ྫɿ1㱹 1㱸2 ɺ 1㱸2 㱹 2㱸1 w ࿈ݴඪ४ܗʢDPOKVODUJWFOPSNBMGPSN$/'ʣ w جຊબݴͱɺϦςϥϧ·ͨͭҎ্ͷϦςϥϧͷબݴͰ͋Δ w $/'ͱɺجຊબݴͰ͋Δ͔ɺͭҎ্ͷجຊબݴͷ࿈ݴͰ͋Δ w ྫɿ1㱸 1㱹2 ɺ 1㱹2 㱸 2㱹1 w ఆཧ w ͯ͢ͷཧࣜɺಉͳ%/'ͱ$/'Λ࣋ͭ 12
A A ^ B B A A _ B A B _ A A _ B, ¬A B A _ B, ¬B A A, A ! B B derive B from A A ! B ࿈ݴ $POKVODUJPO $POK ୯७Խʢ4JNQMJpDBUJPO 4JNQ Ճ "EEJUJPO "EE બݴࡾஈ๏ %JTKVODUJWF4ZMMPHJTN %4 Ϟʔμεɾϙωϯε .PEVT1POFOT .1 ݅ূ໌ $POEJUJPOBM1SPPG $1
¬A A ೋॏ൱ఆ %PVCMF/FHBUJPO %/ ໃ६ $POUSBEJDUJPO $POUS ؒূ໌ *OEJSFDU1SPPG *1 ϞʔμεɾτϨϯε .PEVT5PMMFOT .5 A ! B, ¬B ¬A ߹͚ʹΑΔূ໌ 1SPPGCZ$BTFT $BTFT A _ B, A ! C, B ! C C Ծݴࡾஈ๏ )ZQPUIFUJDBM4ZMMPHJTN )4 A ! B, B ! C A ! C ߏతδϨϯϚ $POTUSVDUJWF%JMFNNB $% A _ B, A ! C, B ! D C _ D
A B P 2. ¬A A P 3. ¬B P [B ͳͷͰ] 4. ¬¬A P [¬A ͳͷͰ] 5. A DN 6. B 1, 5, MP 7. False 3, 6, Contr 8. ¬A 4-7, IP 9. A 2, 8, MP 10. False 8, 9, Contr 11. B 3, 8-10, IP QED
w ૉཧࣜʢBUPNJDGPSNVMBʣͱɺ߲Ͱ͋ΔҾʹɺड़ޠΛద༻ͨ͠ͷ w ྫɿQ Y B R Z G D w ཧࣜʢXFMMGPSNFEGPSNVMBʣͷఆٛ w ҙͷૉཧࣜཧࣜͰ͋Δ w 8ͱ7͕ཧࣜͰY͕มͰ͋Εɺ࣍ͷࣜཧࣜͰ͋Δ w 8 ɺ8ɺ8㱹7ɺ8㱸7ɺ8ˠ7ɺ㱽Y8ɺ㱼Y8 39
ಉ͡ݶྔࢠͷೖΕସ͑ ݶྔࢠͱ൱ఆ ݁߹ࢠˠ্ͷݶྔࢠ ݶྔࢠͷ x(p(x) q(x)) x p(x) x q(x) x(p(x) q(x)) x p(x) x q(x) x yW y x W x yW y x W ¬( xW) x¬W ¬( xW) x¬W xW(x) yW(x/y) xW(x) yW(x/y)
YU Λߟ͑ͯΈΔ w ߲UதͷมZ͕8 Y தͰݶྔ͞Ε͓ͯΓɺͦͷݶྔൣғதʹ͋ΔYࣗ ༝Ͱ͋ΔͨΊɺ8 Y தͷมYΛ߲UͰஔ͖͑Δ͜ͱग़དྷͳ͍ w ஔ͖͑Λڐ͢ͱ㱽ZQ Z Z ͱͳΔ w Q Y Z ΛZYͷͰ͋Δͱ͢Δͱɺ㱽ZQ Z Z ZZͷͰ͋Δͱ ͳͬͯ͠·͏ 67
B 㱹Q C w ۭઅͱɺθϩݸͷϦςϥϧͷબݴͷ͜ͱͰ˘Ͱද͢ w અܗࣜͱɺͯ͢ͷݶྔࢠ͕શশݶྔࢠͰɺࣗ༝มͷͳ͍ף಄࿈ݴඪ४ܗͷ͜ͱ w ྫ w Q Y ͷઅܗࣜɺ㱼YQ Y w Q Y 㱸R Z C ͷઅܗࣜɺ㱼Y㱼Z Q Y 㱸R Z C w અܗࣜͷू߹දݱ w ू߹දݱͰɺઅܗࣜΛɺ֤جຊબݴͷू߹Ͱද͢ w ྫɿ㱼Y㱼Z Q Y 㱸R Z C \Q Y R Z C ^ͱͳΔ 88
C 㱸Q C Ͱ͋ΔͷͰॆෆೳ w ͭ·ΓɺઅܗࣜͰද͢ͱ\Q Y Q C ^ͱͳΓɺҰͭͷϦςϥϧͷม Yʹྑ͍ײ͡ͰΛೖ͢Δͱɺಋग़نଇΛ༻͍Δ͜ͱ͕ग़དྷΔ w มʹྑ͍ײ͡ʹɺ·ͨผͷมΛೖ͢ΔͨΊͷํ๏ͱͯ͠ɺೖ ͱ୯ҰԽͱ͍͏ख๏͕͋Γɺͦͷख๏ʹ͍ͭͯઆ໌͢Δ 101
L В Џ \^ͱઃఆͯ͠ɺεςοϓ 4ВLΛܭࢉ͠ɺ݁Ռ͕୯߲ू߹Ͱ͋ΕВLΛ4ͷ࠷൚୯ҰԽࢠͰ͋Δͱͯ͠ܭࢉΛऴྃɻͦ ͏Ͱͳ͚Εɺ%LΛ4ВLͷෆҰகू߹ͱͯ͠εςοϓ %L͕มWͱ߲UΛؚΈɺW͕Uதʹग़ݱ͠ͳ͍ͳΒɺ߹ВL ВL\WU^Λܭࢉ͠ɺ L L ͱઃఆͯ͠εςοϓɻͦ͏Ͱͳ͚Εɺ4୯ҰԽෆೳͰ͋Δͱͯ͠ܭࢉΛऴྃ 114
^ͷ࠷൚୯ҰԽࢠΛٻΊΑ В Џ 4В 4Џ 4Ͱ4୯߲ू߹Ͱͳ͍ͷͰܭࢉଓߦɻ% \Y H Z ^ͱܭࢉ มYɺ%ͷ߲H Z தʹग़ݱ͠ͳ͍ͷͰܭࢉଓߦ В В\YH Z ^ \YH Z ^ͱܭࢉ 4В \Q H Z G Z Q H Z [ ^୯߲ू߹Ͱͳ͍ͷͰܭࢉଓߦ % \G Z [^ͱܭࢉ ม[ɺ%ͷ߲G Z தʹग़ݱ͠ͳ͍ͷͰܭࢉଓߦ В В\[G Z ^ \YH Z [G Z ^ͱܭࢉ 4В \Q H Z G Z ^୯߲ू߹ͳͷͰɺ࠷൚୯ҰԽࢠΛВ \YH Z [G Z ^ͱͯ͠ܭࢉऴྃ 115
ͱɺR Z B 㱹Q Z ͷͭͷઅͷಋग़نଇΛٻΊΑ w ͭͷࣜR C Y ɺR C B ͱɺͭΊͷࣜͷR Z B ͕୯ҰԽՄೳͰ࠷൚ ୯ҰԽࢠВ \YB ZC^ͱͳΓɺ୯ҰԽ݁ՌR C B ͱͳΔ w ͭͷ͔ࣜΒR C B ͱͳΔૉཧࣜΛআͯ͠୯ҰԽͨ݁͠Ռ͕ɺ Q Y В Q B ͱͳΔ w ಉ༷ʹɺͭΊͷ͔ࣜΒQ Z В Q C ͱͳΔ w ಋग़نଇΑΓQ B 㱹Q C ͱ͍͏৽ͨͳઅ͕ಘΒΕΔ 120
w ྫɿB BࣹతɺB Bඇࣹత w ਪҠؔ w B3CɺC3Dͷͱ͖ʹB3D͕ΓཱͭͳΒਪҠతɺͦ͏Ͱͳ͍ͳΒඇਪҠత w ྫɿB C C DB DͱΓཱͭͷͰ ਪҠతɺB㱠CɺC㱠DඇਪҠత w ରশؔ w B3Cͷͱ͖ʹɺC3B͕ΓཱͭͳΒରশతɺͦ͏Ͱͳ͍ͳΒඇରশత w ྫɿB CରশతɺB Cඇରশత 123
Z 㱸Q Z [ ˠQ Y [ w ඇରশతɿ㱼Y㱼Z Q Y Z ˠQ Z Y w ೋ߲͕ؔඇࣹత͔ͭਪҠతͳΒɺඇରশతͰ͋Δ͜ͱΛূ໌ͤΑ 124 W = ඇࣹత ਪҠత ඇରশతɹͳͷͰ ¬W = ඇࣹత ਪҠత ¬ඇରশతɹͱͳΔ ͜͜Ͱɺ¬ඇରশత = ¬ x y(p(x, y) ¬p(y, x)) = x y¬(p(x, y) ¬p(y, x)) = x y(p(x, y) p(y, x)) ؔ ཧࣜ અܗࣜ ඇࣹత x¬p(x, x) ¬p(x, x) ਪҠత x y z(p(x, y) p(y, z) p(x, y)) ¬p(u, v) ¬p(v, w) p(u, w) ¬ඇରশత x y(p(x, y) p(y, x)) p(a, b) ͱ p(b, a) ͳͷͰɺઅܗࣜҎԼͷΑ͏ʹͳΔ [
BOOL | FUN | CALL | LET | FIX | IF VAR := alphabet (alphabet | num)* NUM := number BOOL := true | false FUN := fun VAR = EXPR . CALL := EXPR EXPR LET := let VAR = EXPR in EXPR . FIX := fix VAR = EXPR . IF := if EXPR then EXPR else EXPR .
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