Intersection types fit well with resource control Silvia Ghilezan, Jelena Ivetić, Pierre Lescanne, Silvia Likavec University of Novi Sad, École normale supérieure de Lyon, Unviersità di Torino TYPES 2015, 18 May S. Ghilezan, J. Ivetić, P. Lescanne, S. Likavec (University Intersection of Novi&Sad, resource École control normale supérieure de Lyon, TYPES Unviersità 2015, 18 di May Torino) 1/9 From practice to theory With resource control we know when a variable is duplicated or erased. For a compiler maker: duplication requires pointers and memory management, it should be used only when necessary, erasure allows to free memory explicitly. For a proof theorist: duplication corresponds to contraction, erasure corresponds to thinning. S. Ghilezan, J. Ivetić, P. Lescanne, S. Likavec (University Intersection of Novi&Sad, resource École control normale supérieure de Lyon, TYPES Unviersità 2015, 18 di May Torino) 2/9 Well formed terms The set Λr of terms are of the following forms: x λx.M MN x <xx12 M x M Variable Abstraction Application Duplication Erasure Axiom → Introduction → Elimination Contraction Thinning Moreover there are constraints on variables: variables occur once and only once. A substitution is defined for this calculus. S. Ghilezan, J. Ivetić, P. Lescanne, S. Likavec (University Intersection of Novi&Sad, resource École control normale supérieure de Lyon, TYPES Unviersità 2015, 18 di May Torino) 3/9 The reduction rules of λr Rules (λx.M)N → M||[N/ /x]| (β) (γ1 ) (γ2 ) (γ3 ) (ω1 ) (ω2 ) (ω3 ) x <xx12 x x (λy .M) → λy .x <xx12 M (MN) → (x <xx12 M)N, if x1 , x2 ∈ 6 Fv (N) x 1 (MN) → M(x <x2 N), if x1 , x2 ∈ 6 Fv (M) <xx12 <xx12 λx.(y M) → y (λx.M), x 6= y (x M)N → x (MN) M(x N) → x (MN) (γω1 ) x <xx12 (y M) → y (x <xx12 M), y 6= x1 , x2 (γω2 ) x <xx12 (x1 M) → M||[x/ /x2 ]| S. Ghilezan, J. Ivetić, P. Lescanne, S. Likavec (University Intersection of Novi&Sad, resource École control normale supérieure de Lyon, TYPES Unviersità 2015, 18 di May Torino) 4/9 Structural equivalences (ε1 ) x (y M) (ε2 ) x <xx12 M y (ε3 ) x <z (y <uv M) (ε4 ) x <xx12 (y <yy12 M) ≡λr ≡λr ≡λr ≡λr y x x y (x M) <xx21 M <yu (y <zv M) <yy12 (x <xx12 M), x 6= y1 , y2 , y 6= x1 , x2 S. Ghilezan, J. Ivetić, P. Lescanne, S. Likavec (University Intersection of Novi&Sad, resource École control normale supérieure de Lyon, TYPES Unviersità 2015, 18 di May Torino) 5/9 Rules of λ∩r Typing is syntax directed. (Ax) x :σ`x :σ Γ, x : α ` M : σ (→I ) Γ ` λx.M : α → σ n Γ ` M : ∩i=1 τi → σ Fitness ∆0 ` N : τ0 . . . ∆n ` N : τn (→E ) > Γ, ∆0 u ∆1 u ... u ∆n ` MN : σ Γ, x : α, y : β ` M : σ (Cont) x Γ, z : α ∩ β ` z <y M : σ Γ`M :σ (Thin) Γ, x : > ` x M : σ → is introduced by (→I ) ∩ is introduced by (Cont) > is introduced by (Thin). S. Ghilezan, J. Ivetić, P. Lescanne, S. Likavec (University Intersection of Novi&Sad, resource École control normale supérieure de Lyon, TYPES Unviersità 2015, 18 di May Torino) 6/9 Role of variables Variable have several roles with a type associated to this role. Roles role placeholder variables duplicated variables erased variables type strict intersection > The role of a variable may change by reduction. S. Ghilezan, J. Ivetić, P. Lescanne, S. Likavec (University Intersection of Novi&Sad, resource École control normale supérieure de Lyon, TYPES Unviersità 2015, 18 di May Torino) 7/9 Equivalence between strong normalisation and typeablity SN ⇒ Typeability in Λr ∩ Typeability in Λr ∩ ⇒ SN The proof uses the correspondence between typing rules and roles of variables. S. Ghilezan, J. Ivetić, P. Lescanne, S. Likavec (University Intersection of Novi&Sad, resource École control normale supérieure de Lyon, TYPES Unviersità 2015, 18 di May Torino) 8/9 Conclusion The way variables are introduced =⇒ the way terms are typed. No irrelevant intersection types are introduced. The flexibility on the type (as in →E ) comes from the choice in invoking the axiom. S. Ghilezan, J. Ivetić, P. Lescanne, S. Likavec (University Intersection of Novi&Sad, resource École control normale supérieure de Lyon, TYPES Unviersità 2015, 18 di May Torino) 9/9
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