Computational interpretations of intuitionistic logic Sequent term calculi for intuitionistic logic λ Gtz -calculus with intersection types Computational interpretations of logics Silvia Ghilezan University of Novi Sad, Serbia Belgrade, January 30, 2009 S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic Sequent term calculi for intuitionistic logic λ Gtz -calculus with intersection types Outline of the talk - first part Computational interpretations of intuitionistic logic 1 Axiomatic System - Combinatory Logic Natural Deduction - λ -calculus Sequent calculus - ? Sequent term calculi for intuitionistic logic 2 λ LJ -calculus ¯ λ -calculus λ Gtz -calculus λ Gtz -calculus with intersection types 3 Calculi with gen. application and explicit substitution Ongoing work S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic Axiomatic System - Combinatory Logic Sequent term calculi for intuitionistic logic Natural Deduction - λ -calculus λ Gtz -calculus with intersection types Sequent calculus - ? Computational interpretations of intuitionistic logic Curry-Howard-de Brujin-Lambek correspondence logic vs term calculus types as formulae – terms as proofs – terms as programs ⊢ A ⇔ ⊢ t : A axiomatic (Hilbert) system (axioms/Modus Ponens) Combinatory Logic (combinators/application) 1930s Schönfinkel, Curry natural deduction (introduction/elimination) λ calculus (abstraction/application) 1940s Church sequent calculus (right/left introduction/cut) various attempts λ calculus (abstraction/application/substitution) 1970s S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic Axiomatic System - Combinatory Logic Sequent term calculi for intuitionistic logic Natural Deduction - λ -calculus λ Gtz -calculus with intersection types Sequent calculus - ? Axiomatic (Hilbert style) system - Combinatory Logic ( Ax 1 ) ⊢ A → A ( Ax 2 ) ⊢ A → ( B → A ) ( Ax 3 ) ⊢ ( A → ( B → C )) → (( A → B ) → ( A → C )) ⊢ A → B ⊢ A ( MP ) ⊢ B S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic Axiomatic System - Combinatory Logic Sequent term calculi for intuitionistic logic Natural Deduction - λ -calculus λ Gtz -calculus with intersection types Sequent calculus - ? Axiomatic (Hilbert style) system - Combinatory Logic ( Ax 1 ) ⊢ I : A → A ( Ax 2 ) ⊢ K : A → ( B → A ) ( Ax 3 ) ⊢ S : ( A → ( B → C )) → (( A → B ) → ( A → C )) ⊢ t : A → B ⊢ s : A ( MP ) ts : ⊢ B S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic Axiomatic System - Combinatory Logic Sequent term calculi for intuitionistic logic Natural Deduction - λ -calculus λ Gtz -calculus with intersection types Sequent calculus - ? Natural Deduction - λ -calculus ( axiom ) Γ , A ⊢ A Γ ⊢ A → B Γ ⊢ A ( → elim ) Γ ⊢ B Γ , A ⊢ B ( → intr ) Γ ⊢ A → B ⊢ A ⇔ ⊢ t : A S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic Axiomatic System - Combinatory Logic Sequent term calculi for intuitionistic logic Natural Deduction - λ -calculus λ Gtz -calculus with intersection types Sequent calculus - ? Natural Deduction - λ -calculus ( axiom ) Γ , x : A ⊢ x : A Γ ⊢ t : A → B Γ ⊢ s : A ( → elim ) ( app ) Γ ⊢ ts : B Γ , x : A ⊢ t : B ( → intr ) ( abs ) Γ ⊢ λ x . t : A → B ⊢ A ⇔ ⊢ t : A S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic Axiomatic System - Combinatory Logic Sequent term calculi for intuitionistic logic Natural Deduction - λ -calculus λ Gtz -calculus with intersection types Sequent calculus - ? Sequent calculus - ? ( axiom ) Γ , A ⊢ A Γ ⊢ A Γ , B ⊢ C ( → left ) Γ , A → B ⊢ C Γ , A ⊢ B ( → right ) Γ ⊢ A → B Γ ⊢ A Γ , A ⊢ B ( cut ) Γ ⊢ B S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic λ LJ -calculus ¯ Sequent term calculi for intuitionistic logic λ -calculus λ Gtz -calculus with intersection types λ Gtz -calculus Sequent calculus intuitionistic logic Pottinger, Zucker 1970s comparing cut-elimination to proof normalization Gallier [1991] Mints [1996] Barendregt, Ghilezan [2000]: λ LJ -calculus But in these, terms do not encode derivations. Herbelin [1995]: ¯ λ -calculus - developed the idea of making terms explicitly represent sequent calculus derivations. Computation over terms reflects cut-elimination Espírito Santo [2006]: λ Gtz -calculus S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic λ LJ -calculus ¯ Sequent term calculi for intuitionistic logic λ -calculus λ Gtz -calculus with intersection types λ Gtz -calculus λ LJ -calculus Barendregt and Ghilezan term calculus λ -calculus type system LJ Γ ⊢ Γ , B ⊢ A C ( axiom ) ( → left ) Γ A ⊢ A Γ , A → B ⊢ C Γ , A ⊢ B Γ ⊢ A Γ , A ⊢ B ( → right ) ( cut ) Γ ⊢ : A → B Γ ⊢ : B S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic λ LJ -calculus ¯ Sequent term calculi for intuitionistic logic λ -calculus λ Gtz -calculus with intersection types λ Gtz -calculus λ LJ -calculus Barendregt and Ghilezan term calculus λ -calculus - natural deduction term structure type system LJ - sequent types structure Γ ⊢ t : A Γ , x : B ⊢ s : C ( axiom ) ( → left ) Γ x : A ⊢ x : A Γ , y : A → B ⊢ s [ x := yt ] : C Γ , x : A ⊢ t : B Γ ⊢ t : A Γ , x : A ⊢ s : B ( → right ) ( cut ) Γ ⊢ ( λ x . t ) : A → B Γ ⊢ s [ x := t ] : B S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic λ LJ -calculus ¯ Sequent term calculi for intuitionistic logic λ -calculus λ Gtz -calculus with intersection types λ Gtz -calculus From λ LJ to ¯ λ -calculus λ LJ -calculus: Using a subsystem λ LJ cf Gentzen’s Hauptsatz (cut-elimination) theorem is easily proved! But, the Curry-Howard correspondence fails... u �→ yz �→ λ x . yz u �→ λ x . u �→ λ x . yz . S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic λ LJ -calculus ¯ Sequent term calculi for intuitionistic logic λ -calculus λ Gtz -calculus with intersection types λ Gtz -calculus From λ LJ to ¯ λ -calculus λ LJ -calculus: Using a subsystem λ LJ cf Gentzen’s Hauptsatz (cut-elimination) theorem is easily proved! But, the Curry-Howard correspondence fails... u �→ yz �→ λ x . yz u �→ λ x . u �→ λ x . yz . ¯ λ -calculus of Herbelin introduction of explicit substitution λ x . ( u � u = yz � ) ( λ x . u ) � u = yz � restriction of the sequent logic LJ - LJT : (Γ; ⊢ A ) i (Γ; B ⊢ A ) introduction of a new constructor - list of arguments instead of (( yu 1 ) ... u n ) the applicative part is y [ u 1 ; ... ; u n ] . S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic λ LJ -calculus ¯ Sequent term calculi for intuitionistic logic λ -calculus λ Gtz -calculus with intersection types λ Gtz -calculus ¯ λ -calculus Syntax: (Terms) t , u , v ::= xl | λ x . t | tl | t � x = v � [ ] | t :: l | l @ l ′ | l � x = t � l , l ′ ::= (Lists) Reduction rules: ( β cons ) λ x . u ( v :: l ) → u � x = v � l ( β nil ) λ x . u [ ] → λ x . u ( C var ) ( tl ) l ′ → t ( l @ l ′ ) ( C cons ) ( t :: l )@ l ′ → t :: ( l @ l ′ ) ( C nil ) [ ]@ l → l ( S yes ) ( xl ) � x = v � → vl � x = v � ( S no ) ( yl ) � x = v � → yl � x = v � ( S λ ) ( λ y . u ) � x = v � → λ y . ( u � x = v � ) ( S nil ) [ ] � x = v � → [ ] ( S cons ) ( u :: l ) � x = v � → u � x = v � :: l � x = v � . S.Ghilezan Computational interpretations of logics
Computational interpretations of intuitionistic logic λ LJ -calculus ¯ Sequent term calculi for intuitionistic logic λ -calculus λ Gtz -calculus with intersection types λ Gtz -calculus ¯ λ - simple types Γ , x : A ; . : A ⊢ ( . l ) : B ( Ax ) ( Cont ) Γ ; . : A ⊢ ( . [ ]) : A Γ , x : A ; ⊢ xl : B Γ; ⊢ t : A Γ ; . : B ⊢ ( . l ) : C Γ , x : A ; ⊢ t : B ( → R ) ( → L ) Γ; ⊢ λ x . t : A → B Γ ; . : A → B ⊢ ( . ( t :: l )) : C Γ ; . : A ⊢ ( . l ) : C Γ ; . : C ⊢ ( . l ′ ) : B Γ; ⊢ t : A Γ ; . : A ⊢ ( . l ) : B ( C H 2 ) ( C H 1 ) Γ ; . : A ⊢ ( . l @ l ′ ) : B Γ; ⊢ tl : B Γ; ⊢ t : A Γ , x : A ; ⊢ u : B Γ; ⊢ t : C Γ , x : C ; . : A ⊢ ( . l ) : B ( C M 1 ) ( C M 2 ) Γ; ⊢ u � x = t � : B Γ ; . : A ⊢ ( . l � x = t � ) : B Curry-Howard correspondence: normal forms of ¯ λ correspond to cut-free proofs in LJT . S.Ghilezan Computational interpretations of logics
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