories Differential Combinator - Aiom D2 [D2] Product rule: D ( ; (f g)) = (Id ); (D (f ) g) + (Id ); (f D (g)) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 21 / 29

63 (Monoidal) Differential Categories (Monoidal) Differential Categories Differential Combinator - Aiom D2 [D2] Product rule: D ( ; (f g)) = (Id ); (D (f ) g) + (Id ); (f D (g)) The tensor of two functions on the same arguments is morally the product of two functions. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 21 / 29

64 (Monoidal) Differential Categories (Monoidal) Differential Categories Differential Combinator - Aiom D3 [D3] Linear maps: D (ε A ; f ) = (Id e A ); f Giulio Manzonetto (RU) Differential Model Theory 11/06/11 22 / 29

65 (Monoidal) Differential Categories (Monoidal) Differential Categories Differential Combinator - Aiom D3 [D3] Linear maps: D (ε A ; f ) = (Id e A ); f The derivative of a map which is linear is constant. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 22 / 29

66 (Monoidal) Differential Categories (Monoidal) Differential Categories Differential Combinator - Aiom D4 [D4] The chain rule: D (δ;!f ; g) = (Id ); (D (f ) δ;!f ); D (g) Giulio Manzonetto (RU) Differential Model Theory 11/06/11 23 / 29

67 (Monoidal) Differential Categories (Monoidal) Differential Categories Differential Combinator - Aiom D4 [D4] The chain rule: D (δ;!f ; g) = (Id ); (D (f ) δ;!f ); D (g) The derivative of the composite of f and g is the derivative of f composed with the derivative of g Giulio Manzonetto (RU) Differential Model Theory 11/06/11 23 / 29

68 (Monoidal) Differential Categories Differential Combinator - a simpler characterization The whole differential structure is generated by the derivative of the identity!!a Id!A!A A!A D (Id!A )!A!Id A f Id A!Id A f!a f!a A!A D (f ) d A = D (Id!A ) is a deriving transformation, namely a transformation (natural in A) satisfying [D1-D4] rephrased.!a Giulio Manzonetto (RU) Differential Model Theory 11/06/11 24 / 29

69 (Monoidal) Differential Categories Differential Categories - a simpler characterization This is the reason why the differential bo is bottomless : Differential Category - Def. 2 A differential category is an additive symmetric monoidal category with a deriving transformation d A : A!A!A. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 25 / 29

70 (Monoidal) Differential Categories Eamples of Differential Categories Finiteness Spaces, Sets and relations + the bag functor M f ( ) (finite multisets): d A : A!A!A : a 0, [a 1,..., a n ] [a 0, a 1,..., a n ] Sup-lattices + dual of the free -algebra (see later) Vector spaces op K + opposite of free commutative algebra monad Convenient differential category Harmer-McCusker s category of games Other categories of games... Giulio Manzonetto (RU) Differential Model Theory 11/06/11 26 / 29

71 (Monoidal) Differential Categories Differential Storage Categories Storage modality The comonad (!, δ, e) is a storage modality if: it is symmetric monoidal, (!A,, e) commutative monoids, the comonoid is a morphism of the coalgebras for the comonad. Differential storage category A storage differential category is a differential category if: it has products, it has a storage modality. Theorem [Blute, Cockett, Seely 09] The cokleisly C! of a (monoidal closed) differential storage category C is a Cartesian (closed) differential category. What s that? We will see after the break... Giulio Manzonetto (RU) Differential Model Theory 11/06/11 27 / 29

72 (Monoidal) Differential Categories Differential Storage Categories Storage modality The comonad (!, δ, e) is a storage modality if: it is symmetric monoidal, (!A,, e) commutative monoids, the comonoid is a morphism of the coalgebras for the comonad. Differential storage category A storage differential category is a differential category if: it has products, it has a storage modality. Theorem [Blute, Cockett, Seely 09] The cokleisly C! of a (monoidal closed) differential storage category C is a Cartesian (closed) differential category. What s that? We will see after the break... Giulio Manzonetto (RU) Differential Model Theory 11/06/11 27 / 29

73 Conclusion Conclusions We have: Defined the resource calculus Λ r, Shown the relationship with the differential λ-calculus, Eplained the notion of a (monoidal) differential category (not enough to model differential λ-calculus!) After the break we will: Introduced the Cartesian closed differential categories, Show that they model the differential λ-calculus the resource calculus, Give types/untyped models modeling the Taylor epansion, Give a canonical construction SMC Differential CCC, Apply the construction to categories of games full abstraction of MRel for Resource PCF. Giulio Manzonetto (RU) Differential Model Theory 11/06/11 28 / 29

74 Conclusion Thanks for your attention! Questions? Giulio Manzonetto (RU) Differential Model Theory 11/06/11 29 / 29