✤ ✜ On Continuous Normalization ✣ ✢ Klaus Aehlig Felix Joachimski Mathematisches Institut LMU M¨ unchen { aehlig | joachski } @mathematik.uni-muenchen.de CSL 2002 http://www.mathematik.uni-muenchen.de/ ˜ { aehlig | joachski } On Continuous Normalization CSL 2002 1 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Some history • Cut-elimination: a cut ¬ A ∨ ¬ B A ∧ B is replaced by cuts on A and B . ⊥ • This process is defined by induction on (semi-formal) derivations. • Want to separate the operational definition from the proof-theoretical analysis (well-foundedness of the derivation involves ordinal notation systems and strong means). • In order to compute the normalized derivation in a primitive recursive way • Mints 1978 introduced repetition rule ∗ Γ ⊢ A to compute the last rule of the Γ ⊢ A normalized derivation (“Please wait; your proof will soon be computed (hopefully)”) • This procedure even applies to non-wellfounded derivations (it is continuous). ∗ which has the subformula property! On Continuous Normalization CSL 2002 2 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Goal • Transfer ideas to λ -calculus (as Schwichtenberg 1998 and others did). On Continuous Normalization CSL 2002 3 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Goal • Transfer ideas to λ -calculus (as Schwichtenberg 1998 and others did). • Interesting, because some λ -terms (like Y ) have infinite “normal forms”, or are even diverging (like Ω ). On Continuous Normalization CSL 2002 3 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Goal • Transfer ideas to λ -calculus (as Schwichtenberg 1998 and others did). • Interesting, because some λ -terms (like Y ) have infinite “normal forms”, or are even diverging (like Ω ). • Need a coinductive λ -calculus to write out the normal forms! On Continuous Normalization CSL 2002 3 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Goal • Transfer ideas to λ -calculus (as Schwichtenberg 1998 and others did). • Interesting, because some λ -terms (like Y ) have infinite “normal forms”, or are even diverging (like Ω ). • Need a coinductive λ -calculus to write out the normal forms! ✬ ✩ Thus: define and analyze a primitive recursive normalization function ( ) β : Λ (co) → Λ co R ✫ ✪ On Continuous Normalization CSL 2002 3 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Goal • Transfer ideas to λ -calculus (as Schwichtenberg 1998 and others did). • Interesting, because some λ -terms (like Y ) have infinite “normal forms”, or are even diverging (like Ω ). • Need a coinductive λ -calculus to write out the normal forms! ✬ ✩ Thus: define and analyze a primitive recursive normalization function ( ) β : Λ (co) → Λ co R ✫ ✪ In particular, explain why and how many repetition rules are needed. On Continuous Normalization CSL 2002 3 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Terms Terms. Inductive and coinductive λ -calculus in de Bruijn-notation ( x ∈ N ). Λ ∋ r, s ::= x | rs | λr On Continuous Normalization CSL 2002 4 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Terms Terms. Inductive and coinductive λ -calculus in de Bruijn-notation ( x ∈ N ). Λ ∋ r, s ::= x | rs | λr r, s ::= co x | rs | λr Λ co ∋ On Continuous Normalization CSL 2002 4 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Terms Terms. Inductive and coinductive λ -calculus in de Bruijn-notation ( x ∈ N ). Λ ∋ r, s ::= x | rs | λr r, s ::= co x | rs | λr Λ co ∋ Λ R ∋ r, s ::= x | rs | λr | R r On Continuous Normalization CSL 2002 4 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Terms Terms. Inductive and coinductive λ -calculus in de Bruijn-notation ( x ∈ N ). Λ ∋ r, s ::= x | rs | λr r, s ::= co x | rs | λr Λ co ∋ Λ R ∋ r, s ::= x | rs | λr | R r | βr On Continuous Normalization CSL 2002 4 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Terms Terms. Inductive and coinductive λ -calculus in de Bruijn-notation ( x ∈ N ). Λ ∋ r, s ::= x | rs | λr r, s ::= co x | rs | λr Λ co ∋ Λ R ∋ r, s ::= x | rs | λr | R r | βr r, s ::= co x | rs | λr | R r | βr Λ co ∋ R On Continuous Normalization CSL 2002 4 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Terms Terms. Inductive and coinductive λ -calculus in de Bruijn-notation ( x ∈ N ). Λ ∋ r, s ::= x | rs | λr r, s ::= co x | rs | λr Λ co ∋ Λ R ∋ r, s ::= x | rs | λr | R r | βr r, s ::= co x | rs | λr | R r | βr Λ co ∋ R Examples. Θ := tt with t = λλ. 0 . 110 On Continuous Normalization CSL 2002 4 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Terms Terms. Inductive and coinductive λ -calculus in de Bruijn-notation ( x ∈ N ). Λ ∋ r, s ::= x | rs | λr r, s ::= co x | rs | λr Λ co ∋ Λ R ∋ r, s ::= x | rs | λr | R r | βr r, s ::= co x | rs | λr | R r | βr Λ co ∋ R Examples. Θ := tt with t = λλ. 0 . 110 = “ λtλf ( f ( ttf )) ” On Continuous Normalization CSL 2002 4 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Terms Terms. Inductive and coinductive λ -calculus in de Bruijn-notation ( x ∈ N ). Λ ∋ r, s ::= x | rs | λr r, s ::= co x | rs | λr Λ co ∋ Λ R ∋ r, s ::= x | rs | λr | R r | βr r, s ::= co x | rs | λr | R r | βr Λ co ∋ R Examples. Θ := tt with t = λλ. 0 . 110 = “ λtλf ( f ( ttf )) ” := λ. ( λ. 1 . 00)( λ. 1 . 00) = “ λf. ( λx ( f ( xx )))( λx ( f ( xx ))) ” Y On Continuous Normalization CSL 2002 4 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Terms Terms. Inductive and coinductive λ -calculus in de Bruijn-notation ( x ∈ N ). Λ ∋ r, s ::= x | rs | λr r, s ::= co x | rs | λr Λ co ∋ Λ R ∋ r, s ::= x | rs | λr | R r | βr r, s ::= co x | rs | λr | R r | βr Λ co ∋ R Examples. Θ := tt with t = λλ. 0 . 110 = “ λtλf ( f ( ttf )) ” := λ. ( λ. 1 . 00)( λ. 1 . 00) = “ λf. ( λx ( f ( xx )))( λx ( f ( xx ))) ” Y Y co rY co := = r ( r ( r . . . r r On Continuous Normalization CSL 2002 4 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Terms Terms. Inductive and coinductive λ -calculus in de Bruijn-notation ( x ∈ N ). Λ ∋ r, s ::= x | rs | λr r, s ::= co x | rs | λr Λ co ∋ Λ R ∋ r, s ::= x | rs | λr | R r | βr r, s ::= co x | rs | λr | R r | βr Λ co ∋ R Examples. Θ := tt with t = λλ. 0 . 110 = “ λtλf ( f ( ttf )) ” := λ. ( λ. 1 . 00)( λ. 1 . 00) = “ λf. ( λx ( f ( xx )))( λx ( f ( xx ))) ” Y Y co rY co := = r ( r ( r . . . r r Observational equality. r ≃ k s iff the outermost k constructors are identical. E.g., 1 λλ 0 ≃ 2 1 λ (0 2) . Equality. r = s iff r ≃ k s for all k . On Continuous Normalization CSL 2002 4 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Continuous normalization Substitution. r [ s ] is well-defined On Continuous Normalization CSL 2002 5 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Continuous normalization Substitution. r [ s ] is well-defined and continuous with identity as modulus of continuity, i.e., r ≃ k r ′ ∧ s ≃ k s ′ ⇒ r [ s ] ≃ k r ′ [ s ′ ] . On Continuous Normalization CSL 2002 5 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Continuous normalization Substitution. r [ s ] is well-defined and continuous with identity as modulus of continuity, i.e., r ≃ k r ′ ∧ s ≃ k s ′ ⇒ r [ s ] ≃ k r ′ [ s ′ ] . Reduction. → is the compatible closure of ( λr ) s → r [ s ] . On Continuous Normalization CSL 2002 5 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
Continuous normalization Substitution. r [ s ] is well-defined and continuous with identity as modulus of continuity, i.e., r ≃ k r ′ ∧ s ≃ k s ′ ⇒ r [ s ] ≃ k r ′ [ s ′ ] . Reduction. → is the compatible closure of ( λr ) s → r [ s ] . Normal forms. NF ∋ r, s ::= co x� r | λr | R r | βr . On Continuous Normalization CSL 2002 5 History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions
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