Analysis-directed semantics

Transcription

1 Analysis-directed semantics Dominic Orchard Imperial College London work in progress

2 Syntax directed e.g. (untyped) λ-calculus to reduction relation (λx. e 1 ) e 2 e 1 [x/e 2 ] e 1 e 1 e 1 e 2 e 1 e 2 2

3 Syntax-and-type directed e.g. simply-typed λ-calculus to CCCs [ Γ e : τ ] C([Γ], [τ]) i.e. : [Γ] [τ] with [Γ] C [τ] C e.g. [[ ` e 1 :! ]] = f :[[ ]]! [[ ` e 2 : ]] = g :[[ ]]! [[ ` e 1 e 2 : ]] = app hf,gi :[[ ]]! [[,v : ` e : ]] = f :[[ ]]! [[ ` v.e :! ]] = f :[[ ]]! 3

4 Syntax-and-type directed [ Γ1 e1 : t1 ] = [ Γ2 e2 : t2 ] t1 = t2 Γ1 = Γ2 see signature of interpretation [_] : (e : term) * ((Γ, τ) : ty(e)) C([Γ],[τ]) syntax type analysis 4

5 (Syntax-and-)analysis directed [_] : (e : term) * (i : analysis(e)) D i e.g. simple-typed λ-calculus with effect system [ Γ e : τ, F ] (Γ T F τ) 5

6 Constructing analysisdirected semantics Analysis domain A, semantic domain D Define F : A D to be structure preserving (homomorphism) between A and D Gives a design framework for A and D Equations in A map to equations in D 6

7 Context Work on coeffects (with Tomas Petricek & Alan Mycroft) [ Γ? R e : τ ] C(DR[Γ], [τ]) A core quantitative coeffect calculus (Brunel, Gaboardi, Mazza, Zdancewic), ESOP 2013 Bounded linear types (Ghica and Smith), ESOP 2013 Work on effects (Shinya Katsumata, parametric effect monads ) [ Γ e : τ! F ] C([Γ], MF[τ]) All define analysis-directed semantics (and leverage this for soundness) 7

8 Effect systems Γ e : τ, F monoid (F,, ) [abs] Γ, x : σ e : τ, F F Γ λx. e : σ τ, [var] x : σ Γ Γ x : σ, [app] F Γ e 1 : σ τ, G Γ e 2 : σ, H Γ e 1 e 2 : τ, G H F [write] Γ e : τ, F (x : ref τ) Γ Γ x := e : (), F {W(x)} [read] (x : ref τ) Γ Γ!x : τ, {R(x)} 8

9 Effect systems married to monads Γ e : T F τ monoid (F,, ) [abs] Γ, x : σ e : T F τ Γ λx. e : T (σ T F τ) [var] x : σ Γ Γ x : T σ [app] Γ e 1 : T G (σ T F τ) Γ e 2 : T H σ Γ e 1 e 2 : T (G H F) τ 9 The marriage of effects and monads (Wadler & Thiemann, 2003)

10 <4 A9 Unifying effect-analysis and semantics monoid homomorphism M : [C, C ] F : Set T : F [C, C ] seq id μ : M M M η : 1C M : F F F : 1 F T F T G = T (F G) 1 C = T F F t / F 1 ; / F T T [C, C] [C, C] / [C, C] µ T id + T [C, C] μf,g : TF TG T(F G) η : 1 T 10 Katsumata, Parametric Effect Monads and Semantics of Effect Systems, POPL 2014

11 Equations Identities preserved (trivial) T F = T (F ) = T F T = T F 1 = T F For lax, have the diagram: For strict monoids analogues of monad laws T T F O t ; o T F µ F,; T F µ F,; T F O T ; o T F ; T F 1 C T F T ; ot F ; T F 11

12 Corresponding equations [ Γ1 e1 : t1, F] = [ Γ2 e2 : t2, G ] t1 = t2 Γ1 = Γ2 (F = G) When considering equations on semantics? [ Γ1 e1 : t1, F] = [ Γ2 e2 : t2, G ] proof tree for (F = G) implies semantic laws e.g. [ Γ1 e1 : t1, F ] = [ Γ2 e2 : t2, F ] µ F,; T F O if η is the only way to introduce 12 T F T ; ot F ; T F

13 Bounded linear logic analysis Reuse bounds on free-variables x : σ? n Core rules (with sub-coeffects) [abs] Γ, x : σ? s e : τ Γ λx.e : σ τ [app] Specialised structural rules s Γ1 e1 : σ τ s [var] Γ1, s * Γ2 e1 e2 : τ x : σ? 1 x : σ Γ2 e2 : σ [weak] Γ e : τ Γ, x : σ? 0 e : τ [contr] Γ1, x : σ? a, y : σ? b, Γ2 e : τ Γ1, z : σ? a+b, Γ2 e[z/x,z/y] : τ 13

14 Bounded linear logic analysis (abs) (app) ` ( v.x+v+v)(x+y). variable x is used three x:z,v : Z?h1, 2i ` x + v + v : Z x:z?h1i ` ( v.x + v + v) :Z 2! Z.. x 0 : Z,y : Z?h1, 1i ` x 0 + y : Z ( ) x:z,x0 :Z,y:Z?h1i (2 h1, 1i) ` ( v.x + v + v) (x 0 + y) :Z (contr) x:z,x0 :Z,y:Z?h1, 2, 2i ` ( v.x + v + v) (x 0 + y) :Z x:z,y:z?h3, 2i ` ( v.x + v + v) (x + y) :Z ( v.x + v + v) (x + y) x +(x + y)+(x + y) h i ` x:z,y:z?h3, 2i ` x +(x + y)+(x + y) :Z 14

15 BLL-directed semantics Bounded reuse (exponent) Dn A = A n = <A1,..,An> D : N [C,C] Monoid and scalar-vector multiplication (monoid action) on N + : N N N contraction 0 : 1 N weakening 1 : 1 N variables * : N N n N n * : N N N composition Γ1 e1 : σ τ s Γ1, s * Γ2 e1 e2 : τ Γ2 e2 : σ s * (x1 : t1? r1,, xn : tn? rn) = x1 : t1? s * r1,, xn : tn? s * rn 15 will treat as a vector s * <r1,, rn> = <s * r1,, s * rn>

16 BLL-directed semantics Structure preserving D : N [C,C] s * <r1,, rn> N N n * N n <s * r1,, s * rn> D D n δ n D n Ds <Dr1,, Drn> [C,C] [C,C] n n [C,C] n <D(s * r1),, D(s * rn)> δ n <DsDr1,, DsDrn> 16 δ n : (D(s * r1) D(s * rn)) (DsDr1 DsDrn)

17 BLL-directed semantics δ n : (D(s * r1) D(s * rn)) (DsDr1 DsDrn) Coeffect-parameterised comonad δ : Ds*r A DsDr A s*r copies turned into s copies of r copies ε : D1 A A use one copy 17

18 Coeffect-directed semantics [contr] Γ1, x : σ? a, y : σ? b, Γ2 e : τ Γ1, z : σ? a+b, Γ2 e : τ (N N) D D [C,C] [C,C] + m <-, -> N D [C, C] Δr,s : D (r + s) A Dr A Ds A 18

19 Coeffect-directed semantics BLL analysis as coeffect analysis in general one shaped annotation [abs] [abs] [abs] Γ, x : σ? s e : τ Γ λx.e : σ τ Γ, x : σ? R s e : τ Γ? R λx.e : σ τ Γ, x : σ? R s e : τ Γ? R λx.e : σ τ s s s e.g. distributed resources [abs] Γ, x : σ? {gps, db} e : τ gps Γ? {db} λx.e : σ τ 19

20 Coeffect-directed semantics [abs] Γ, x : σ? R s e : τ Γ? R λx.e : σ τ s let D = uncurry D i.e. D : I C C composes binary ops (I C) (I C) D D C C m I C D C mr,s : Dr A Ds B D (r s) (A B) 20

21 Constructing analysisdirected semantics Analysis domain A, semantic domain D Define F : A D to be structure preserving (homomorphism) between A and D Gives a design framework for A and D Equations in A map to equations in D 21

22 Corresponding equations Use algebraic solver on analysis domain A (e.g., I use Prover 9) Rest of proof not corresponding to A usually naturally and universality Tactic generator! Build into theorem prover? 22

23 Thanks!