Using session types as an effect system (slides)

Transcription

1 Using session types as an effect system (slides) Effects in a pi Dominic Ochad, Nobuko Yoshida dochad.co.uk PLACES Apil 18th 2015

2 Effect systems descibe side-effect behaviou λ-calculus as pototype (app) H Γ M : σ τ, F Γ N : σ, G Γ M N : τ, F G H e.g. Γ 2 := : unit, {ead R1, wite R2} Session types descibe communication behaviou π-calculus as pototype e.g. (ecv) Γ, x : τ; Δ, c : S P Γ; Δ, c :?[τ].s c?(x).p c :?[int].![int] c?(x).c!<x+1> Ae they elated?

3 λ-calculus + simple types + state + effect systems embedding this wok* π-calculus + sessions + session types T Expessive powe of session types! Session types genealise causal effect systems π-calculus with effect system fo fee! Concuent effect semantics via π- calculus! Compile into π-calculus P

4 Vaiable agent x Stoe c, x P get put Stoe c, x V c

5 Vaiable agent def Stoe(c, x) = c {get : c! x.stoe c, x, put : c?(y).stoe c, y, stop : 0} in Stoe c, i x Stoe c, V x P get put Stoe c, V V c

6 Vaiable agent def Stoe(c, x) = c {get : c! x.stoe c, x, put : c?(y).stoe c, y, stop : 0} in Stoe c, i Seve get (c)(x).p = (c get).c?(x).p put (c) V.P = (c put).c! V.P stop = (c stop).0 Client e.g. incement stoe def Stoe(c, x) = in (get(c)(x).put(c) x+1.stop Stoe c, i )

7 λ-calculus + simple types + state + effect systems embedding embedding π-calculus + sessions + session types

8 Γ M : τ, F Effect calculus abs Γ, x : σ M : τ, F F Γ λx.m : σ τ, Γ M : σ, F Γ, x : σ N : τ, G Γ let x = M in N : τ, F G app H Γ M : σ τ, F Γ M N : τ, F G H Γ N : σ, G va x : σ Γ Γ x : σ, monoid (F,, )

9 Effect calculus fo state Γ V : τ, [ ] Γ put V : (), [put τ] Γ get : τ, [get τ] (List {put t, get t t τ}, ++, [ ]) e.g. incement stoe Γ let x = get in put (x + 1) : int, [get int, put int]

10 Γ ; Δ P π-calculus with sessions ` ` h i (ecv),x: ;,c: S ` P ;,c:?[ ].S c?(x).p (send) ; ;`e : ;,c: S ` P ` ;,c:![ ].S` `hc!hei.p i (banch) ;,c: S i ` P i ;,c:&[ l : S] ` c { l : P } (select) ;,c: S ` P ;,c: [l : S] ` c l.p e.g. incement stoe get (c)(x).p = c get. c?(x).p put (c) V.P = c put. c! V.P c : [get :?[int]. [put :![int].end ]] get(c)(x).put(c) x+1.0 cf. effects [get int, put int]

11 Sessions as effects! Effect handle pocess [e.g., vaiable agent]! [cf. Baue, Petna Pogamming with algebaic effects and handles. ]! Effect channel [a session channel fo communicating with handle]!! whose session type is (encoding of) effect annotation! Theading effect channel though contol flow of encoding! [cf. state e, s e, s o monadic semantics a M b ]

12 State effect annotations as session types [] = end (get τ) : F = [get :? τ. F ] (get τ) : F = [put :! τ. F ]

13 Embedding (mid) Γ M : τ, F eff = Γ, :! τ, eff : F ei,eo νei, eo. ( Γ M : τ, F ei! eff.eo(c)) (top) Γ M : τ, F = Γ, :! τ νeff. ( Γ M : τ, F eff H(eff)) (low) ei,eo Γ M : τ, F = g. Γ, :! τ, ei :? F g, eo :! g eceive effect channel send effect channel

14 Embedding (zeoth-ode) Γ x : τ, ei,eo = ei?(c).! x.eo! c whee g. Γ ; :! τ, ei :? g, eo :! g ei?(c).! x.eo! c Γ let x = M in N : τ, F G ei,eo = ei, a q a, eo ν q, a. ( M q?(x). N ) whee h. h. ei, a q :! σ, ei :? F h, a :! h M q h G h x : σ ; :! τ, a :? G h, eo :! h N a, eo

15 Embedding (zeoth-ode) Γ x : τ, ei,eo = ei?(c).! x.eo! c whee g. Γ ; :! τ, ei :? g, eo :! g ei?(c).! x.eo! c Γ let x = M in N : τ, F G ei,eo = ei, a q a, eo ν q, a. ( M q?(x). N ) whee h. h. ei, a q :! σ, ei :? F G h, a :! G h M q h G h x : σ ; :! τ, a :? G h, eo :! h N a, eo

16 Embedding (highe-ode) Must embed latent effects F σ τ σ τ =![? σ ].![! τ ]. end F σ τ =![? σ ].![? F G ].![! G ].![! τ ]. end send channel which can eceive effect channel fo latent effects send channel which can send effect channel fo continuation

17 Embedding (highe-ode) Γ λx. M : σ τ, = F ei,eo a,b ν y. (ei?(c).eo! c.! y.*y?(p, a, b, q).p?(x). M ) q g, h. :![? σ,? F h,! h,! τ ], ei :? g, eo :! g ν y. (ei?. ) Γ M N : τ, F G H ei,eo = ei, a ν q,s,a,b,p. ( M N q?(y).s?(ag).y! p, b, eo,.p! ag ) q a, b s

18 Example Γ, :! int, eff : [get int, put int] Γ let x = get in put (x + 1) : int, [get int, put int] eff J K Γ let x = νeff. ( Γ let x Va(eff, 0)) eff Soundness ` M N :,F J K;( :!J K.end, e : JF K) ` JMK e JNK e

19 An application Effect-infomed optimisations, e.g. implicit paallelism if then Γ M : σ, and Γ N : t, F L let x M in (let y N in P ) M ei,eo = q, s, eb. (JMK q L N M ei,eb s q?(x).s?(y).l P M eb,eo ) Use this to give semantics of concuent effects! e.g., non-intefeence, atomicity via sessions

20 ! Conclusion Sessions and session types expessive enough to encode effects with a causal effect system! Pe effect notion [e.g., state, counting, I/O]:! effect mapping, handle, encoding opeations! Extended to case and fix Set-based effect systems ecoveed by tansfoming causal Thanks! Moe details in ou PLACES 15 pape; see dochad.co.uk