停止性問題と数 / Halting problem and number

by Takahiro Uchiyama

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ఀࢭੑ໰୊ͱ਺ !IJHIXJEF Y෺ޠୈY໷ ʮ਺ʯ

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ϓϩάϥϜͷ ఀࢭੑ໰୊ ͱ͸

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ʮϓϩάϥϜʹσʔλΛ༩͑ͯಈ ͔ͨ͠ͱ͖ɺ༗ݶ࣌ؒ಺ʹಈ࡞͕ ఀࢭ͢Δ͔Ͳ͏͔ʯΛ൑ఆ͢Δͱ ͍͏໰୊ ݁৓ߒʮϓϩάϥϚͷ਺ֶʯ1

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͜Ε࣮͸ ʮܭࢉෆೳʯͱ ূ໌͞Ε͍ͯΔ

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ূ໌

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ఀࢭ͢Δ͔Ͳ͏͔ ൑ఆ͢Δ ϓϩάϥϜ IBMU ͕ ॻ͚ͨͱԾఆͯ͠ def halt?(program, data) # 真に驚くべき実装を発見したが、 # ここに記すには余白が狭すぎる [true, false].sample end

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IBMU ͕USVFͳΒಈ͖ଓ͚ͯ GBMTFͳΒఀࢭ͢ΔॲཧΛॻ͘ def self_check(program) result = halt?(program, program) puts "The result of 'halt?' is #{result}" if result while true puts "Never halting!!" end else puts "Just halting" end end

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def halt?(program, data) # 真に驚くべき実装を発見したが、 # ここに記すには余白が狭すぎる [true, false].sample end def self_check(program) result = halt?(program, program) puts "The result of 'halt?' is #{result}" if result while true puts "Never halting!!" end else puts "Just halting" end end self_check($stdin) ϓϩάϥϜ શମ૾

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% ruby halt_checker.rb < halt_checker.rb ࣗ෼ࣗ਎Λ ධՁ͢ΔͱͲ͏ͳΔʁ

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The result of 'halt?' is true Never halting!! Never halting!! Never halting!! Never halting!! Never halting!! Never halting!! Never halting!! Never halting!! Never halting!! Never halting!! --snip-- The result of 'halt?' is false Just halting ໃ६͢Δ

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Αͬͯ ʮఀࢭ͢Δ͔൑ఆ͢Δ ϓϩάϥϜ͕ॻ͚Δʯ ͱ͍͏Ծఆ͸ޡΓ എཧ๏

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ࠓճͷςʔϚͱ ؔ܎ͳͦ͞͏ ʘ ?P? ʗ

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ࠓճͷ ςʔϚ

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ʮ਺ʯ

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ಈࢺԽ

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਺͑Δ

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< >DPVOU

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਺͑ΒΕΔ

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<શͯͷ੔਺>DPVOU

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਺͑ΒΕΔ

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ʲ਺͑ʳʲΒΕΔʳ DPVOUBCMF

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<શͯͷ੔਺>͸ DPVOUBCMF͔

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DPVOUBCMF USVF

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Χ΢ϯλϒϧͱ͸ ʮू߹ͷཁૉ͕༗ݶݸͰ͋Δ͔ɺ·ͨ͸ू ߹ͷ͢΂ͯͷཁૉΛҎ্ͷ੔਺ͱҰରҐஔ ʹରԠ෇͚ΒΕΔʯͱ͖ɺͦͷू߹͸Χ΢ ϯλϒϧ DPVOUBCMF Ͱ͋Δͱఆٛ͠·͢ɻ ݁৓ߒʮϓϩάϥϚͷ਺ֶʯ1

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શͯͷ੔਺͸൪߸Λ͚ͭΒΕΔ ੔਺ ൪߸

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<ੈͷதʹଘࡏ͢Δ શͯͷϓϩάϥϜ>͸ DPVOUBCMF͔ʁ

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ϓϩάϥϜͰ࢖͑Δจࣈ͸ DPVOUBCMFͰ͋Δ a b c d e f g h i j k l m n o p q r s t u v w x y z A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 0 1 2 3 4 5 6 7 8 9 ! " # % & ' ( ) + , - . / : ; < = > ? [ \ ] ^ _ ` { | } ~

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/छྨͷจࣈΛ Oࣈ෼࢖ͬͯ࡞ΕΔ ϓϩάϥϜͷ૯਺͸/?Oݸ

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จࣈͰ࡞ΕΔϓϩάϥϜ/ݸ จࣈͰ࡞ΕΔϓϩάϥϜ/?ݸ จࣈͰ࡞ΕΔϓϩάϥϜ/?ݸ ݁ہશͯͷϓϩάϥϜʹ ൪߸Λ෇͚ΒΕΔ

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<ੈͷதʹଘࡏ͢Δ શͯͷϓϩάϥϜ>͸ DPVOUBCMFͩͬͨ

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DPVOUBCMFͰ ͳ͍΋ͷͱ͸ʁ

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< Ҏ্ҎԼͷ શͯͷ࣮਺>͸ DPVOUBCMF͔

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GBMTF

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ূ໌

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ͷ܁ฦ͠ ͷ܁Γฦ͠ ˽ʸ ͷ܁Γฦ͠ ϥϯμϜ ʙͷશͯͷ࣮਺ྻΛ ༻ҙͨ͠ͱԾఆͯ͠

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͜͜ʹଘࡏ͠ͳ͍ ࣮਺Λ࡞Δʹ͸ʁ

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ͷ܁ฦ͠ ͷ܁Γฦ͠ ˽ʸ ͷ܁Γฦ͠ ϥϯμϜ ֤ྻͷ֤ܻΛର֯ઢʹ૸ࠪͯ͠ ͍ͯ͘͠ͱඞͣ৽͍࣮͠਺͕Ͱ͖Δ

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ͳͥͳΒ શͯͷྻͱ ඞͣͲ͔ܻ͕͜ ҧ͏ͨΊ

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ΑͬͯԾఆ͸ޡΓ <Ҏ্ҎԼͷ શͯͷ࣮਺>͸ DPVOUBCMFͰ͸ͳ͍

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Χϯτʔϧͷର֯ઢ࿦๏ ͱ͍͏Β͍͠ (FPSH'FSEJOBOE   -VEXJH1IJMJQQ$BOUPS (FSNBOZ

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͛͢ʔ

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͔͜͜Β͕ ຊ୊

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ϓϩάϥϜͷ ఀࢭੑ໰୊ͷূ໌͸ Χϯτʔϧͷର֯ઢ࿦๏Λ ࣮͸ར༻͍ͯͨ͠ͱ͍͏࿩

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ର֯ઢ࿦๏ͱ͸ͭ·Γ ͜͏͍͏͜ͱ

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9Λू߹ͱ͠ɺ9Λ9ͷ΂͖ू߹ͱ͢Δɻ ͞ΒʹСΛ9͔Β9΁ͷࣸ૾ͱ͢Δɻ9ͷ ෦෼ू߹:Λ XJLJQFEJBʮΧϯτʔϧͷର֯ઢ࿦๏ʯ {x 2 X : x / 2 (x)} ʹΑΓఆٛ͢ΔͱɺС Y :ͱͳΔY㱨9͸ଘ ࡏ͠ͳ͍ɻ

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ू߹ࣗવ਺ ΂͖ू߹ʙͷ੔਺ͷྻ ࣸ૾DPVOU͢Δ ର֯ઢ্ʹ૸ࠪͯ͠৽͍͠ྻΛ࡞Δ ࣸ૾Ͱ͖ͳ͍ DPVOUBCMFͰͳ͍

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ఀࢭੑ໰୊ʹ ద༻͢Δͱ

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ू߹શͯͷϓϩάϥϜ ˺ࣗવ਺ ΂͖ू߹ϓϩάϥϜΛೖྗ஋ͱͨ͠ϓϩ άϥϜಉ࢜ͷ૊Έ߹Θͤ ࣸ૾ఀࢭ͢Δ͔൑ఆ͢Δ ఀࢭ͢Δͱ൑ఆͨ͠Β ఀࢭ͠ͳ͍ϓϩάϥϜΛ࡞Δ ఀࢭ൑ఆͰ͖ͳ͍ DPVOUBCMFͰͳ͍

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ͪͳΈʹ ͜͏΍ͬͯఀࢭੑ໰୊Λ ূ໌ͨ͠ͷ͸͜ͷਓ "MBO.BUIJTPO5VSJOH 6,