Coherence Generalises Duality

Transcription

1 Coherence Generalises Duality Marco Carbone, Sam Lindley, Fabrizio Montesi, Carsten Schürmann, Philip Wadler University of Lisbon Tuesday 23 January

2 CONCUR,

3 CONCUR,

4 FORTE,

5 ABCD,

6 Part I CP: Classical Processes 6

7 Types A, B, C::= output types input types X type variable X dual of type variable A B output A then behave as B A B input A then behave as B A B select from A or B A B offer choice of A or B?A client request!a server accept 1 unit for unit for 0 unit for unit for 7

8 Duals (X) = X (X ) = X (A B) = A B (A B) = A B (A B) = A B (A B) = A B (!A) =?A (?A) =!A 1 = = 1 0 = = 0 8

9 CP: Old and new Old P Γ, x : A Q, x : A x y x : A, y : A AXIOM νx:a. (P Q) Γ, CUT New P Γ, x : A Q, y : A x y A x : A, y : A AXIOM (νx A y) (P Q) Γ, CUT 9

10 CP: Processes P Γ, x : A Q, y : A x y A x : A, y : A AXIOM (νx A y) (P Q) Γ, P Γ, y : A Q, x : B x[y].(p Q) Γ,, x : A B P Γ, x : A x[inl].p Γ, x : A B 1 P Γ, x : A Q Γ, x : B x.case(p, Q) Γ, x : A B P Γ, y : A?x[y].P Γ, x :?A? P Γ, y : A, x : B x(y).p Γ, x : A B P Γ, x : B x[inr].p Γ, x : A B 2 P?Γ, y : A!x(y).P?Γ, x :!A! P Γ P Γ, y :?A, z :?A P Γ, x :?A WEAKEN P {x/y, x/z} Γ, x :?A CONTRACT x[] x : 1 1 CUT P Γ x().p Γ, x : (no rule for 0) x.case() Γ, x : 10

11 Duplicating a bit P x def = x[y].(y[inl].y[] x[inl].x[]) Q x def = x[y].(y[inr].y[] x[inr].x[]) y[] y : 1 1 y[inl].y[] y :1 1 x[] x : x[inl].x[] x:1 1 y[] y : y[inr].y[] y :1 1 x[] x : x[inr].x[] x:1 1 2 P x x : (1 1) (1 1) w().p x w :, x : (1 1) (1 1) Q x x : (1 1) (1 1) w().q x w :, x : (1 1) (1 1) w.case(w().p x, w().q x ) w :, x : (1 1) (1 1) x(w).w.case(w().p x, w().q x ) x : ( ) ((1 1) (1 1)) 11

12 Two-buyer protocol 12

13 Two-buyer protocol b 1 (b 1).s[s ].(b 1 s name s(s ).b 1 [b 1].(s b 1cost s(s ).b 2 [b 2].(s b 2cost b 1 (b 1).b 2 [b 2].(b 1 b 2cost b 2.case(s[inl].b 2 (b 2).s[s ].(b 2 s addr b 1 ().b 2 ().s[]), s[inr].b 1 ().b 2 ().s[]))))) b 1 : name (cost (cost )), b 2 : cost (cost ((addr ) )), s : name (cost (cost ((addr 1) 1))) 13

14 Part II GCP: Globally-governed Classical Processes 14

15 GCP: Coherence x A y x : A, y : A AXIOM G (x i : A i ) i, y : C H Γ, (x i : B i ) i, y : D x y(g).h Γ, (x i : A i B i ) i, y : C D G Γ, x : A, (y i : C i ) i H Γ, x : B, (y i : D i ) i x ỹ.case(g, H) Γ, x : A B, (y i : C i D i ) i G x : A, (y i : B i ) i!x ỹ(g) x :?A, (y i :!B i ) i?! x y (x i : 1) i, y : 1 x ỹ.case() Γ, x : 0, (y i : ) i 0 15

16 GCP: Processes (P i Γ i, x i : A i ) i G (x i : A i ) i (ν xã : G) ( P ) Γ CCUT 16

17 From GCP to CP Global cut as binary cut (ν xã : G) ( P ) def = (νx 1 A 1 y 1 ) ( P 1 (νx n A n y n ) ( P n G {ỹ/ x}) ) Global types as processes x A y def = x y A x y(g).h def = x 1 (u 1 ). x n (u n ).y[v].( G {ũ/ x, v/y} H ) (ỹ fresh) x ỹ.case(g, H) def = x.case(y 1 [inl]. y n [inl]. G, y 1 [inr]. y n [inr]. H )!x ỹ(g) def =!x(u).?y 1 [v 1 ].?y n [v n ]. G {u/x, ṽ/ỹ} x y def = x 1 (). x n ().y[] x ỹ.case() def = x.case() (ũ, v, u, ṽ fresh) 17

18 η-expansion x y A B x : A B, y : A B AXIOM = u v A u : A, v : A AXIOM x y B x : B, y : B AXIOM y[v].(u v A x y B ) u : A, x : B, y : A B x(u).y[v].(u v A x y B ) x : A B, y : A B 18

19 From duality to coherence Duality G u : A, v : C H x : B, y : D y[v].( G H ) u : A, x : B, y : C D x(u).y[v].( G H ) x : A B, y : C D Coherence G u 1 : A 1, u 2 : A 2, v : C H x 1 : B 1, x 2 : B 2, y : D y[v].( G H ) u 1 : A 1, x 1 : B 1, u 2 : A 2, x 2 : B 2, y : C D x 2 (u 2 ).y[v].( G H ) u 1 : A 1, x 1 : B 1, x 2 : A 2 B 2, y : C D x 1 (u 1 ).x 2 (u 2 ).y[v].( G H ) x 1 : A 1 B 1, x 2 : A 2 B 2, y : C D 19

20 From GCP to CP Theorem 1 (Type preservation from GCP to CP) 1. If P Γ in GCP, then P Γ in CP. 2. If G Γ in GCP, then G Γ in CP. Theorem 2 (Simulation of GCP in CP) 1. If P Γ and P Q in GCP, then P Q in CP. 2. If P Γ and P η Q in GCP, then P η Q in CP. 3. If G Γ and G η H in GCP, then G η H in CP. 4. If P Γ and P Q in GCP, then P = + Q in CP. Theorem 3 (Reflection of CP in GCP) If P Γ in GCP and P Q in CP, then there exists Q such that P = Q in GCP and Q = Q in CP. 20

21 From CP to GCP Binary cut as global cut (νx A y) (P Q) = (νx, y : x A y) ( P Q ) Theorem 4 (Type preservation from CP to GCP) If P Γ, then P Γ. Theorem 5 (Simulation of CP in GCP) 1. If P Γ and P Q in CP, then P Q in GCP. 2. If P Γ and P Q in CP, then P + Q in GCP. 21

22 Two-buyer protocol B 1 S(B name 1 S). S B 1 (S cost B 1 ). S B 2 (S cost B 2 ). B 1 B 2 (B cost 1 B 2 ). B 2 S.case(B 2 S(B addr 2 S).(B 1, B 2 ) S, (B 1, B 2 ) S)) B 1 : name (cost (cost 1)), B 2 : cost (cost ((addr 1) 1)), S : name (cost (cost ((addr ) ))) 22

23 Part III MCP: Multiparty Classical Processes 23

24 Types A, B, C::= output types input types X type variable X dual of type variable A z B output A then behave as B A z B input A then behave as B A z B select from A or B A z B offer choice of A or B? z A client request! z A server accept 1 z unit for z unit for 0 z unit for z unit for 24

25 MCP: Coherence A = B x A y B x : A, y : B AXIOM G (x i :A i ) i, y :C H Γ, (x i :B i ) i, y :D x y(g).h Γ, (x i :A i y B i ) i, y :C x D G 1 Γ, x:a, (y i :C i ) i G 2 Γ, x:b, (y i :D i ) i x ỹ.case(g 1, G 2 ) Γ, x:a ỹ B, (y i :C i x D i ) i G x:a, (y i :B i ) i!x ỹ(g) x:?ỹa, (y i :! x B i ) i!? x y (x i :1 y ) i, y : x 1 x ỹ.case() Γ, x:0ỹ, (y i : x ) i 0 25

26 MCP: Processes A = B (P i Γ i, x i : A i ) i G ( x i : A i ) i x A y B x : A, y : B AXIOM (ν xã : G) ( P ) Γ P Γ, y : A Q, x : B x z [y].(p Q) Γ,, x : A z B P Γ, x : A x z [inl].p Γ, u : A z B 1 P Γ, x : A Q Γ, x : B x z.case(p, Q) Γ, x : A z B P Γ, y : A, x : B x z (y).p Γ, x : A z B P Γ, x : B x z [inr].p Γ, x : A z B 2 P?Γ, y : A!x z (y).p?γ, x :! z A! P Γ, y : A?x z [y].p Γ, x :? z A? P Γ P Γ, x :? z A WEAKEN P Γ, y :? w A, z :? w A P [x/y, x/z] Γ, x :? w A CONTRACT CCUT x z [] x : 1 z 1 P Γ x z ().P Γ, x : z no rule for 0 x z.case() Γ, x : z 26

27 Two-buyer protocol B 1 S(B name 1 S). S B 1 (S cost B 1 ). S B 2 (S cost B 2 ). B 1 B 2 (B cost 1 B 2 ). B 2 S.case(B 2 S(B addr 2 S).(B 1, B 2 ) S, (B 1, B 2 ) S)) B 1 : name S (cost S (cost B 2 1 S )), B 2 : cost S (cost B 1 ((addr S 1 S ) S 1 S ), S : name B 1 (cost B 1 (cost B 2 ((addr B 2 B 1,B 2 ) B 2 B 1,B 2 ))) 27

28 Bibliography Marco Carbone, Fabrizio Montesi, Carsten Schrmann, Nobuko Yoshida. Multiparty Types as Coherence Proofs. CONCUR, Marco Carbone, Sam Lindley, Fabrizio Montesi, Carsten Schrmann, Philip Wadler. Coherence Generalises Duality: a logical explanation of multiparty session types. CONCUR, Luis Caires and Jorge Perez. Multiparty session types within a canonical binary theory, and beyond. FORTE,