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Lambda Calculus
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USAMI Kosuke
November 03, 2018
Science
650
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Lambda Calculus
※ Docswell に移行しました
https://www.docswell.com/s/usami-k/5DE4M9-lambda-calculus
USAMI Kosuke
November 03, 2018
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Transcript
ϥϜμܭࢉͷ Ӊࠤݟެี ୈ3ճ ֶ༵ؔ ༑ͷձ
ࣗݾհ • Ӊࠤݟެีʢ@usamik26ʣ • େֶֶ࣌ઐ߈ʢɺԠ༻ཧʣ • ݱࡏϓϩάϥϚʢϑΣϯϦϧגࣜձࣾɺiOS ΞϓϦ։ൃʣ
ϥϜμܭࢉʢλ-calculusʣͱ • ܭࢉػՊֶͰग़ͯ͘Δମܥͷͻͱͭ • ϓϩάϥϛϯάݴޠͷܕγεςϜͱͷؔ࿈ • ؔܕϓϩάϥϛϯάΛࢧ͑Δཧ
ϥϜμه๏ • ֶͰΘΕΔؔͷදه๏ • ϥϜμه๏ʹΑΔؔͷදه๏
ϥϜμه߸ͷҙຯ߹͍ • ϥϜμه๏ʹ͓͚Δ ɺ ͷΑ͏ͳҙຯ߹͍ͷه߸ • ͦ͏ࢥͬͯฒͯΈΔͱͳΜͱͳ͘ࣅ͍ͯΔؾ͕͢Δ
ϥϜμه๏ɿؔͷద༻ • ϥϜμه๏ͰؔʹΛೖ͢Δ • ه๏ͷൺֱɿؔ ʹΛೖ͢Δ
ϥϜμه๏ɿଟมؔ • ଟมؔͷϥϜμه๏ • ͜Ε1มؔͷΈ߹ΘͤͱಉҰࢹͰ͖ΔʢΧϦʔԽʣ
ϥϜμه๏ɿߴ֊ؔ • ʹؔ Λ2ճద༻͢Δؔͷྫ • ΧϦʔԽͯ͠ॻ͚
ϥϜμࣜʢλ-termʣͷಋೖ • ϥϜμه๏ͦͷͷΛɺΑΓநతʹऔΓѻ͏ • ͦͷͨΊʹɺه߸ྻͱͯ͠ͷϥϜμࣜΛಋೖ͢Δ
ϥϜμࣜͷఆٛ • Մࢉແݶೱͷू߹ ͕༩͑ΒΕ͍ͯΔͱ͢Δ • ҎԼͰ࠶ؼతʹఆٛ͞ΕΔه߸ྻͷू߹ ͷཁૉΛϥϜμࣜ ͱݺͿʢͳ͓ɺׅހҰఆͷϧʔϧͰऔΓ֎͢͜ͱ͕Ͱ͖Δʣ • (1)
ͷͱ͖ • (2) ͷͱ͖ • (3) ͷͱ͖
ϥϜμࣜͱϥϜμه๏ • มͷू߹Λҙຯ͢Δ • (1) ɺมϥϜμࣜͰ͋Δ͜ͱΛҙຯ͢Δ • (2) ɺؔͷϥϜμه๏ʹରԠ͢Δ •
(3) ɺؔద༻ͷϥϜμه๏ʹରԠ͢Δ
ϥϜμܭࢉ • ͜͏ͯ͠ఆٛ͞ΕͨϥϜμࣜͷੑ࣭ΛݟΔ͜ͱͰɺϥϜμه๏ Ͱهड़͞Εͨؔͷͳ͢ੈքΛݚڀ͢Δͷ͕ɺϥϜμܭࢉͷཧ Ͱ͋Δ • ಛʹɺݸʑͷؔʢ ͱ͔ ͱ ͔ʣͷੑ࣭ʹΑΒͣɺؔͷੈքʹࡏ͢Δੑ࣭Λߟ͢Δ
ϥϜμࣜʹ͍ͭͯͷิ • ه߸ྻͱͯ͠ఆٛ͞ΕͨϥϜμ͕ࣜɺৗʹϥϜμه๏ʹରԠ͠ ͯҙຯΛ࣋ͭͱݶΒͳ͍ʢϥϜμه๏ͱͯ͋͠Γ͑ͳ͍͕ࣜ Ͱ͖ͯ͠·͏ʣ • ͔͠͠ɺ·ͣͦΕΛڐ༰ͯٞ͢͠Δํ͕ݟ௨͕͠ྑ͘ͳΔ • ͦͷޙɺܕʢؔͷఆٛҬҬʹ૬͢Δ֓೦ʣΛಋೖ͢Δ ͜ͱͰɺϥϜμه๏ͱͷରԠͷ͕ٞͰ͖Δ
ϕʔλม • ҎԼͰ࠶ؼతʹఆٛ͞ΕΔϥϜμࣜͷมΛϕʔλมͱݺͿ • (1) • (2) ͳΒ ͓Αͼ •
͜͜Ͱ ɺϥϜμࣜ ͷதͷม ΛϥϜμ ࣜ ʹஔ͖͑ͨϥϜμࣜͷ͜ͱͰ͋Δʢೖʣ
ೖʹ͍ͭͯͷิ • ϥϜμࣜ ͷதͷม ʹ͍ͭͯɺ ͱ͍͏෦ࣜ ͕͋Δͱ͖ Λଋറมɺͦ͏Ͱͳ͍ͱ͖ࣗ༝มͱݺͿ • ೖ
Λߟ͑Δͱ͖ɺ ͷࣗ༝ม ͷଋറ มͰͳ͍ͷͱ͢Δ • ·ͨ ͷଋറม Ҏ֎ͷมʹஔ͖͑ͯߟ͑ΔʢҰ ൠʹɺϥϜμࣜͷଋറมΛͦͷࣜʹݱΕͳ͍ผͷมʹஔ͖ ͑ͨࣜΛɺݩͷࣜͱಉҰࢹ͢Δʣ
ϕʔλมͷྫ ֤ϥϜμࣜͷࣗ༝มଞͷࣜͷଋറมͰͳ͍ͷͱ͢Δ
Church-Rosser ͷఆཧ • ϕʔλมͷ͔ͨ͠Ұ௨ΓͰͳ͍ • Church-Rosser ͷఆཧɿͻͱͭͷϥϜμ͔ࣜΒϕʔλมͰ ಘΒΕͨ;ͨͭͷϥϜμࣜɺԿ͔ϕʔλมΛߦ͏͜ͱ Ͱɺಉ͡ϥϜμࣜʹͰ͖Δ
ਖ਼نܗ • Church-Rosser ͷఆཧʹΑΓɺϥϜμࣜ ͔Β࢝Ίͯ༗ݶ ͷεςοϓͰࢭ·Δʢࣜதʹ ͕ͳ͘ͳΔʣϕʔλมྻ͕͋ Δͱ͖ɺ࠷ऴ݁ՌҰக͢Δ • ͜ͷ࠷ऴ݁ՌΛ
ͷਖ਼نܗͱݺͿ • ༩͑ΒΕͨؔʢϥϜμه๏ʣΛػցతʹॲཧͯ͠ਖ਼نܗΛಘ Δͷ͕ɺؔܕϓϩάϥϛϯάݴޠʹ͓͚ΔܭࢉͰ͋Δʢܭࢉ ػՊֶͷ༻ޠͰධՁͱݺͿʣ
͞ΒͳΔɾɾɾ • ͜͜·ͰͷϥϜμܭࢉɺܕͳ͠ϥϜμܭࢉͱݺΕΔ • ܕΛಋೖͨ͠ɺܕ͖ͭϥϜμܭࢉͱ͍͏ମܥ͕͋Δ • ܕ͖ͭϥϜμܭࢉɺϥϜμه๏ͱͷରԠ͕Ͱ͖Δ • ܕ͖ͭϥϜμܭࢉɺܕΛରɺϥϜμࣜΛࣹͱͯ͠ɺݍΛͳ ͢
• ͜ΕʹΑΓɺݍͷ͕ٞԠ༻Ͱ͖Δ