Relational Graph Models, Taylor Expansion and Extensionality

Transcription

1 Relational Graph Models, Taylor Expansion and Extensionality Domenico Ruoppolo Giulio Manzonetto Laboratoire d Informatique de Paris Nord Université Paris-Nord Paris 13 (France) MFPS XXX Ithaca, New York 15 th June 2014

2 Relational Graph Models, Taylor Expansion and Extensionality Domenico Ruoppolo Giulio Manzonetto Laboratoire d Informatique de Paris Nord Université Paris-Nord Paris 13 (France) MFPS XXX Ithaca, New York 15 th June 2014

3 Observational equivalence of programs

4 Observational equivalence of programs P 1 P 2

5 Observational equivalence of programs C C P 1 P 2

6 Observational equivalence of programs C C P 1 P 2 C(P 1 ) C(P 2 )

7 Observational equivalence of programs C C P 1 O P 2 C(P 1 ) C(P 2 ) O

8 Observational equivalence of programs C C P 1 hnf P 2 C(P 1 ) C(P 2 ) hnf s

9 Morris s observational equivalence for the untyped λ-calculus M nf N for all context C( ) C(M) has a β-normal form iff C(N) has a β-normal form

10 Characterizing observational equivalences Observational equivalences denotational semantics syntactic characterizations

11 Resource-sensitive characterizations for Morris Observational equivalences Morris s equivalence??? Resource-sensitive denotational semantics??? Resource-sensitive syntactic characterizations

12 x 1 x 2 x n [[M]] black box o

13 x 1 x 2 x n x 1 x 1 x n x 1 o What about resource usage of the execution?

14 Characterizing observational equivalences Observational equivalences denotational semantics syntactic characterizations

15 Characterizing observational equivalences Observational equivalences Böhm-trees denotational semantics syntactic characterizations [ Barendregt 1981 ]

16 Characterizing Morris s equivalence observational equivalences Morris s equivalence denotational semantics syntactic characterizations [ Morris 1968, Hyland 1975 ]

17 Characterizing Morris s equivalence observational equivalences Morris s equivalence denotational semantics extensional Böhm trees syntactic characterizations [ Hyland 1975, Levy 1976 ]

18 Characterizing Morris s equivalence observational equivalences Morris s equivalence filter model D cdz denotational semantics extensional Böhm trees syntactic characterizations [ Coppo, Dezani & Zacchi 1987 ]

19 Böhm tree Output of the program The key idea Approximations of the β-normal form of the term

20 Böhm tree Output of the program The key idea Approximations of the β-normal form of the term Taylor expansion Resource management of the program T (MN) = k N M [ k times {}}{ N,...,N ] [ Ehrhard & Regnier 2003 ]

21 (Linear) Resource Calculus t ::= x λx.t t b b ::= [t 1,...,t n ] where n 0

22 (Linear) Resource Calculus Terms t ::= x λx.t t b b ::= [t 1,...,t n ] where n 0

23 (Linear) Resource Calculus t ::= x λx.t t b b ::= [t 1,...,t n ] where n 0

24 (Linear) Resource Calculus t ::= x λx.t t b b ::= [t 1,...,t n ] where n 0 Bags

25 (Linear) Resource Calculus t ::= x λx.t t b b ::= [t 1,...,t n ] where n 0 (λxλy.t)[s 11,s 12,s 13 ][s 21 ] x y x y x x o

26 (Linear) Resource Calculus t ::= x λx.t t b b ::= [t 1,...,t n ] where n 0 (λxλy.t)[s 11,s 12,s 13 ][s 21 ] x y x y x x o

27 (Linear) Resource Calculus If the number of occurrences of x in t equals k } (λx.t)[s 1,...,s k ] β t {s p(1) /x 1,...,s p(k) /x k p S k

28 (Linear) Resource Calculus If the number of occurrences of x in t equals k } (λx.t)[s 1,...,s k ] β t {s p(1) /x 1,...,s p(k) /x k p S k Seeking linear substitution? Then take non-determinism, too!

29 Taylor Expansion T : λ-terms infinite sums of resource terms

30 Taylor Expansion T : λ-terms infinite sums of resource terms T (MN) = k N M [ k times {}}{ N,...,N ]

31 Taylor Expansion T : λ-terms infinite sums of resource terms T (MN) = k N M [ k times {}}{ N,...,N ] I was lying! Definition (Taylor expansion) T (x) = x T (λx.m) = t T (M) λx.t T (MN) = k N,t T (M),s 1,...,s k T (N) t [ ] s 1,...,s k

32 Taylor Expansion T : λ-terms infinite sums of resource terms Example T (λy.xyy) = n,k N λy.x [ { n }} times { y,...,y k times ][ {}}{ y,...,y ]

33 Taylor Expansion T : λ-terms infinite sums of resource terms Theorem (Ehrhard & Regnier 2008) For every λ-term M nf β ( T (M) ) = T (BT(M)).

34 Taylor Expansion T : λ-terms infinite sums of resource terms Theorem (Ehrhard & Regnier 2008) For every λ-term M nf β T (M) = T BT(M)

35 Resource-sensitive characterizations for Morris Observational equivalences Morris s equivalence??? Resource-sensitive denotational semantics??? Resource-sensitive syntactic characterizations

36 Relational Semantics The cartesian closed category MRel: objects: sets A, B,... morphisms: A B M f (A) B Kleisli of Rel, self-dual -autonomous category, provided with the exponential comonad! = M f.

37 Relational Semantics The cartesian closed category MRel: objects: sets A, B,... morphisms: A B M f (A) B Kleisli of Rel, self-dual -autonomous category, provided with the exponential comonad! = M f.

38 Relational Semantics The cartesian closed category MRel: objects: sets A, B,... morphisms: A B M f (A) B Kleisli of Rel, self-dual -autonomous category, provided with the exponential comonad! = M f.

39 Relational Semantics The cartesian closed category MRel: objects: sets A, B,... morphisms: A B M f (A) B Kleisli of Rel, self-dual -autonomous category, provided with the exponential comonad! = M f. x 1 x 1 x 1 o x n x n x 1

40 Relational Semantics The cartesian closed category MRel: objects: sets A, B,... morphisms: A B M f (A) B Reflexive objects in MRel, i.e. retracts [D D] D

41 Relational Semantics The cartesian closed category MRel: objects: sets A, B,... morphisms: A B M f (A) B Reflexive objects in MRel, i.e. retracts [D D] = M f (D) D D

42 Relational Graph Models Definition (relational graph model) D = (D,i) is given by an infinite set D ; a total injection i : M f(d) D D. D is extensional when i is bijective.

43 Relational Graph Models Definition (relational graph model) D = (D,i) is given by an infinite set D ; a total injection i : M f(d) D D. D is extensional when i is bijective. Reminder (Plotkin s graph model) D = (D,i) is given by an infinite set D ; a total injection i : P f (D) D D.

44 rgm as Non Idempotent Intersection Type Systems To a rgm D = (D, i) we associate T D : σ ::= α µ σ I D : µ ::= ω σ σ µ

45 rgm as Non Idempotent Intersection Type Systems To a rgm D = (D, i) we associate T D : σ ::= α µ σ I D : µ ::= ω σ σ µ atomic types correspond to atoms of D

46 rgm as Non Idempotent Intersection Type Systems To a rgm D = (D, i) we associate T D : σ ::= α µ σ I D : µ ::= ω σ σ µ atomic types correspond to atoms of D intersections correspond to multisets of elements of D

47 rgm as Non Idempotent Intersection Type Systems To a rgm D = (D, i) we associate T D : σ ::= α µ σ I D : µ ::= ω σ σ µ atomic types correspond to atoms of D intersections correspond to multisets of elements of D arrow types correspond to elements of [D D]=M f(d) D

48 rgm as Non Idempotent Intersection Type Systems To a rgm D = (D, i) we associate T D : σ ::= α µ σ I D : µ ::= ω σ σ µ atomic types correspond to atoms of D intersections correspond to multisets of elements of D arrow types correspond to elements of [D D]=M f(d) D The injection i : M f (D) D D induces an equivalence σ τ

49 rgm as Non Idempotent Intersection Type Systems For a closed λ-term M M D = { σ D M : σ }

50 rgm as Non Idempotent Intersection Type Systems For a closed λ-term M M D = { σ D M : σ } Theorem (approximation) [ M D = BT(M) ] D

51 rgm as Non Idempotent Intersection Type Systems For a closed λ-term M M D = { σ D M : σ } Theorem ( New approximation) [ ] D [ M D = T (M) = BT(M) ] D It is here that we replace the traditional Böhm tree -like approximations with the new Taylor -like one!

52 Full Abstraction Theorem (that we are looking for) Let D be a rgm with possibly some more hypothesis. The following are equivalent for M, N closed λ-terms: 1. M D = N D 2. M nf N

53 Full Abstraction Theorem (that we are looking for) Let D be a rgm with possibly some more hypothesis. The following are equivalent for M, N closed λ-terms: 1. M D N D 2. M nf N

54 Full Abstraction Lemma (characterization of β-normalizability) A closed λ-term M has a β-normal form iff in every rgm D preserving ω-polarities, D M : σ for some type σ such that ω / + σ.

55 Full Abstraction Lemma (characterization of β-normalizability) A closed λ-term M has a β-normal form iff in every rgm D preserving ω-polarities, D M : σ for some type σ such that ω / + σ.

56 Full Abstraction Lemma (characterization of β-normalizability) A closed λ-term M has a β-normal form iff in every rgm D preserving ω-polarities, D M : σ for some type σ such that ω / + σ.

57 Full Abstraction Lemma (characterization of β-normalizability) A closed λ-term M has a β-normal form iff in every rgm D preserving ω-polarities, D M : σ for some type σ such that ω / + σ.

58 Full Abstraction Theorem (full abstraction for Morris s) Let D be an extensional rgm preserving ω-polarities. The following are equivalent for M, N closed λ-terms: 1. M D N D 2. M nf N

59 Full Abstraction Theorem (full abstraction for Morris s) Let D be an extensional rgm preserving ω-polarities. The following are equivalent for M, N closed λ-terms: 1. M D N D 2. M nf N Examples [Bucciarelli & others 2007] D ε ω ε ε fully abstract for hnf D fully abstract for Morris!

60 Full Abstraction Theorem (full abstraction for Morris s) Let D be an extensional rgm preserving ω-polarities. The following are equivalent for M, N closed λ-terms: 1. M D N D 2. M nf N Examples [Bucciarelli & others 2007] D ε ω ε ε fully abstract for hnf D fully abstract for Morris!

61 Full Abstraction Theorem (full abstraction for Morris s) Let D be an extensional rgm preserving ω-polarities. The following are equivalent for M, N closed λ-terms: 1. M D N D 2. M nf N Examples [Bucciarelli & others 2007] D ε ω ε ε fully abstract for hnf D fully abstract for Morris!

62 Full Abstraction Theorem (full abstraction for Morris s) Let D be an extensional rgm preserving ω-polarities. The following are equivalent for M, N closed λ-terms: 1. M D N D 2. M nf N Examples [Bucciarelli & others 2007] D ε ω ε ε fully abstract for hnf D fully abstract for Morris!

63 Full Abstraction Theorem (full abstraction for Morris s) Let D be an extensional rgm preserving ω-polarities. The following are equivalent for M, N closed λ-terms: 1. M D N D 2. M nf N Examples [Bucciarelli & others 2007] D ε ω ε ε fully abstract for hnf D fully abstract for Morris!

64 Full Abstraction Theorem (full abstraction for Morris s) Let D be an extensional rgm preserving ω-polarities. The following are equivalent for M, N closed λ-terms: 1. M D N D 2. M nf N Examples [Bucciarelli & others 2007] D ε ω ε ε fully abstract for hnf D fully abstract for Morris!

65 Full Abstraction Theorem (full abstraction for Morris s) Let D be an extensional rgm preserving ω-polarities. The following are equivalent for M, N closed λ-terms: 1. M D N D 2. M nf N Examples [Bucciarelli & others 2007] D ε ω ε ε fully abstract for hnf D fully abstract for Morris!

66 Full Abstraction Theorem (full abstraction for Morris s) Let D be an extensional rgm preserving ω-polarities. The following are equivalent for M, N closed λ-terms: 1. M D N D 2. M nf N Examples [Bucciarelli & others 2007] D ε ω ε ε fully abstract for hnf D fully abstract for Morris!

67 Fine! What else? Observational equivalences Morris s equivalence extensional rgm preserving ω-polarity Resource-sensitive denotational semantics??? Resource-sensitive syntactic characterizations

68 Fine! What else? Observational equivalences Morris s equivalence extensional rgm preserving ω-polarity Resource-sensitive denotational semantics T η (M) Resource-sensitive syntactic characterizations

69 Extensional Taylor Expansion We seek an analogous of [Ehrhard & Regnier 2008] commutativity theorem nf β T (M) = T BT(M) taking η-reduction into account. Something like nf η nf β T (M) = T BT e (M)

70 Extensional Taylor Expansion The core problem: how to η-reduce resource terms? A paradigmatic instance of the problem: λx.y [ n times {}}{ x,..., x ] η y? n N

71 Extensional Taylor Expansion The core problem: how to η-reduce resource terms? A paradigmatic instance of the problem: λx.y [ ] η y?

72 Extensional Taylor Expansion The core problem: how to η-reduce resource terms? A paradigmatic instance of the problem: λx.y [ ] η y? λx.y [] is a term of the sum nf β T (λx.y x)

73 Extensional Taylor Expansion The core problem: how to η-reduce resource terms? A paradigmatic instance of the problem: λx.y [ ] η y? but also λx.y [] is a term of the sum nf β T (λx.y x) λx.y [] is a term of the sum nf β T (λx.y y)

74 Extensional Taylor Expansion Moral of the story: η-conversion of resource approximants has a global behavior.

75 Extensional Taylor Expansion Moral of the story: η-conversion of resource approximants has a global behavior. Definition (extensional Taylor expansion) Let M be a λ-term. Then T η (M) = nf η ll ( nf β T (M) )

76 Extensional Taylor Expansion Moral of the story: η-conversion of resource approximants has a global behavior. Definition (extensional Taylor expansion) Let M be a λ-term. Then T η (M) = nf η ll ( nf β T (M) )

77 Extensional Taylor Expansion Moral of the story: η-conversion of resource approximants has a global behavior. Definition (extensional Taylor expansion) Let M be a λ-term. Then T η (M) = nf η ll ( nf β T (M) ) Example λx.y [] x η(x) as a term in the sum nf ( ) β T (λx.y x) λx.y [] y ( ) is a term in the sum nf β T (λx.y y)

78 Extensional Taylor Expansion Moral of the story: η-conversion of resource approximants has a global behavior. Definition (extensional Taylor expansion) Let M be a λ-term. Then T η (M) = nf η ll ( nf β T (M) ) Example λx.y [] x η(x) as a term in the sum nf ( ) β T (λx.y x) λx.y [] y ( ) is a term in the sum nf β T (λx.y y)

79 Extensional Taylor Expansion Moral of the story: η-conversion of resource approximants has a global behavior. Definition (extensional Taylor expansion) Let M be a λ-term. Then T η (M) = nf η ll ( nf β T (M) ) Example λx.y [] x η(x) as a term in the sum nf ( ) β T (λx.y x) λx.y [] y ( ) is a term in the sum nf β T (λx.y y)

80 Extensional Taylor Expansion Theorem For every λ-term M T η (M) = T BT η (M) Here BT η (M) is a coinductive version of the extensional Böhm tree, that characterizes nf, but not nf. So do our T η (M). Better than nothing!

81 Conclusion We introduced a class of models fully abstract for nf extensional rgm preserving ω-polarity We introduced a syntactic mathematical model of nf T η (M)

82 Conclusion & Future Work We introduced a class of models fully abstract for nf extensional rgm preserving ω-polarity Future works: refining the notion of preservation of ω-polarity in order to get all relational models fully abstract for nf, along the line of [Breuvart LICS14] We introduced a syntactic mathematical model of nf T η (M) Future works: looking for a similar model of nf

83 Thanks for your attention.