Lambda Calculus - Speaker Deck

Lambda Calculus

by USAMI Kosuke

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ϥϜμܭࢉͷ࿩ Ӊࠤݟެี ୈ3ճ ؔ੢೔༵਺ֶ ༑ͷձ

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ࣗݾ঺հ • Ӊࠤݟެีʢ@usamik26ʣ • େֶ࣌୅͸਺ֶઐ߈ʢ୅਺ɺԠ༻਺ཧʣ • ݱࡏ͸ϓϩάϥϚʢϑΣϯϦϧגࣜձࣾɺiOS ΞϓϦ։ൃʣ

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ϥϜμܭࢉʢλ-calculusʣͱ͸ • ܭࢉػՊֶͰग़ͯ͘Δମܥͷͻͱͭ • ϓϩάϥϛϯάݴޠͷܕγεςϜͱͷؔ࿈ • ؔ਺ܕϓϩάϥϛϯάΛࢧ͑Δཧ࿦

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ϥϜμه๏ • ਺ֶͰ࢖ΘΕΔؔ਺ͷදه๏ • ϥϜμه๏ʹΑΔؔ਺ͷදه๏

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ϥϜμه߸ͷҙຯ߹͍ • ϥϜμه๏ʹ͓͚Δ ͸ɺ ΍ ͷΑ͏ͳҙຯ߹͍ͷه߸ • ͦ͏ࢥͬͯฒ΂ͯΈΔͱͳΜͱͳ͘ࣅ͍ͯΔؾ͕͢Δ

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ϥϜμه๏ɿؔ਺ͷద༻ • ϥϜμه๏Ͱؔ਺ʹ஋Λ୅ೖ͢Δ • ه๏ͷൺֱɿؔ਺ ʹ஋Λ୅ೖ͢Δ

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ϥϜμه๏ɿଟม਺ؔ਺ • ଟม਺ؔ਺ͷϥϜμه๏ • ͜Ε͸1ม਺ؔ਺ͷ૊Έ߹ΘͤͱಉҰࢹͰ͖ΔʢΧϦʔԽʣ

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ϥϜμه๏ɿߴ֊ؔ਺ • ʹؔ਺ Λ2ճద༻͢Δؔ਺ͷྫ • ΧϦʔԽͯ͠ॻ͚͹

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ϥϜμࣜʢλ-termʣͷಋೖ • ϥϜμه๏ͦͷ΋ͷΛɺΑΓந৅తʹऔΓѻ͏ • ͦͷͨΊʹɺه߸ྻͱͯ͠ͷϥϜμࣜΛಋೖ͢Δ

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ϥϜμࣜͷఆٛ • Մࢉແݶೱ౓ͷू߹ ͕༩͑ΒΕ͍ͯΔͱ͢Δ • ҎԼͰ࠶ؼతʹఆٛ͞ΕΔه߸ྻͷू߹ ͷཁૉΛϥϜμࣜ ͱݺͿʢͳ͓ɺׅހ͸ҰఆͷϧʔϧͰऔΓ֎͢͜ͱ͕Ͱ͖Δʣ • (1) ͷͱ͖ • (2) ͷͱ͖ • (3) ͷͱ͖

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ϥϜμࣜͱϥϜμه๏ • ͸ม਺ͷू߹Λҙຯ͢Δ • (1) ͸ɺม਺͸ϥϜμࣜͰ͋Δ͜ͱΛҙຯ͢Δ • (2) ͸ɺؔ਺ͷϥϜμه๏ʹରԠ͢Δ • (3) ͸ɺؔ਺ద༻ͷϥϜμه๏ʹରԠ͢Δ

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ϥϜμܭࢉ • ͜͏ͯ͠ఆٛ͞ΕͨϥϜμࣜͷੑ࣭ΛݟΔ͜ͱͰɺϥϜμه๏ Ͱهड़͞Εͨؔ਺ͷͳ͢ੈքΛݚڀ͢Δͷ͕ɺϥϜμܭࢉͷཧ ࿦Ͱ͋Δ • ಛʹɺݸʑͷؔ਺ʢ ͱ͔ ͱ ͔ʣͷੑ࣭ʹΑΒͣɺؔ਺ͷੈքʹ಺ࡏ͢Δੑ࣭Λߟ࡯͢Δ

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ϥϜμࣜʹ͍ͭͯͷิ଍ • ه߸ྻͱͯ͠ఆٛ͞ΕͨϥϜμ͕ࣜɺৗʹϥϜμه๏ʹରԠ͠ ͯҙຯΛ࣋ͭͱ͸ݶΒͳ͍ʢϥϜμه๏ͱͯ͋͠Γ͑ͳ͍͕ࣜ Ͱ͖ͯ͠·͏ʣ • ͔͠͠ɺ·ͣ͸ͦΕΛڐ༰ͯٞ͠࿦͢Δํ͕ݟ௨͕͠ྑ͘ͳΔ • ͦͷޙɺܕʢؔ਺ͷఆٛҬ΍஋Ҭʹ૬౰͢Δ֓೦ʣΛಋೖ͢Δ ͜ͱͰɺϥϜμه๏ͱͷରԠͷٞ࿦͕Ͱ͖Δ

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ϕʔλม׵ • ҎԼͰ࠶ؼతʹఆٛ͞ΕΔϥϜμࣜͷม׵Λϕʔλม׵ͱݺͿ • (1) • (2) ͳΒ͹ ͓Αͼ • ͜͜Ͱ ͸ɺϥϜμࣜ ͷதͷม਺ ΛϥϜμ ࣜ ʹஔ͖׵͑ͨϥϜμࣜͷ͜ͱͰ͋Δʢ୅ೖʣ

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୅ೖʹ͍ͭͯͷิ଍ • ϥϜμࣜ ͷதͷม਺ ʹ͍ͭͯɺ ͱ͍͏෦෼ࣜ ͕͋Δͱ͖ Λଋറม਺ɺͦ͏Ͱͳ͍ͱ͖ࣗ༝ม਺ͱݺͿ • ୅ೖ Λߟ͑Δͱ͖ɺ ͷࣗ༝ม਺͸ ͷଋറ ม਺Ͱ͸ͳ͍΋ͷͱ͢Δ • ·ͨ ͷଋറม਺͸ Ҏ֎ͷม਺ʹஔ͖׵͑ͯߟ͑ΔʢҰ ൠʹɺϥϜμࣜͷଋറม਺ΛͦͷࣜʹݱΕͳ͍ผͷม਺ʹஔ͖ ׵͑ͨࣜΛɺݩͷࣜͱಉҰࢹ͢Δʣ

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ϕʔλม׵ͷྫ ֤ϥϜμࣜͷࣗ༝ม਺͸ଞͷࣜͷଋറม਺Ͱ͸ͳ͍΋ͷͱ͢Δ

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Church-Rosser ͷఆཧ • ϕʔλม׵ͷ͔ͨ͠͸Ұ௨ΓͰ͸ͳ͍ • Church-Rosser ͷఆཧɿͻͱͭͷϥϜμ͔ࣜΒϕʔλม׵Ͱ ಘΒΕͨ;ͨͭͷϥϜμࣜ͸ɺԿ౓͔ϕʔλม׵Λߦ͏͜ͱ Ͱɺಉ͡ϥϜμࣜʹͰ͖Δ

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ਖ਼نܗ • Church-Rosser ͷఆཧʹΑΓɺϥϜμࣜ ͔Β࢝Ίͯ༗ݶ ͷεςοϓͰࢭ·Δʢࣜதʹ ͕ͳ͘ͳΔʣϕʔλม׵ྻ͕͋ Δͱ͖ɺ࠷ऴ݁Ռ͸Ұக͢Δ • ͜ͷ࠷ऴ݁ՌΛ ͷਖ਼نܗͱݺͿ • ༩͑ΒΕͨؔ਺ʢϥϜμه๏ʣΛػցతʹॲཧͯ͠ਖ਼نܗΛಘ Δͷ͕ɺؔ਺ܕϓϩάϥϛϯάݴޠʹ͓͚ΔܭࢉͰ͋Δʢܭࢉ ػՊֶͷ༻ޠͰ͸ධՁͱݺͿʣ

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͞ΒͳΔ࿩୊ɾɾɾ • ͜͜·ͰͷϥϜμܭࢉ͸ɺܕͳ͠ϥϜμܭࢉͱݺ͹ΕΔ • ܕΛಋೖͨ͠ɺܕ͖ͭϥϜμܭࢉͱ͍͏ମܥ͕͋Δ • ܕ͖ͭϥϜμܭࢉ͸ɺϥϜμه๏ͱͷରԠ͕Ͱ͖Δ • ܕ͖ͭϥϜμܭࢉ͸ɺܕΛର৅ɺϥϜμࣜΛࣹͱͯ͠ɺݍΛͳ ͢ • ͜ΕʹΑΓɺݍ࿦ͷٞ࿦͕Ԡ༻Ͱ͖Δ