Generalized Developments in -calculus (SLIDES) Article February - PDF document

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/251230477 Generalized Developments in λ -calculus (SLIDES) Article · February 2009 CITATIONS READS 0 121 2 authors: Yiorgos Stavrinos George Koletsos National and Kapodistrian University of Athens National Technical University of Athens 19 PUBLICATIONS 39 CITATIONS 31 PUBLICATIONS 121 CITATIONS SEE PROFILE SEE PROFILE All content following this page was uploaded by Yiorgos Stavrinos on 22 May 2014. The user has requested enhancement of the downloaded file.

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion Generalized Developments in λ -calculus Yiorgos Stavrinos ( joint work with George Koletsos ) February 6, 2009 Yiorgos Stavrinos: Generalized Developments in λ -calculus 1/17

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion Outline Generalized developments � λ -calculi with types � Embedding � Finiteness of gen. developments � Conclusion � Yiorgos Stavrinos: Generalized Developments in λ -calculus 2/17

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion • V = { x , y , z , . . . } infinite set of variables Λ: set of λ -terms M ::= x | λ x . P | PQ − → β P [ Q / x ] β -reduction: ( λ x . P ) Q � �� � � �� � redex contractum • Generalized development generalized β -redex ( g-redex ): ( λ x 1 . . . λ x n . P ) Q 1 . . . Q n F a set of g-redex occurrences in M ∆ ∆ 1 ∆ 2 M − → β M 1 − → β M 2 − → β . . . Yiorgos Stavrinos: Generalized Developments in λ -calculus 3/17

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion • V = { x , y , z , . . . } infinite set of variables Λ: set of λ -terms M ::= x | λ x . P | PQ − → β P [ Q / x ] β -reduction: ( λ x . P ) Q � �� � � �� � redex contractum • Generalized development generalized β -redex ( g-redex ): ( λ x 1 . . . λ x n . P ) Q 1 . . . Q n F a set of g-redex occurrences in M ∆ ∈F ∆ 1 ∆ 2 ( M , F ) − → β M 1 − → β M 2 − → β . . . Yiorgos Stavrinos: Generalized Developments in λ -calculus 3/17

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion • V = { x , y , z , . . . } infinite set of variables Λ: set of λ -terms M ::= x | λ x . P | PQ − → β P [ Q / x ] β -reduction: ( λ x . P ) Q � �� � � �� � redex contractum • Generalized development generalized β -redex ( g-redex ): ( λ x 1 . . . λ x n . P ) Q 1 . . . Q n F a set of g-redex occurrences in M ∆ ∈F ∆ 1 ∈F 1 ∆ 2 ( M , F ) − → β ( M 1 , F 1 ) − → β M 2 − → β . . . Yiorgos Stavrinos: Generalized Developments in λ -calculus 3/17

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion • V = { x , y , z , . . . } infinite set of variables Λ: set of λ -terms M ::= x | λ x . P | PQ − → β P [ Q / x ] β -reduction: ( λ x . P ) Q � �� � � �� � redex contractum • Generalized development generalized β -redex ( g-redex ): ( λ x 1 . . . λ x n . P ) Q 1 . . . Q n F a set of g-redex occurrences in M ∆ ∈F ∆ 1 ∈F 1 ∆ 2 ∈F 2 ( M , F ) − → β ( M 1 , F 1 ) − → β ( M 2 , F 2 ) − → β . . . Yiorgos Stavrinos: Generalized Developments in λ -calculus 3/17

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion • V = { x , y , z , . . . } infinite set of variables Λ: set of λ -terms M ::= x | λ x . P | PQ − → β P [ Q / x ] β -reduction: ( λ x . P ) Q � �� � � �� � redex contractum • Generalized development generalized β -redex ( g-redex ): ( λ x 1 . . . λ x n . P ) Q 1 . . . Q n F a set of g-redex occurrences in M ∆ ∈F ∆ 1 ∈F 1 ∆ 2 ∈F 2 ( M , F ) − → β ( M 1 , F 1 ) − → β ( M 2 , F 2 ) − → β . . . • Λ ′ : set of indexed λ -terms M ::= x | λ x . P | PQ | ( λ 0 x 1 . . . λ 0 x n . P ) Q 1 . . . Q n � �� � generalized indexed redex β 0 -reduction: ( λ 0 x 1 . . . λ 0 x n . P ) Q 1 . . . Q n − → β 0 ( λ 0 x 2 . . . λ 0 x n . P [ Q 1 / x ]) Q 2 . . . Q n Yiorgos Stavrinos: Generalized Developments in λ -calculus 3/17

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion Example β -reductions: (( λ x .λ y . x ) λ x . x )(( λ x . xx )( λ x . xx )) − → β (( λ x .λ y . x ) λ x . x )(( λ x . xx )( λ x . xx )) − → β . . . (( λ x .λ y . x ) λ x . x )(( λ x . xx )( λ x . xx )) − → β ( λ y .λ x . x )(( λ x . xx )( λ x . xx )) − → β λ x . x gen. developments: ∗ − → β 0 ( λ y .λ x . x )(( λ x . xx )( λ x . xx )) (( λ 0 x .λ y . x ) λ x . x )(( λ 0 x . xx )( λ x . xx )) ∗ (( λ 0 x .λ 0 y . x ) λ x . x )(( λ 0 x . xx )( λ x . xx )) − → β 0 ( λ 0 y .λ x . x )(( λ x . xx )( λ x . xx )) − → β 0 λ x . x Yiorgos Stavrinos: Generalized Developments in λ -calculus 4/17

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion Gen. developments are simulated by β 0 -reduction To any term M ∈ Λ and a set F of g-redex occurrences in M we associate a term M ′ ∈ Λ ′ where all g-redex occurrences in F are indexed, i.e. x . P ) � x . P ) � if M ≡ . . . ( λ� Q . . . where ( λ� Q ∈ F M ′ ≡ . . . ( λ 0 � x . P ) � then Q . . . Proposition ∗ ( M , F ) − → β ( M n , F n ) Gen. development : Λ � � ∗ M ′ M ′ Λ ′ β 0 -reduction : − → β 0 n Theorem (finiteness of gen. developments) All gen. developments of M ∈ Λ are finite and end with the same term. Equivalently : ∗ − → β 0 is strongly normalizable and has the Church-Rosser property . Yiorgos Stavrinos: Generalized Developments in λ -calculus 5/17

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion Gen. developments are simulated by β 0 -reduction To any term M ∈ Λ and a set F of g-redex occurrences in M we associate a term M ′ ∈ Λ ′ where all g-redex occurrences in F are indexed, i.e. x . P ) � x . P ) � if M ≡ . . . ( λ� Q . . . where ( λ� Q ∈ F M ′ ≡ . . . ( λ 0 � x . P ) � then Q . . . Proposition ∗ ( M , F ) − → β ( M n , F n ) Gen. development : Λ � � ∗ M ′ M ′ Λ ′ β 0 -reduction : − → β 0 n Theorem (finiteness of gen. developments) All gen. developments of M ∈ Λ are finite and end with the same term. Equivalently : ∗ − → β 0 is strongly normalizable and has the Church-Rosser property . Yiorgos Stavrinos: Generalized Developments in λ -calculus 5/17

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion Simply typed λ -calculus, λ → Types σ ::= α | σ → σ , where α ∈ { α 1 , α 2 , . . . } Contexts Γ = { x 1 : σ 1 , . . . , x k : σ k } (Ax.) Γ , x : σ ⊢ x : σ Γ , x : σ ⊢ M : τ Γ ⊢ M : σ → τ Γ ⊢ N : σ ( → E) ( → I) Γ ⊢ MN : τ Γ ⊢ λ x . M : σ → τ ∗ Γ ⊢ M : σ M − → β N Subject reduction property : Γ ⊢ N : σ Theorem If Γ ⊢ M : σ , then M is strongly normalizable and has the Church-Rosser property . Proof: By the method of reducibility: strong normalization [Tait 1967, Girard 1971], C-R property [Koletsos 1985] Yiorgos Stavrinos: Generalized Developments in λ -calculus 6/17

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion Intersection types, system D Types σ ::= α | σ → σ | σ ∩ σ , where α ∈ { α 1 , α 2 , . . . } D := λ → + ( ∩ I) , ( ∩ E) Γ ⊢ M : σ 1 ∩ σ 2 Γ ⊢ M : σ Γ ⊢ M : τ ( ∩ I) ( ∩ E) i = 1 , 2 Γ ⊢ M : σ ∩ τ Γ ⊢ M : σ i Theorem 1. Γ ⊢ M : σ if and only if M is strongly normalizable . 2. If Γ ⊢ M : σ , then M has the Church-Rosser property . Proof: Characterizing the strongly normalizable terms [Pottinger 1980], C-R property [Koletsos-Stavrinos 1997] Yiorgos Stavrinos: Generalized Developments in λ -calculus 7/17

Outline Generalized developments λ -calculi with types Embedding Finiteness of gen. developments Conclusion The sets Λ C , Λ C ! Add to the variables of Λ a set C = { c 1 , c 2 , . . . , c i , . . . } of new variables. So now the set of variables is V C = { x , y , z , . . . } ∪ C . x ∈ V C \ C , c i ∈ C Λ C : x . P ) � M ::= x | λ x . P | c i PQ | ( λ� Q , where x ,� Λ C ! : the set of λ -terms M ∈ Λ C s.t. the c i ’s occur at most once in M Example ∈ Λ C , but λ x . c i yx ∈ Λ C c i , λ x . yx / ( λ x .λ y . x ) y , ( λ x .λ y . x ) yz , c i (( λ x .λ y . x ) y ) z ∈ Λ C ∈ Λ C ! , but ( λ x . c 3 yx )( λ x . c 1 yx ) ∈ Λ C ! ( λ x . c 1 yx )( λ x . c 1 yx ) / Lemma If M ∈ Λ C and M ∗ → β N, then N ∈ Λ C . − Yiorgos Stavrinos: Generalized Developments in λ -calculus 8/17