ラムダ計算入門 / Introduction of Lambda Calculus

型勉強会 #2 2017/08/09

Transcript

1. ϥϜμܭࢉೖ໳
   ܕษڧձ #2 2017/08/09
   

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2. ࢀߟจݙ: TAPL
   

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3. ϥϜμܭࢉ
   • ͢΂ͯͷܭࢉ͕ؔ਺ఆٛͱؔ਺ద༻ʹؼண͞ΕΔମܥ
   
   • ߏจ͸ɺม਺ɺϥϜμந৅ (ແ໊ؔ਺)ɺؔ਺ద༻ͷ

   3ͭͷ߲ͷΈ͔Βߏ੒͞ΕΔ
   
   • ؔ਺ܕݴޠͷΞηϯϒϦͱݺ͹ΕͯΔΒ͍͠
   
   • ࠓճ͸ܕແ͠ϥϜμܭࢉΛઆ໌͠·͢

   (୯७ܕ෇͖ϥϜμܭࢉ͸࣍ճ……ʁ)
   

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4. t ::=
   x
   λx. t
   t t
   ม਺
   ϥϜμந৅
   ؔ਺ద༻
   λx. λy. x (λz. x z) y
   ྫ:
   

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5. Q ਅِ஋ͱ͔਺஋ͱ͔͕ແ͍……ʁ
   

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6. Q ਅِ஋ͱ͔਺஋ͱ͔͕ແ͍……ʁ
   A ϥϜμந৅Ͱදݱ͢Ε͹͍͍͡Όͳ͍
   

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7. ਅِ஋ͷදݱ
   tru = λt. λf. t
   fls = λt. λf. f
   → true ͱΈͳͤΔ
   → false ͱΈͳͤΔ
   

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8. ྫ͑͹ if ࣜͬΆ͍ϥϜμந৅Λ͜ͷΑ͏ʹఆٛͰ͖Δ
   test = λl. λm. λn. l m n
   e.g. test t v w 

   t ͕ tru ͷ࣌ v Λɺfls ͷ࣌ w Λฦ͢
   

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9. ɹ test tru v w
   → (λl. λm. λn. l m n) tru v w
   → (λm. λn. tru m n) v w
   → (λn. tru v n) w
   → tru v w
   → (λt. λf. t) v w
   → (λf. v) w
   → v
   t = tru ͷ࣌ͷධՁͷྲྀΕ
   

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10. ੔਺஋ͷදݱ
    c0 = λs. λz. z
    c1 = λs. λz. s z
    c2 = λs. λz. s (s z)
    c3 = λs. λz. s (s (s z))
    …
    → 0
    
    → 1
    
    → 2
    
    → 3
    

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11. plus = λm. λn. λs. λz. m s (n s z)
    ଍͠ࢉΛ͢ΔϥϜμந৅
    

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12. ɹ plus c1 c3
    → (λm. λn. λs. λz. m s (n s z)) c1 c3
    → (λn. λs. λz. c1 s (n s z)) c3
    → λs. λz. c1 s (c3 s z)
    → λs. λz. (λs’. λz’. s’ z’) s (c3 s z)
    → λs. λz. (λz’. s z’) (c3 s z)
    → λs. λz. s (c3 s z)
    → λs. λz. s ((λs’’. λz’’. s’’ (s’’ (s’’ z’’)) s z)
    → λs. λz. s ((λz’’. s (s (s z’’)) z)
    → λs. λz. s (s (s (s z)))
    → c4
    ධՁͷྲྀΕ
    

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13. β-؆໿
    • [x → t2] t12
    ͸ t12
    தͷ x ͷࣗ༝ͳग़ݱΛ͢΂ͯ t2
    Ͱஔ͖׵
    ͑Δ͜ͱʹΑͬͯಘΒΕΔ߲
    
    • ͭ·Γؔ਺ద༻ͷධՁ
    (λx. t12) t2 → [x → t2] t12
    

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14. α-ม׵
    • ม਺໊ΛҟͳΔม਺໊ʹม׵͢Δ͜ͱ
    
    • ม਺໊͸ॏཁͰ͸ͳ͍ͷͰม׵ޙ΋ಉҰͷ߲ͱͳΔ
    
    • ͨͩ͠ผͷม਺໊ͱ͔ͿΒͳ͍Α͏ʹม׵͠ͳ͍ͱ

    ҎԼͷྫͷΑ͏ʹม׵લޙͰҟͳͬͯ͠·͏
    
    • ز͔ͭͷίϯύΠϥ͸ม਺ΛҰҙʹ͢ΔͨΊʹα-ม׵Λߦ͏
    λx. x x → λy. y y
    λx. x y → λy. y y
    

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15. ߲ͷ໊ແ͠දݱ
    • ม਺໊͸ͳΜͰ΋ྑ͍
    
    • ͦΕͳΒ࠷ॳ͔ΒҰҙʹͳΔදݱΛબ୒͢Ε͹ྑ͍
    
    • ࣗવ਺ k ΛɺͦΕΛғΉ k ൪໨ͷ λ ʹΑͬͯଋറ͞ΕΔ
    ม਺ͱ͢Δ
    λx. x (λy. x y) → λ. 0 (λ. 1 0)
    

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16. ࣮૷ͯ͠Έͨ
    https://github.com/ktsn/lambda-calculus
    

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