Classical Combinatory Logic Karim Nour LAMA - Equipe de logique - PDF document

Classical Combinatory Logic Karim Nour LAMA - Equipe de logique Universit´ e de Savoie 73376 Le Bourget du Lac France e-mail: nour@univ-savoie.fr and some helpful discussions with Ren´ e David 1

Content 1. (Ordinary) Combinatory Logic 2. Classical Combinatory Logic (Karim Nour) 3. Strong Normalization (Ren´ e David) 2

Combinatory Logic CL 3

The λ -calculus • The terms t := x | λx.t | ( t t ) • The reduction rule ( λx.u v ) → u [ x := v ] • The typing rules Γ , x : A ⊢ x : A Γ , x : A ⊢ u : B Γ ⊢ λx.u : A → B Γ ⊢ u : A → B Γ ⊢ v : A Γ ⊢ ( u v ) : B 4

The combinatory logic • The terms T := x | K | S | ( T T ) • The reduction rules ( K U V ) ⊲ U ( S U V W ) (( U W ) ( V W )) ⊲ • The typing rules Γ , x : A ⊢ x : A Γ ⊢ K : A → ( B → A ) Γ ⊢ S : ( A → ( B → C )) → (( A → B ) → ( A → C )) Γ ⊢ U : A → B Γ ⊢ V : A Γ ⊢ ( U V ) : B 5

CL � λ -calculus The coding ψ ( x ) = x ψ ( K ) = λx.λy.x ψ ( S ) = λx.λy.λz. (( x z ) ( y z )) ψ (( U V )) = ( ψ ( U ) ψ ( V )) Theorem • If Γ ⊢ c U : A , then Γ ⊢ λ ψ ( U ) : A . • If U ⊲ V , then ψ ( U ) → + ψ ( V ). 6

λ -calculus � CL The coding φ ( x ) = x φ ( λx.t ) = l x ( φ ( t )) φ (( u v )) = ( φ ( u ) φ ( v )) where l x ( x ) = I = ( S K K ) l x ( T ) = ( K T ) if x �∈ V ar ( T ) l x (( U V )) = ( S l x ( U ) l x ( V )) Theorem • If Γ ⊢ λ t : A , then Γ ⊢ c φ ( t ) : A . • ( l x ( U ) V ) ⊲ ∗ U [ x := V ]. • If u → w v , then φ ( u ) ⊲ + φ ( v ). 7

Classical Combinatory Logic CCL 8

Add new combinators with appropriate typing rules? • ¬¬ A → A • ( ¬ A → A ) → A and ⊥→ A • ( ¬ A → ¬ B ) → ( B → A ) and ⊥→ A Problem The reduction rule is complicated and artificial . 9

Code an existing classical calculus? • The λµ -calculus (Parigot) This reduction rule is problematic since it must somehow “code” the translation. ( µa.u v ) → µa.u [ a := ∗ v ] • The λµ ˜ µ -calculus (Curien and Herbelin) Same problem with the reduction rule � λx.u, v.w � → � v, µx. � u, w �� • The λ Sym -calculus (Barbanera and Berardi) OK 10

The types of λ Sym -calculus • Two sets of variables A = { a, b, ... } and A ⊥ = { a ⊥ , b ⊥ , ... } . • The set of m -types A ::= a | a ⊥ | A ∧ A | A ∨ A • The set of types C ::= A |⊥ • For an m -type A , A ⊥ is defined by: ( a ) ⊥ = a ⊥ and ( a ⊥ ) ⊥ = a ( A ∧ B ) ⊥ = A ⊥ ∨ B ⊥ and ( A ∨ B ) ⊥ = A ⊥ ∧ B ⊥ (and thus A ⊥⊥ = A ). 11

The terms and the typing rules of λ Sym -calculus t := x | � t, t � | σ 1 ( t ) | σ 2 ( t ) | λx.t | t ⋆ t Γ , x : A ⊢ x : A Γ ⊢ u : A Γ ⊢ v : B Γ ⊢ � u, v � : A ∧ B Γ ⊢ t : A Γ ⊢ t : B Γ ⊢ σ 1 ( t ) : A ∨ B Γ ⊢ σ 2 ( t ) : A ∨ B Γ ⊢ u : A ⊥ Γ , x : A ⊢ t : ⊥ Γ ⊢ v : A Γ ⊢ λx.t : A ⊥ Γ ⊢ u ⋆ v : ⊥ 12

The reduction rules of λ Sym -calculus ( λx.u ) ⋆ v → u [ x := v ] v ⋆ ( λx.u ) → u [ x := v ] � u, v � ⋆ σ 1 ( w ) → u ⋆ w � u, v � ⋆ σ 2 ( w ) → v ⋆ w σ 1 ( w ) ⋆ � u, v � → w ⋆ u σ 2 ( w ) ⋆ � u, v � → w ⋆ v λx. ( u ⋆ x ) (1) → u λx. ( x ⋆ u ) (1) → u u [ x := v ] (2) → v (1) x �∈ Fv ( u ) (2) If u and v with type ⊥ , x occurs only once in u and u � = x . Lemma If Γ ⊢ λ s u : A and u → ∗ v , then Γ ⊢ λ s v : A . 13

The terms of CCL T := x | K | S | C | P | Q 1 | Q 2 | ( T T ) | T ⋆ T 14

The typing rules of CCL Γ , x : A ⊢ x : A Γ ⊢ K : A ⊥ ∨ ( B ∨ A ) Γ ⊢ S : ( A ∧ ( B ∧ C ⊥ )) ∨ (( A ∧ B ⊥ ) ∨ ( A ⊥ ∨ C )) Γ ⊢ C : ( A ∧ B ) ∨ (( A ∧ B ⊥ ) ∨ A ⊥ ) Γ ⊢ P : A ⊥ ∨ ( B ⊥ ∨ ( A ∧ B )) Γ ⊢ Q 1 : A ⊥ ∨ ( A ∨ B ) Γ ⊢ Q 2 : B ⊥ ∨ ( A ∨ B ) Γ ⊢ U : A ⊥ ∨ B Γ ⊢ V : A Γ ⊢ ( U V ) : B Γ ⊢ U : A ⊥ Γ ⊢ V : A Γ ⊢ U ⋆ V : ⊥ 15

The typing rules of CCL Γ , x : A ⊢ x : A Γ ⊢ K : A → ( B → A ) Γ ⊢ S : ( A → ( B → C )) → (( A → B ) → ( A → C )) Γ ⊢ C : ( A → B ⊥ ) → (( A → B ) → A ⊥ ) Γ ⊢ P : A → ( B → ( A ∧ B )) Γ ⊢ Q 1 : A → ( A ∨ B ) Γ ⊢ Q 2 : B → ( A ∨ B ) Γ ⊢ U : A → B Γ ⊢ V : A Γ ⊢ ( U V ) : B Γ ⊢ U : A ⊥ Γ ⊢ V : A Γ ⊢ U ⋆ V : ⊥ 16

The reduction rules of CCL ( K U V ) ⊲ U ( S U V W ) (( U W ) ( V W )) ⊲ ( C U V ) ⋆ W ( U W ) ⋆ ( V W ) ⊲ W ⋆ ( C U V ) ( U W ) ⋆ ( V W ) ⊲ ( P U V ) ⋆ ( Q 1 W ) ⊲ U ⋆ W ( P U V ) ⋆ ( Q 2 W ) ⊲ V ⋆ W ( Q 1 W ) ⋆ ( P U V ) ⊲ W ⋆ U ( Q 2 W ) ⋆ ( P U V ) ⊲ W ⋆ V ( C ( K U ) I ) (1) ⊲ U ( C I ( K U )) (1) ⊲ U W [ x := ( C ( K U ) ( K V ))] U ⋆ V (2) ⊲ (1) I = ( S K K ) ( ⊢ cc I : A ⊥ ∨ A ) (2) If W has type ⊥ Lemma If Γ ⊢ cc U : A and U ⊲ ∗ V , then Γ ⊢ cc V : A . 17

The coding: λ Sym � CCL φ ( x ) = x φ ( λx.t ) = l x ( φ ( t )) φ ( u ⋆ v ) = φ ( u ) ⋆ φ ( v ) φ ( � u, v � ) = ( P φ ( u ) φ ( v )) φ ( σ i ( t )) = ( Q i φ ( t )) where l x ( x ) = I l x ( T ) = ( K T ) if x �∈ V ar ( T ) and T � = T 1 ⋆ T 2 l x (( U V )) = ( S l x ( U ) l x ( V )) l x ( U ⋆ V ) = ( C l x ( U ) l x ( V )) Theorem • If Γ ⊢ λ s t : A , the Γ ⊢ cc φ ( t ) : A . • If u → w v , then φ ( u ) ⊲ + φ ( v ). 18

Some definitions π i t = λx. ( t⋆σ i ( x )) and π i 1 ...i n t = π i 1 ...π i n t [ u, v ] = λx. ( u ⋆ � v, x � ) Lemma • π i � u 1 , u 2 � → ∗ u i . • If Γ ⊢ λ s t : A 1 ∧ A 2 , then Γ ⊢ λ s π i t : A i . • If Γ ⊢ λ s u : A ⊥ ∨ B and Γ ⊢ λ s v : A , then Γ ⊢ λ s [ u, v ] : B . 19

Example Let K = λx. ( π 1 x ⋆ π 22 x ). • ⊢ λ c K : A ⊥ ∨ ( B ∨ A ) x : A ∧ ( B ⊥ ∧ A ⊥ ) x : A ∧ ( B ⊥ ∧ A ⊥ ) π 2 x : B ⊥ ∧ A ⊥ π 22 x : A ⊥ π 1 x : A π 1 x ⋆ π 22 x : ⊥ K : A ⊥ ∨ ( B ∨ A ) • [[ K , u ] , v ] → ∗ u [[ K , u ] , v ] [ λy. ( u ⋆ π 2 y ) , v ] → → λz.u ⋆ z → u 20

The coding: CCL � λ Sym ψ ( x ) = x ψ ( K ) = λx. ( π 1 x ⋆ π 22 x ) ψ ( S ) = λx. ([[ π 1 x, π 122 x ] , [ π 12 x, π 122 x ]] ⋆ π 222 x ) ψ ( C ) = λx. ([ π 1 x, π 22 x ] ⋆ [ π 12 x, π 22 x ]) ψ ( P ) = λx. ( � π 1 x, π 12 x � ⋆ π 22 x ) ψ ( Q 1 ) = λx. ( σ 1 ( π 1 x ) ⋆ π 2 x ) ψ ( Q 2 ) = λx. ( σ 2 ( π 1 x ) ⋆ π 2 x ) ψ (( U V )) = [ ψ ( U ) , ψ ( V )] ψ ( U ⋆ V ) = ψ ( U ) ⋆ ψ ( V ) Theorem • If Γ ⊢ cc U : A , then Γ ⊢ λ s ψ ( U ) : A . • If U ⊲ V , then ψ ( U ) → + ψ ( V ). 21

Strong Normalization ( SN ) ( Ren´ e David) 22

Proofs of SN for λ Sym 1. The original one: using candidates of re- ducibility defined by a fix point and thus not arithmetical . 2. A proof (by P. Battyanyi) adapted from our proof for the symmetric λµ -calculus and the λµ ˜ µ -calculus. 3. A proof using CCL ? 23

The problem We would like 1. SN of CCL ? SN of λ Sym )? 2. ( SN of CCL ⇒ But • If u → w v , then φ ( u ) ⊲ + φ ( v ). does not implies 2. 24

The solution V SN : stronger notion of SN . 1. V SN of CCL ? 2. ( φ ( t ) ∈ V SN t ∈ SN )? ⇒ 25