Call-by-Value in a Basic Logic for Interaction

ion of Γ, x : N `M : N nat× σ (γ × nat)× σ M ∀σ. Abstraction of Γ, x : N `M : Nion of Γ, x : N `M : N nat× σ (γ × nat)× σ M ∀σ. nat× σ′ (γ × nat)× σ′ γ × σ nat× (γ × σ′) Abstraction of Γ, x : N `M : Nion of Γ, x : N `M : N nat× σ (γ × nat)× σ M ∀σ. nat× τ γ × σ ∀σ.∃φ.∀τ . φ× σ (φ× nat)× τ nat× (φ× τ) λx:X.M Abstraction of Γ, x : N `M : Nion of Γ, x : N `M : N nat× σ (γ × nat)× σ M ∀σ. nat× τ γ × σ ∀σ.∃φ.∀τ . φ× σ (φ× nat)× τ nat× (φ× τ) λx:X.M cps(Γ ` λx:X.M) = Λσ. λk. k C(Γ) (Λτ . λk1.Λφx. λx. (〈a, 〈~z, t〉〉 7→ 〈〈~z, a〉, t〉) (cps(Γ, x : X `M) τ k1)) Correctness Theorem. Suppose `M : N andM −→cbv n, where n is a value. Then cps( `M) unit K = (〈~z, s〉 7→ 〈n, s〉)(K unit ?) for any closed continuation K of type Kunit(N). unit× σ nat× σ Jcps( `M)K Correctness Theorem. Suppose `M : N andM −→cbv n, where n is a value. Then cps( `M) unit K = (〈~z, s〉 7→ 〈n, s〉)(K unit ?) for any closed continuation K of type Kunit(N). unit× σ nat× σ Jcps( `M)K Lemma. LetM be a source term well-typed in context Γ. Then, for all σ and all closed K such that cps(Γ `M) σ K is well-typed, we have cps(Γ `M) σ K = M :σ K. Lemma. IfM −→cbv N thenM :σ K = N :σ K for any σ and closed K of the appropriate type. Contraction To translate the full source calculus, we need contraction on variables of type Aφ(X). Value Types A,B ::= α | nat | unit | A×B | 0 | A+B Computation Types X,Y ::= ⊥ | A ·X ( Y | ∀α.X Conclusion CPS translations for call-by-name and call-by-value can be refined to target a low-level computation calculus. • fully specified translation to low-level language • interface specification • separate compilation • exposes low-level details, e.g. closure conversion • soundness proof managable • value/computation-separation à la defunctionalization Further work • space bounds / optimisation using ∀αCA.X • understand control flow data • relation to call-by-value games • fully abstract compilation