Untyped Normalization-by-Evaluation Klaus Aehlig Felix - - PowerPoint PPT Presentation

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Untyped Normalization-by-Evaluation Klaus Aehlig Felix - - PowerPoint PPT Presentation

Jun 22, 2023 304 likes •1.05k views

Untyped Normalization-by-Evaluation Klaus Aehlig Felix Joachimski Mathematisches Institut LMU M unchen { aehlig , joachski } @mathematik.uni-muenchen.de Tallinn, 17.4.4 Klaus Aehlig and Felix Joachimski. Operational

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 1

✤ ✣ ✜ ✢

Untyped Normalization-by-Evaluation

Klaus Aehlig Felix Joachimski Mathematisches Institut LMU M¨ unchen

{aehlig,joachski}@mathematik.uni-muenchen.de

Tallinn, 17.4.4

Klaus Aehlig and Felix Joachimski. Operational aspects of untyped normalization by evaluation. Accepted by Mathematical Structures in Computer Science, 2003. http://www.mathematik.uni-muenchen.de/˜aehlig

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 2

Motivation

✬ ✫ ✩ ✪

[[·]]ξ : Terms − → Semantics (evaluation function) ↓ : Semantics − → Normal forms (inverse) ↑ : Variables − → Semantics (valuation) ✄ Soundness: r → s ⇒ [[r]]ξ = [[s]]ξ ✄ Reproduction: r in normal form ⇒ ↓[[r]]↑ = r

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 2

Motivation

✬ ✫ ✩ ✪

[[·]]ξ : Terms − → Semantics (evaluation function) ↓ : Semantics − → Normal forms (inverse) ↑ : Variables − → Semantics (valuation) ✄ Soundness: r → s ⇒ [[r]]ξ = [[s]]ξ ✄ Reproduction: r in normal form ⇒ ↓[[r]]↑ = r r =β s ∈ NFβ

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 2

Motivation

✬ ✫ ✩ ✪

[[·]]ξ : Terms − → Semantics (evaluation function) ↓ : Semantics − → Normal forms (inverse) ↑ : Variables − → Semantics (valuation) ✄ Soundness: r → s ⇒ [[r]]ξ = [[s]]ξ ✄ Reproduction: r in normal form ⇒ ↓[[r]]↑ = r r =β s ∈ NFβ = ⇒ r →∗ s

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 2

Motivation

✬ ✫ ✩ ✪

[[·]]ξ : Terms − → Semantics (evaluation function) ↓ : Semantics − → Normal forms (inverse) ↑ : Variables − → Semantics (valuation) ✄ Soundness: r → s ⇒ [[r]]ξ = [[s]]ξ ✄ Reproduction: r in normal form ⇒ ↓[[r]]↑ = r r =β s ∈ NFβ = ⇒ r →∗ s = ⇒ ↓[[r]]↑

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 2

Motivation

✬ ✫ ✩ ✪

[[·]]ξ : Terms − → Semantics (evaluation function) ↓ : Semantics − → Normal forms (inverse) ↑ : Variables − → Semantics (valuation) ✄ Soundness: r → s ⇒ [[r]]ξ = [[s]]ξ ✄ Reproduction: r in normal form ⇒ ↓[[r]]↑ = r r =β s ∈ NFβ = ⇒ r →∗ s = ⇒ ↓[[r]]↑ = ↓[[s]]↑

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 2

Motivation

✬ ✫ ✩ ✪

[[·]]ξ : Terms − → Semantics (evaluation function) ↓ : Semantics − → Normal forms (inverse) ↑ : Variables − → Semantics (valuation) ✄ Soundness: r → s ⇒ [[r]]ξ = [[s]]ξ ✄ Reproduction: r in normal form ⇒ ↓[[r]]↑ = r r =β s ∈ NFβ = ⇒ r →∗ s = ⇒ ↓[[r]]↑ = ↓[[s]]↑ = s

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 3

Example — The typed λ-calculus

✄ Terms: Λ with simple types

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 3

Example — The typed λ-calculus

✄ Terms: Λ with simple types ✄ Semantics: [[ι]] := NFι, [[ρ → σ]] := [[ρ]] → [[σ]]

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 3

Example — The typed λ-calculus

✄ Terms: Λ with simple types ✄ Semantics: [[ι]] := NFι, [[ρ → σ]] := [[ρ]] → [[σ]] ✄ Evaluation: [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λ λX[[r]]ξ,x:=X

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 3

Example — The typed λ-calculus

✄ Terms: Λ with simple types ✄ Semantics: [[ι]] := NFι, [[ρ → σ]] := [[ρ]] → [[σ]] ✄ Evaluation: [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λ λX[[r]]ξ,x:=X ✄ ↑ρ : {x r | r normal} → [[ρ]] and ↓ρ : [[ρ]] → NFρ

★ ✧ ✥ ✦

↑ιr := r, ↑ρ→σr := λ λX ↑σ(r ↓ρX), ↓ιr := r, ↓ρ→σR := λx ↓σ(R ↑ρx), x new

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 3

Example — The typed λ-calculus

✄ Terms: Λ with simple types ✄ Semantics: [[ι]] := NFι, [[ρ → σ]] := [[ρ]] → [[σ]] ✄ Evaluation: [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λ λX[[r]]ξ,x:=X ✄ ↑ρ : {x r | r normal} → [[ρ]] and ↓ρ : [[ρ]] → NFρ

★ ✧ ✥ ✦

↑ιr := r, ↑ρ→σr := λ λX ↑σ(r ↓ρX), ↓ιr := r, ↓ρ→σR := λx ↓σ(R ↑ρx), x new ✄ Properties: ↓ρ→σλ λXR

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 3

Example — The typed λ-calculus

✄ Terms: Λ with simple types ✄ Semantics: [[ι]] := NFι, [[ρ → σ]] := [[ρ]] → [[σ]] ✄ Evaluation: [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λ λX[[r]]ξ,x:=X ✄ ↑ρ : {x r | r normal} → [[ρ]] and ↓ρ : [[ρ]] → NFρ

★ ✧ ✥ ✦

↑ιr := r, ↑ρ→σr := λ λX ↑σ(r ↓ρX), ↓ιr := r, ↓ρ→σR := λx ↓σ(R ↑ρx), x new ✄ Properties: ↓ρ→σλ λXR = λx ↓σ( (λ λXR)↑ρx)

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 3

Example — The typed λ-calculus

✄ Terms: Λ with simple types ✄ Semantics: [[ι]] := NFι, [[ρ → σ]] := [[ρ]] → [[σ]] ✄ Evaluation: [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λ λX[[r]]ξ,x:=X ✄ ↑ρ : {x r | r normal} → [[ρ]] and ↓ρ : [[ρ]] → NFρ

★ ✧ ✥ ✦

↑ιr := r, ↑ρ→σr := λ λX ↑σ(r ↓ρX), ↓ιr := r, ↓ρ→σR := λx ↓σ(R ↑ρx), x new ✄ Properties: ↓ρ→σλ λXR = λx ↓σ( (λ λXR)↑ρx) = λx ↓σ(R[X := ↑ρx]).

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 3

Example — The typed λ-calculus

✄ Terms: Λ with simple types ✄ Semantics: [[ι]] := NFι, [[ρ → σ]] := [[ρ]] → [[σ]] ✄ Evaluation: [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λ λX[[r]]ξ,x:=X ✄ ↑ρ : {x r | r normal} → [[ρ]] and ↓ρ : [[ρ]] → NFρ

★ ✧ ✥ ✦

↑ιr := r, ↑ρ→σr := λ λX ↑σ(r ↓ρX), ↓ιr := r, ↓ρ→σR := λx ↓σ(R ↑ρx), x new ✄ Properties: ↓ρ→σλ λXR = λx ↓σ( (λ λXR)↑ρx) = λx ↓σ(R[X := ↑ρx]). (↑ρ→σr)S

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 3

Example — The typed λ-calculus

✄ Terms: Λ with simple types ✄ Semantics: [[ι]] := NFι, [[ρ → σ]] := [[ρ]] → [[σ]] ✄ Evaluation: [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λ λX[[r]]ξ,x:=X ✄ ↑ρ : {x r | r normal} → [[ρ]] and ↓ρ : [[ρ]] → NFρ

★ ✧ ✥ ✦

↑ιr := r, ↑ρ→σr := λ λX ↑σ(r ↓ρX), ↓ιr := r, ↓ρ→σR := λx ↓σ(R ↑ρx), x new ✄ Properties: ↓ρ→σλ λXR = λx ↓σ( (λ λXR)↑ρx) = λx ↓σ(R[X := ↑ρx]). (↑ρ→σr)S = (λ λX ↑σ(r↓ρX))S

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 3

Example — The typed λ-calculus

✄ Terms: Λ with simple types ✄ Semantics: [[ι]] := NFι, [[ρ → σ]] := [[ρ]] → [[σ]] ✄ Evaluation: [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λ λX[[r]]ξ,x:=X ✄ ↑ρ : {x r | r normal} → [[ρ]] and ↓ρ : [[ρ]] → NFρ

★ ✧ ✥ ✦

↑ιr := r, ↑ρ→σr := λ λX ↑σ(r ↓ρX), ↓ιr := r, ↓ρ→σR := λx ↓σ(R ↑ρx), x new ✄ Properties: ↓ρ→σλ λXR = λx ↓σ( (λ λXR)↑ρx) = λx ↓σ(R[X := ↑ρx]). (↑ρ→σr)S = (λ λX ↑σ(r↓ρX))S = ↑σ(r↓ρS).

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 3

Example — The typed λ-calculus

✄ Terms: Λ with simple types ✄ Semantics: [[ι]] := NFι, [[ρ → σ]] := [[ρ]] → [[σ]] ✄ Evaluation: [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λ λX[[r]]ξ,x:=X ✄ ↑ρ : {x r | r normal} → [[ρ]] and ↓ρ : [[ρ]] → NFρ

★ ✧ ✥ ✦

↑ιr := r, ↑ρ→σr := λ λX ↑σ(r ↓ρX), ↓ιr := r, ↓ρ→σR := λx ↓σ(R ↑ρx), x new ✄ Properties: ↓ρ→σλ λXR = λx ↓σ( (λ λXR)↑ρx) = λx ↓σ(R[X := ↑ρx]). (↑ρ→σr)S = (λ λX ↑σ(r↓ρX))S = ↑σ(r↓ρS). ↓ι↑ιr

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 3

Example — The typed λ-calculus

✄ Terms: Λ with simple types ✄ Semantics: [[ι]] := NFι, [[ρ → σ]] := [[ρ]] → [[σ]] ✄ Evaluation: [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λ λX[[r]]ξ,x:=X ✄ ↑ρ : {x r | r normal} → [[ρ]] and ↓ρ : [[ρ]] → NFρ

★ ✧ ✥ ✦

↑ιr := r, ↑ρ→σr := λ λX ↑σ(r ↓ρX), ↓ιr := r, ↓ρ→σR := λx ↓σ(R ↑ρx), x new ✄ Properties: ↓ρ→σλ λXR = λx ↓σ( (λ λXR)↑ρx) = λx ↓σ(R[X := ↑ρx]). (↑ρ→σr)S = (λ λX ↑σ(r↓ρX))S = ↑σ(r↓ρS). ↓ι↑ιr = r.

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 4

From properties to axioms

✄ Properties. ↓ρ→σλ λXR = λx ↓σ( (λ λXR)↑ρx) = λx ↓σ(R[X := ↑ρx]). (↑ρ→σr)S = (λ λX ↑σ(r↓ρX))S = ↑σ(r↓ρS). ↓ι↑ιr = r.

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 4

From properties to axioms

✄ Properties. ↓ρ→σλ λXR = λx ↓σ( (λ λXR)↑ρx) = λx ↓σ(R[X := ↑ρx]). (↑ρ→σr)S = (λ λX ↑σ(r↓ρX))S = ↑σ(r↓ρS). ↓ι↑ιr = r.

✓ ✒ ✏ ✑

Discard types — we only need to know whether function or not

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 4

From properties to axioms

✄ Properties. ↓ρ→σλ λXR = λx ↓σ( (λ λXR)↑ρx) = λx ↓σ(R[X := ↑ρx]). (↑ρ→σr)S = (λ λX ↑σ(r↓ρX))S = ↑σ(r↓ρS). ↓ι↑ιr = r.

✓ ✒ ✏ ✑

Discard types — we only need to know whether function or not ✄ Axioms. ↓λ λXR = λx ↓(R[X := ↑x]). (↑r)S = ↑(r↓S). ↓↑r = r.

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 5

(Just another) 2-level-calculus

✄ Terms. Λ ∋ r, s, t ::= x | rs | λxr,

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 5

(Just another) 2-level-calculus

✄ Terms. Λ ∋ r, s, t ::= x | rs | λxr, Λ1 ∋ r, s, t ::= x | rs | λxr

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 5

(Just another) 2-level-calculus

✄ Terms. Λ ∋ r, s, t ::= x | rs | λxr, Λ1 ∋ r, s, t ::= x | rs | λxr | ↓R

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 5

(Just another) 2-level-calculus

✄ Terms. Λ ∋ r, s, t ::= x | rs | λxr, Λ1 ∋ r, s, t ::= x | rs | λxr | ↓R Λ2 ∋ R, S, T ::= X | RS | λXR

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 5

(Just another) 2-level-calculus

✄ Terms. Λ ∋ r, s, t ::= x | rs | λxr, Λ1 ∋ r, s, t ::= x | rs | λxr | ↓R Λ2 ∋ R, S, T ::= X | RS | λXR | ↑r

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 5

(Just another) 2-level-calculus

✄ Terms. Λ ∋ r, s, t ::= x | rs | λxr, Λ1 ∋ r, s, t ::= x | rs | λxr | ↓R Λ2 ∋ R, S, T ::= X | RS | λXR | ↑r ✄ Reductions. (λXR)S →β R[X := S] (↑r)S →a ↑(r↓S) ↓λXR →d λx↓(R[X := ↑x]), x new ↓↑r →d r

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 5

(Just another) 2-level-calculus

✄ Terms. Λ ∋ r, s, t ::= x | rs | λxr, Λ1 ∋ r, s, t ::= x | rs | λxr | ↓R Λ2 ∋ R, S, T ::= X | RS | λXR | ↑r ✄ Reductions. (λXR)S →β R[X := S] (↑r)S →a ↑(r↓S) ↓λXR →d λx↓(R[X := ↑x]), x new ↓↑r →d r ✄ This yields an orthogonal higher order rewrite system ⇒ confluence.

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X.

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ.

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ]

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ = [[r[x := s]]]ξ

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ = [[r[x := s]]]ξ ✄ Reproduction. ↓[[r]]↑ = r for normal forms r. ↓[[x r ]]↑

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ = [[r[x := s]]]ξ ✄ Reproduction. ↓[[r]]↑ = r for normal forms r. ↓[[x r ]]↑ = ↓((↑x)[[ r ]]↑)

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ = [[r[x := s]]]ξ ✄ Reproduction. ↓[[r]]↑ = r for normal forms r. ↓[[x r ]]↑ = ↓((↑x)[[ r ]]↑) →∗

a ↓↑(x↓[[

r ]]↑)

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ = [[r[x := s]]]ξ ✄ Reproduction. ↓[[r]]↑ = r for normal forms r. ↓[[x r ]]↑ = ↓((↑x)[[ r ]]↑) →∗

a ↓↑(x↓[[

r ]]↑) →d x↓[[ r ]]↑

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ = [[r[x := s]]]ξ ✄ Reproduction. ↓[[r]]↑ = r for normal forms r. ↓[[x r ]]↑ = ↓((↑x)[[ r ]]↑) →∗

a ↓↑(x↓[[

r ]]↑) →d x↓[[ r ]]↑ →∗

s x

r.

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ = [[r[x := s]]]ξ ✄ Reproduction. ↓[[r]]↑ = r for normal forms r. ↓[[x r ]]↑ = ↓((↑x)[[ r ]]↑) →∗

a ↓↑(x↓[[

r ]]↑) →d x↓[[ r ]]↑ →∗

s x

r. ↓[[λxr]]↑

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ = [[r[x := s]]]ξ ✄ Reproduction. ↓[[r]]↑ = r for normal forms r. ↓[[x r ]]↑ = ↓((↑x)[[ r ]]↑) →∗

a ↓↑(x↓[[

r ]]↑) →d x↓[[ r ]]↑ →∗

s x

r. ↓[[λxr]]↑ = ↓λX.[[r]]↑,x:=X

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ = [[r[x := s]]]ξ ✄ Reproduction. ↓[[r]]↑ = r for normal forms r. ↓[[x r ]]↑ = ↓((↑x)[[ r ]]↑) →∗

a ↓↑(x↓[[

r ]]↑) →d x↓[[ r ]]↑ →∗

s x

r. ↓[[λxr]]↑ = ↓λX.[[r]]↑,x:=X →d λx↓[[r]]↑,x:=X[X := ↑x]

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ = [[r[x := s]]]ξ ✄ Reproduction. ↓[[r]]↑ = r for normal forms r. ↓[[x r ]]↑ = ↓((↑x)[[ r ]]↑) →∗

a ↓↑(x↓[[

r ]]↑) →d x↓[[ r ]]↑ →∗

s x

r. ↓[[λxr]]↑ = ↓λX.[[r]]↑,x:=X →d λx↓[[r]]↑,x:=X[X := ↑x] = λx↓[[r]]↑

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ = [[r[x := s]]]ξ ✄ Reproduction. ↓[[r]]↑ = r for normal forms r. ↓[[x r ]]↑ = ↓((↑x)[[ r ]]↑) →∗

a ↓↑(x↓[[

r ]]↑) →d x↓[[ r ]]↑ →∗

s x

r. ↓[[λxr]]↑ = ↓λX.[[r]]↑,x:=X →d λx↓[[r]]↑,x:=X[X := ↑x] = λx↓[[r]]↑ →∗

s λxr.

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 6

Evaluation

✄ Definition. [[x]]ξ := ξx, [[rs]]ξ := [[r]]ξ[[s]]ξ, [[λxr]]ξ := λX.[[r]]ξ,x:=X. ✄ Simulation. r →β s = ⇒ [[r]]ξ →β [[s]]ξ. [[(λxr)s]]ξ = (λX[[r]]ξ,x:=X)[[s]]ξ →β [[r]]ξ,x:=X[X := [[s]]ξ] = [[r]]ξ,x:=[[s]]ξ = [[r[x := s]]]ξ ✄ Reproduction. ↓[[r]]↑ = r for normal forms r. ↓[[x r ]]↑ = ↓((↑x)[[ r ]]↑) →∗

a ↓↑(x↓[[

r ]]↑) →d x↓[[ r ]]↑ →∗

s x

r. ↓[[λxr]]↑ = ↓λX.[[r]]↑,x:=X →d λx↓[[r]]↑,x:=X[X := ↑x] = λx↓[[r]]↑ →∗

s λxr.

✄ Corollary. r =β s ∈ NFβ = ⇒ ↓[[r]]↑ →∗ s.

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 7

Implementation: Object calculus

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 7

Implementation: Object calculus

Choose de Bruijn terms for simplicity.

data Term = Var Integer | App Term Term | Abs Term deriving Show

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Untyped Normalization by Evaluation Tallinn 17.4.4 7

Implementation: Object calculus

Choose de Bruijn terms for simplicity.

data Term = Var Integer | App Term Term | Abs Term deriving Show

An interface to make lifting easy:

type TERM = Integer -> Term inspect :: TERM -> Term inspect f = f 0 freevar k = \ n -> Var (n + k) boundvar k = \ n -> Var (n - k - 1) apply r s = \ n -> App (r n) (s n) bind f = \ n -> Abs (f (boundvar n) (n + 1))

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 8

Implementation — Metalevel calculus

Terms. ↑r | λXR | RS | X

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Untyped Normalization by Evaluation Tallinn 17.4.4 8

Implementation — Metalevel calculus

Terms. ↑r | λXR | RS | X

data LAM = Up TERM | ABS (LAM -> LAM)

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Untyped Normalization by Evaluation Tallinn 17.4.4 8

Implementation — Metalevel calculus

Terms. ↑r | λXR | RS | X

data LAM = Up TERM | ABS (LAM -> LAM)

Reductions for ↓: ↓↑r →d r ↓λXR →d λx↓(R[X := ↑x])

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 8

Implementation — Metalevel calculus

Terms. ↑r | λXR | RS | X

data LAM = Up TERM | ABS (LAM -> LAM)

Reductions for ↓: ↓↑r →d r ↓λXR →d λx↓(R[X := ↑x])

down :: LAM -> TERM

  • - the d-rules

down (ABS f) = bind (\ x -> (down (f (Up x)))) down (Up r) = r

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 8

Implementation — Metalevel calculus

Terms. ↑r | λXR | RS | X

data LAM = Up TERM | ABS (LAM -> LAM)

Reductions for ↓: ↓↑r →d r ↓λXR →d λx↓(R[X := ↑x])

down :: LAM -> TERM

  • - the d-rules

down (ABS f) = bind (\ x -> (down (f (Up x)))) down (Up r) = r

Reductions for application: (λXR)S →β R[X := S] (↑r)S →a ↑(r↓S)

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

  • hm Types Conclusions

Untyped Normalization by Evaluation Tallinn 17.4.4 8

Implementation — Metalevel calculus

Terms. ↑r | λXR | RS | X

data LAM = Up TERM | ABS (LAM -> LAM)

Reductions for ↓: ↓↑r →d r ↓λXR →d λx↓(R[X := ↑x])

down :: LAM -> TERM

  • - the d-rules

down (ABS f) = bind (\ x -> (down (f (Up x)))) down (Up r) = r

Reductions for application: (λXR)S →β R[X := S] (↑r)S →a ↑(r↓S)

app :: LAM -> LAM -> LAM app (ABS f) r = f r

  • - the beta-rule

app (Up r) s = Up (apply r (down s))

  • - the a-rule
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Untyped Normalization by Evaluation Tallinn 17.4.4 9

Implementation — Evaluation

type Valuation = Integer -> LAM comma :: Valuation -> LAM -> Valuation comma xi r n = if n == 0 then r else xi (n - 1)

ξ, x := R

eval :: Term -> Valuation -> LAM eval (Var n) xi = xi n

ξx

eval (Abs r) xi = ABS (\ a -> eval r (xi ‘comma‘ a))

λX.[[r]]ξ,x:=X

eval (App r s) xi = (eval r xi) ‘app‘ (eval s xi)

[[r]]ξ[[s]]ξ

nbe r = inspect (down (eval r (Up . freevar)))

↓[[r]]↑

= (down (eval r (Up . freevar))) 0

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Untyped Normalization by Evaluation Tallinn 17.4.4 10

Implementation — Examples

k = Abs (Abs (Var 1))

k = λλ1

s = Abs (Abs (Abs (Var 2 ‘App‘ (Var 0) ‘App‘ (Var 1 ‘App‘ (Var 0))))) i = s ‘App‘ k ‘App‘ k

s = λλλ.20(10)

  • mega

= Abs (Var 0 ‘App‘ (Var 0))

ω = λ.00

  • omega = omega ‘App‘ omega

Ω = ωω

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 10

Implementation — Examples

k = Abs (Abs (Var 1))

k = λλ1

s = Abs (Abs (Abs (Var 2 ‘App‘ (Var 0) ‘App‘ (Var 1 ‘App‘ (Var 0))))) i = s ‘App‘ k ‘App‘ k

s = λλλ.20(10)

  • mega

= Abs (Var 0 ‘App‘ (Var 0))

ω = λ.00

  • omega = omega ‘App‘ omega

Ω = ωω

Main> nbe k Abs (Abs (Var 1))

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 10

Implementation — Examples

k = Abs (Abs (Var 1))

k = λλ1

s = Abs (Abs (Abs (Var 2 ‘App‘ (Var 0) ‘App‘ (Var 1 ‘App‘ (Var 0))))) i = s ‘App‘ k ‘App‘ k

s = λλλ.20(10)

  • mega

= Abs (Var 0 ‘App‘ (Var 0))

ω = λ.00

  • omega = omega ‘App‘ omega

Ω = ωω

Main> nbe k Abs (Abs (Var 1)) Main> nbe i Abs (Var 0)

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 10

Implementation — Examples

k = Abs (Abs (Var 1))

k = λλ1

s = Abs (Abs (Abs (Var 2 ‘App‘ (Var 0) ‘App‘ (Var 1 ‘App‘ (Var 0))))) i = s ‘App‘ k ‘App‘ k

s = λλλ.20(10)

  • mega

= Abs (Var 0 ‘App‘ (Var 0))

ω = λ.00

  • omega = omega ‘App‘ omega

Ω = ωω

Main> nbe k Abs (Abs (Var 1)) Main> nbe i Abs (Var 0) Main> nbe (k ‘App‘ i ‘App‘ oomega) Abs (Var 0)

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 11

Standardization

✄ Standard reduction. Head redexes are executed first or never.

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Untyped Normalization by Evaluation Tallinn 17.4.4 11

Standardization

✄ Standard reduction. Head redexes are executed first or never. ✄ Theorem. Reduction sequences can be standardized.

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Untyped Normalization by Evaluation Tallinn 17.4.4 11

Standardization

✄ Standard reduction. Head redexes are executed first or never. ✄ Theorem. Reduction sequences can be standardized. ✄ Corollary. Normalization (= reduction to normal form)

  • need not reduce under meta-abstractions,
  • may follow a call-by-name strategy.
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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 11

Standardization

✄ Standard reduction. Head redexes are executed first or never. ✄ Theorem. Reduction sequences can be standardized. ✄ Corollary. Normalization (= reduction to normal form)

  • need not reduce under meta-abstractions,
  • may follow a call-by-name strategy.

✄ Corollary. We can compute (the defined parts of) B¨

  • hm trees —

normalization for diverging terms.

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Untyped Normalization by Evaluation Tallinn 17.4.4 12

B¨

  • hm trees — Examples

pair = Abs (Abs (Abs (Var 0 ‘App‘ Var 2 ‘App‘ Var 1))) swap = Abs (Abs (Abs (Var 0 ‘App‘ Var 1 ‘App‘ Var 2)))

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 12

B¨

  • hm trees — Examples

pair = Abs (Abs (Abs (Var 0 ‘App‘ Var 2 ‘App‘ Var 1))) swap = Abs (Abs (Abs (Var 0 ‘App‘ Var 1 ‘App‘ Var 2))) Main> nbe (pair ‘App‘ i ‘App‘ oomega) Abs (App (App (Var 0) (Abs (Var 0))) {Interrupted!}

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 12

B¨

  • hm trees — Examples

pair = Abs (Abs (Abs (Var 0 ‘App‘ Var 2 ‘App‘ Var 1))) swap = Abs (Abs (Abs (Var 0 ‘App‘ Var 1 ‘App‘ Var 2))) Main> nbe (pair ‘App‘ i ‘App‘ oomega) Abs (App (App (Var 0) (Abs (Var 0))) {Interrupted!} Main> cut (nbe (pair ‘App‘ i ‘App‘ oomega)) ["01"] Abs (App (App (Var 0) (Abs (Var 0))) (Var (-1)))

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 12

B¨

  • hm trees — Examples

pair = Abs (Abs (Abs (Var 0 ‘App‘ Var 2 ‘App‘ Var 1))) swap = Abs (Abs (Abs (Var 0 ‘App‘ Var 1 ‘App‘ Var 2))) Main> nbe (pair ‘App‘ i ‘App‘ oomega) Abs (App (App (Var 0) (Abs (Var 0))) {Interrupted!} Main> cut (nbe (pair ‘App‘ i ‘App‘ oomega)) ["01"] Abs (App (App (Var 0) (Abs (Var 0))) (Var (-1))) Main> nbe (pair ‘App‘ i ‘App‘ oomega ‘App‘ swap) Abs (App (App (Var 0) {Interrupted!}

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Motivation Example Axioms 2-level Evaluation Object Meta eval skk Standardization B¨

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Untyped Normalization by Evaluation Tallinn 17.4.4 12

B¨

  • hm trees — Examples

pair = Abs (Abs (Abs (Var 0 ‘App‘ Var 2 ‘App‘ Var 1))) swap = Abs (Abs (Abs (Var 0 ‘App‘ Var 1 ‘App‘ Var 2))) Main> nbe (pair ‘App‘ i ‘App‘ oomega) Abs (App (App (Var 0) (Abs (Var 0))) {Interrupted!} Main> cut (nbe (pair ‘App‘ i ‘App‘ oomega)) ["01"] Abs (App (App (Var 0) (Abs (Var 0))) (Var (-1))) Main> nbe (pair ‘App‘ i ‘App‘ oomega ‘App‘ swap) Abs (App (App (Var 0) {Interrupted!} Main> cut (nbe (pair ‘App‘ i ‘App‘ oomega ‘App‘ swap)) ["001"] Abs (App (App (Var 0) (Var (-1))) (Abs (Var 0)))

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Untyped Normalization by Evaluation Tallinn 17.4.4 13

Types — Computing η-long normal forms

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Untyped Normalization by Evaluation Tallinn 17.4.4 13

Types — Computing η-long normal forms

✄ Head expansion. Instead of ↓↑r →d r define ↓↑rρ →d ηρr with ηιr := r and ηρ→σr := λx.ησ(rηρx).

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Untyped Normalization by Evaluation Tallinn 17.4.4 13

Types — Computing η-long normal forms

✄ Head expansion. Instead of ↓↑r →d r define ↓↑rρ →d ηρr with ηιr := r and ηρ→σr := λx.ησ(rηρx). ✄ Theorem. On typed terms → is SN and computes η-long normal forms. ✄ Corollary. Typed NBE (as defined in the beginning) is a model.

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Untyped Normalization by Evaluation Tallinn 17.4.4 14

Conclusions

✄ Normalization by Evaluation needs no types — the information of whether a function or not suffices. ✄ Non-termination: (partial) normal forms are found whenever they exist. ✄ Use Haskell for normalization (= doing the substitutions). ✄ Type-directed NBE as an example.

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Untyped Normalization by Evaluation Tallinn 17.4.4 15

Technical details

Theorem. r =β λ x(x r) & s →∗ ↓[[r]] = ⇒ ∃ t, s.s ։ λ x(x s ) & r =β t & s →∗ ↓[[ t ]]. Proof. r →∗

β λ

x(x t ) ∗

β← λ

x(x r ) by the Church-Rosser prop. of →β, s →∗ ↓[[r]] →∗ λ x(x↓[[ t ]]) s ❀ λ x(x↓[[ t ]]) by standardization, s ։ λ x(x s ) →∗ λ x(x↓[[ t ]]) property of ։.

  • Theorem. ↓[[t]] →∗ λ

x(x R) = ⇒ t →∗ λ x(x t ).