Confluence of the Call-By-Value -calculus Karim NOUR and Khelifa - PDF document

Confluence of the Call-By-Value λµ ∧∨ -calculus Karim NOUR and Khelifa SABER LAMA - Equipe de logique Universit´ e de Chamb´ ery 73376 Le Bourget du Lac e-mail: { knour, ksabe } @univ-savoie.fr 1

Plan 1. The λµ ∧∨ -calculus • The typed system, syntax and reduction rules. • The main properties. 2. The call-by-value calculus • Subformula property • The new reduction rules • Confluence and the parallel reduction 3. Futur work. 2

The λµ ∧∨ calculus Γ , x : A ⊢ x : A ; ∆ ax Γ , x : A ⊢ t : B ; ∆ Γ ⊢ u : A → B ; ∆ Γ ⊢ v : A ; ∆ Γ ⊢ λx.t : A → B ; ∆ → i → e Γ ⊢ ( u v ) : B ; ∆ Γ ⊢ u j : A j ; ∆ Γ ⊢ t : A 1 ∧ A 2 ; ∆ Γ ⊢ � u 1 , u 2 � : A 1 ∧ A 2 ; ∆ ∧ j Γ ⊢ ( t π j ) : A j ; ∆ ∧ j e i Γ ⊢ t : A ; ∆ Γ ⊢ t : B ; ∆ Γ ⊢ ω 1 t : A ∨ B ; ∆ ∨ 1 Γ ⊢ ω 2 t : A ∨ B ; ∆ ∨ 2 i i Γ ⊢ t : A ∨ B ; ∆ Γ , x : A ⊢ u : C ; ∆ Γ , y : B ⊢ v : C ; ∆ ∨ e Γ ⊢ ( t [ x.u, y.v ]) : C ; ∆ Γ ⊢ t : A ; ∆ , a : A Γ ⊢ t : ⊥ ; ∆ , a : A Γ ⊢ ( a t ) : ⊥ ; ∆ , a : A Γ ⊢ µa.t : A ; ∆ 3

The syntax T := X | λ X . T | ( T E ) |�T , T � | ω i T | ( A T ) | µ A . T E := T | π i | [ X . T , X . T ] The reduction rules • ( λx.u v ) ⊲ β u [ x := v ] • ( � t 1 , t 2 � π i ) ⊲ π t i • ( ω i t [ x 1 .u 1 , x 2 .u 2 ]) ⊲ D u i [ x i := t ] • (( t [ x 1 .u 1 , x 2 .u 2 ]) ε ) ⊲ δ ( t [ x 1 . ( u 1 ε ) , x 2 . ( u 2 ε )]) • ( µa.t ε ) ⊲ µ µa.t [ a := ∗ ε ] where t [ a := ∗ ε ] is obtained from t by replacing induc- tively each subterm in the form ( a v ) by ( a ( v ε )) . 4

The main properties • Subject reduction • Subformula property Ph. De Groote • Confluence Ph. De Groote Y. Andou K. Nour and K. Saber • Strong normalization Ph. De Groote R. David and K. Nour K. Nour and K. Saber R. Matthes 5

The Call-By-Value We need the following properties: • Subformula property Each closed normalizable term (in the or- dinary calculus) can be normalized in the Call-By-value calculus to its really normal form. • Confluence • Strong normalization What are the values? V := X | λ X . T | �T , T � | ω 1 V | ω 2 V | ( A T ) 6

The reduction rules • ( λx.t V ) ⊲ β v t [ x := V ] • ( � t 1 , t 2 � π i ) ⊲ π t i • ( ω i V [ x 1 .t 1 , x 2 .t 2 ]) ⊲ D v t i [ x i := V ] • (( t [ x 1 .t 1 , x 2 .t 2 ]) ε ) ⊲ δ ( t [ x 1 . ( t 1 ε ) , x 2 . ( t 2 ε )]) • ( V ( t [ x 1 .t 1 , x 2 .t 2 ])) ⊲ δ ′ v ( t [ x 1 . ( V t 1 ) , x 2 . ( V t 2 )]) • ( µa.t ε ) ⊲ µ µa.t [ a := ∗ ε ]. • ( V µa.t ) ⊲ µ ′ v µa.t [ a := ∗ V ]. • ω i µa.t ⊲ ω i µa.t [ a := ω i ]. 7

Subformula property Notation • t⊲ v t ′ if t is reduced to t ′ by one of reduction rules above. v t ′ denotes the transitive and reflexive • t ⊲ ∗ closure of ⊲ v . Proposition If t is a closed term and t ⊲ ∗ τ where τ is a normal form, then t ⊲ ∗ v τ . Proof This gives the new rules: ⊲ δ ′ v and ⊲ ω i . 8

Confluence Theorem If t ⊲ ∗ v t 1 and t ⊲ ∗ v t 2 , then there exists t 3 such that t 1 ⊲ ∗ v t 3 and t 2 ⊲ ∗ v t 3 . Proof The generalized parallel reductions ⊲ ∗ • ⊲ v ⊂ ≻ ⊂ v • ≻ is strongly confluente 9

Generalized parallel reduction • x ≻ x • If t ≻ t ′ , then λx.t ≻ λx.t ′ • If t ≻ t ′ , then µa.t ≻ µa.t ′ • If t ≻ t ′ , then ( a t ) ≻ ( a t ′ ) • If t ≻ t ′ , then ω i t ≻ ω i t ′ • If t ≻ t ′ and u ≻ u ′ , then � t, u � ≻ � t ′ , u ′ � • If t ≻ t ′ and ε ≻ ε ′ , then ( t ε ) ≻ ( t ′ ε ′ ) 10

• If t ≻ λx.t ′ and V ≻ V ′ , then ( t V ) ≻ t ′ [ x := V ′ ] • If t ≻ � t 1 , t 2 � , then ( t π i ) ≻ t i • If t ≻ ω i V and u i ≻ u ′ i , then ( t [ x 1 , u 1 , x 2 , u 2 ]) ≻ u ′ i [ x i := V ] • If t ≻ µa.t ′ and ε ≻ ε ′ , then ( t ε ) ≻ µa.t ′ [ a := ∗ ε ′ ] • If t ≻ ( u [ x 1 , u 1 , x 2 , u 2 ]), u ≻ u ′ , u i ≻ u ′ i and ε ≻ ε ′ , then ( t ε ) ≻ ( u ′ [ x 1 , ( u ′ 1 ε ′ ) , x 2 , ( u ′ 2 ε ′ )]) 11

• If t ≻ ( u [ x 1 , u 1 , x 2 , u 2 ]), u ≻ u ′ , u i ≻ u ′ i and V ≻ V ′ , then ( V t ) ≻ ( u ′ [ x 1 , ( V ′ u ′ 1 ) , x 2 , ( V ′ u ′ 2 )]) ≻ V ′ and t ≻ µa.u , then ( V • If V t ) ≻ µa.u [ a := ∗ V ] • If t ≻ µa.u , then ω i t ≻ µa.u [ a := ω i ] • If t ≻ µa.u [ a := ∗ [ x.p, y.q ]], u ≻ u ′ , p ≻ p ′ , q ≻ q ′ , and ε ≻ ε ′ , then ( t ε ) ≻ µa.u ′ [ a := ∗ [ x. ( p ′ ε ′ ) , y. ( q ′ ε ′ )]] • If t ≻ µa.u [ a := ∗ [ x.p, y.q ]], u ≻ u ′ , p ≻ p ′ , q ≻ q ′ , and V ≻ V ′ , then ( V t ) ≻ µa.u ′ [ a := ∗ [ x. ( V ′ p ′ ) , y. ( V ′ q ′ )]] 12

Lemmas t ≻ n t ′ denotes that t is reduced to t ′ by n one-steps of ≻ . Lemma • If t ⊲ v t ′ , then t ≻ t ′ . v t ′ , then t ≻ n t ′ • If t ⊲ ∗ • If t ≻ n t ′ , then t ⊲ ∗ v t ′ The key lemma If t ≻ n t 1 and t ≻ m t 2 , then there exists t 3 such that t 1 ≻ m t 3 and t 2 ≻ n t 3 . Proof By induction on t . 13

What about the strong normalization ? Is very difficult in the presence of the µ ′ -rule. • W. Py and R. David: µµ ′ v • K. Nakazawa: µµ ′ v . • R. David and K. Nour: µµ ′ 14