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PPL2016-9-3

Kyoko KADOWAKI
March 08, 2016
Science
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560

 PPL2016-9-3

Agda による定式化された型推論器の拡張と改良

Kyoko KADOWAKI

March 08, 2016
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Transcript

  1. Agda ʹΑΔఆࣜԽ͞Εͨܕਪ࿦ ثͷ֦ுͱվྑ ͓஡ͷਫঁࢠେֶɹ໳࿬߳ࢠɹઙҪ݈Ұ 

  2. 1͸͡Ίʹ •ݚڀ֓ཁ •ݚڀഎܠ •ຊݚڀͷߩݙ •ൃදͷྲྀΕ

  3. ݚڀ֓ཁ w"HEBΛ༻͍ͯɺఆࣜԽ͞Εͨܕਪ࿦ثʹHFOFSJD QSPHSBNNJOH΍ෆ౳੍ࣜ໿ͳͲͷςΫχοΫΛ༻ ͍ͯɺ࣮૷Λ؆໌ʹ͠ɺߏจͷ֦ுΛ؆୯ʹͨ͠ wHFOFSJDQSPHSBNNJOH wෆ౳੍ࣜ໿

  4. Agda ʹ͍ͭͯ wґଘܕΛ༻͍Δ͜ͱͷͰ͖Δɺఆཧূ໌ࢧԉγες Ϝ͓Αͼϓϩάϥϛϯάݴޠ w)BTLFMMͳͲʹจ๏͕ࣅ͍ͯΔ

  5. ґଘܕ(Dependent Types) w஋ʹґଘ͢ΔܕΛͭ͘Δ͜ͱ͕Ͱ͖Δܕ wྫ7FDOܕ O͸JOUܕͷ஋ 
 ௕͕͞OͷϕΫτϧΛ ܕͷϨϕϧͰ දݱ͍ͯ͠Δ wྫ͑͹ɺ0$BNMͷϦετͰ͸௕͕͞ܕϨϕϧͰ͸ࢦ

    ఆͰ͖ͳ͍ MFOHUIؔ਺ͳͲͰऔΓग़͢͜ͱ͕ඞཁ

  6. ఆࣜԽ ೖྗʢܕແ͠ݴޠʣ ग़ྗʢܕ෇͖ݴޠʣ ܕਪ࿦ n = 1 let f x

    = match x with 0 -> “zero” | _ -> “not zero” n : int n = 1 f : int -> string f x = match x with …

  7. ܕ ఆࣜԽ ೖྗʢܕແ͠ݴޠʣ ग़ྗʢܕ෇͖ݴޠʣ ܕਪ࿦ n : int f :

    int -> string

  8. ܕ ఆࣜԽ ೖྗʢܕແ͠ݴޠʣ ग़ྗʢܕ෇͖ݴޠʣ ܕਪ࿦ n : int f :

    int -> string ݕূػߏ νΣοΫ ᶃ

  9. ఆࣜԽ ೖྗʢܕແ͠ݴޠʣ ग़ྗʢܕ෇͖ݴޠʣ ܕਪ࿦ ܕΛফڈ ܕແ͠ݴޠ ಉ͡Ͱ͋Δ͜ͱΛอূ͢Δ  

  10. ݚڀഎܠ wܕਪ࿦Λ͢Δʹ͸VOJpDBUJPOͷػߏ͕ඞཁ͕ͩɺ ී௨ʹ࣮૷͢Δͱม਺ͷॻ͖׵͑ͳͲ͕ඞཁͰɺ "HEBͰ͸Ͱ͖ͳ͍ wͦ͜Ͱ.D#SJEF͸ܕม਺ͷ਺Λ޼ົʹ؅ཧ͢Δܗ ͰVOJpDBUJPOͷػߏΛQVSFGVODUJPOBMʹ࣮ݱͨ͠ w͜ΕʹΑΓɺ"HEBͰܕਪ࿦ثΛͭ͘Δ౔୆͕ग़དྷ ͨ

  11. ຊݚڀͷߩݙ w.D#SJEF͕ࣔͨ͠VOJpDBUJPOͷػߏ<.D#SJEF >Λ֦ு͠ɺ୯ ҰԽͰ͖͍ͯΔূ໌΋ฦ͢ܗͰఆࣜԽͨ͠ʢ11-ʣ w͜ͷVOJpDBUJPOͷػߏΛ΋ͱʹͯ͠ɺ"HEBͰఆࣜԽ͞Εͨܕਪ࿦ث Λ࣮૷ͨ͠ʢ11-ʣ w͜ͷܕਪ࿦ثʹෆ౳੍ࣜ໿Λಋೖ͢Δ͜ͱͰܕม਺Λࢦఆ͢Δͷ Έͷ࣮૷ΑΓ؆୯ʹ࣮૷͢Δํ๏Λࣔ͠ɺͦΕΛܕنଇͷܗͰఆ ࣜԽͨ͠ wVOJpDBUJPOͷػߏΛHFOFSJDQSPHSBNNJOHʹΑΓ೚ҙͷܕʹ

    ֦ு͠ɺܕͷ֦ுΛ؆୯ʹͨ͠ wܕਪ࿦ͷ݁ՌɺಘΒΕͨܕ෇͖߲͔ΒܕΛऔΓআ͘ͱݩͷ߲͕ಘ ΒΕΔ͜ͱ΋ূ໌ͨ͠ wܕਪ࿦ͷ֦ுͷྫͱͯ͠ɺ૊Λಋೖͨ͠


  12. ࠓ೔ͷίϯςϯπ ͸͡Ίʹ VOJpDBUJPO ܕఆٛ HFOFSJDQSPHSBNNJOHͱෆ౳੍ࣜ໿ ܕਪ࿦ ·ͱΊ

  13. 2 unification •McBrideʹΑΔख๏ •unificationͷఆࣜԽ

  14. McBride ʹΑΔ unification ͷػߏ McBride͸ɺܕม਺Λ༗ݶू߹Ͱදݱ͢Δख๏Ͱఀࢭੑ͕໌ Β͔ͳ unificationΛ࣮ݱͨ͠ ܕН B B

      B N ܕม਺ͷू߹ ࠷େͰNݸ ༗ݶू߹ pOJUFTFU N

  15. McBride ʹΑΔ unification ͷػߏ ܕН B JOU   B

    N N ܕม਺͸୅ೖʹΑΓ۩ମԽʢ۩ମతͳܕʹͳΔʣՄೳੑ͕͋Δ

  16. ܕUBˠ BˠB  B B B ܕUBˠ BˠJOU  B

    JOU B

  17. ܕUBˠ BˠB  B B B ܕUBˠ BˠJOU  B

    JOU B VOJpDBUJPO BJOU B JOU B

  18. ܕUBˠ BˠB  B B B ܕUBˠ BˠJOU  B

    JOU B VOJpDBUJPO BJOU B JOU B ❌ ۩ମԽ͞ΕͨܕΛू߹͔Β࡟আ

  19. ܕUBˠ BˠB  B B B ܕUBˠ BˠJOU  B

    JOU B VOJpDBUJPO BJOU B B Bˠ BˠJOU  ܕม਺ͷ਺͕͔֬ʹݮ͍ͬͯΔ

  20. ܕUBˠ BˠB  B B B ܕUBˠ BˠJOU  B

    JOU B VOJpDBUJPO BJOU B B Bˠ BˠJOU  ܕม਺ͷ਺͕͔֬ʹݮ͍ͬͯΔ ܕม਺͕۩ମԽ͞ΕΔͨͼʹɺ۩ମԽ͞Ε͍ͯͳ͍ܕม਺ͷ਺ ͕ͻͱͭݮΔ

  21. ܕUBˠ BˠB  B B B ܕUBˠ BˠJOU  B

    JOU B VOJpDBUJPO BJOU B B Bˠ BˠJOU  ܕม਺ͷ਺͕͔֬ʹݮ͍ͬͯΔ ܕม਺͕۩ମԽ͞ΕΔͨͼʹɺ۩ମԽ͞Ε͍ͯͳ͍ܕม਺ͷ਺ ͕ͻͱͭݮΔ ఀࢭੑ͕໌Β͔

  22. unificationͷఆࣜԽ wઌ΄Ͳࣔͨ͠ख๏ͰɺVOJpDBUJPOΛ"HEB্Ͱ࣮૷Ͱ͖ͨ wܕਪ࿦ثΛఆࣜԽ͢Δʹ͸z͔֬ʹVOJGZ͞Ε͍ͯΔzূ໌ Λฦ͢͜ͱ΋ඞཁ
 ɹࣜʹ͢ΔͱTVCTUМUTVCTUМU
 ʮ୅ೖͨ͠ޙͷܕ͕͔֬ʹಉҰʯ wVOJpDBUJPOͷؔ਺Λ֦ுͯ͠ɺ͜ͷূ໌Λฦ͢Α͏ʹ͠ ͨʢ11-ʣ


  23. 3 ܕఆٛ •ܕɺܕ൑அɺ߲ •ຊݚڀʹ͓͚Δಛघͳදݱ


  24. ܕఆٛ ߲

  25. ܕఆٛ ߲ N൪໨ͷܕม਺ ؔ਺ܕ

  26. ܕఆٛ ߲ ม਺ ϥϜμந৅ ؔ਺ద༻

  27. ܕ൑அ ਪ࿦نଇ

  28. ܕ൑அ ਪ࿦نଇ ʮܕ؀ڥϵͷ΋ͱͰɺ߲.ͷܕ͸НʹͳΔʯ

  29. ܕ൑அ ਪ࿦نଇ

  30. ܕਪ࿦ͷಛघͳදݱ ܕΛܭࢉͯ͠ग़ྗ͢ΔͨΊͷنଇ

  31. ܕਪ࿦ͷಛघͳදݱ ೖྗ ग़ྗ

  32. ܕਪ࿦ͷಛघͳදݱ ᶃ߲.ͱ ܕม਺Λ࠷େͰNݸ΋ͭܕ؀ڥϵ͕ೖྗ͞ΕΔ

  33. ܕਪ࿦ͷಛघͳදݱ ᶄ৽͍͠ܕม਺͕ܕਪ࿦தʹN``NݸׂΓ౰ͯΒΕΔ

  34. ܕਪ࿦ͷಛघͳදݱ ᶅܕม਺ͷ਺ΛN`ݸ·ͰݮΒ͢୅ೖ͕ಘΒΕΔ

  35. ܕਪ࿦ͷಛघͳදݱ ᶇܕม਺ͷ਺͕N`·ͰݮͬͨܕН͕ฦͬͯ͘Δ

  36. unification ͷಛघͳදݱ ܕНͱНΛಉ͡ʹ͢Δ VOJGZ͢Δ ୅ೖМΛಘΔ

  37. unification ͷಛघͳදݱ ೖྗ ग़ྗ

  38. unification ͷಛघͳදݱ ܕม਺Λ࠷େͰN``ݸ࣋ͭܕН ͱН͸
 ܕม਺ΛN``ݸ͔ΒN`ݸ·ͰݮΒ͢୅ೖМʹΑͬͯ VOJpDBUJPO ୯ҰԽ ͞ΕΔ

  39. 4 generic programmingͱ
 ෆ౳੍ࣜ໿ •generic programming •ෆ౳੍ࣜ໿

  40. generic programming wී௨͸ܕΛίϯετϥΫλͰఆٛ͢Δ w͜ͷ৔߹ɺܕΛૢ࡞͢Δؔ਺ʹ͓͍ͯɺܕΛ࠶࣮૷͢Δͨ ͼʹύλʔϯϚονΛ࠶࣮૷͢Δඞཁ͕͋Δ wܕΛHFOFSJDʹੜ੒͢Δ͜ͱͰɺύλʔϯϚονͷ࠶࣮૷ Λආ͚ɺϢʔβ͕ࢦఆͨ͠೚ҙͷܕΛ࠶࣮૷ͳ͘ਪ࿦Ͱ͖ ΔΑ͏ʹͳΔ wຊݚڀͰ͸ɺ3FHVMBS</PPSUΒ >ΛݩʹHFOFSJDͳ

    ܕΛ࣮૷ͨ͠

  41. generic programming ૊ͷߏจ $POT 'TU 4OE Λ૿΍͍ͨ͠ͱ͖ɺ
 ௚ੵܕΛ௥Ճ͢Δඞཁ͕༗Δ

  42. generic programming ͜ͷΑ͏ʹܕΛ૿΍ͯ͠΋ɺ
 VOJpDBUJPO΍ܕʹؔ͢Δิ୊ͷ࣮૷Λมߋ͢Δඞཁ͕ͳ͍

  43. generic programming ߏจΛ૿΍͢ͱ͖ɺ
 ߏจͦͷ΋ͷͷܕਪ࿦ʹूதͰ͖Δ

  44. generic programming ܕΛ#/'ͷΑ͏ͳܗͰදݱ͢Δ

  45. generic programming ܕΛ#/'ͷΑ͏ͳܗͰදݱ͢Δ

  46. generic programming ܕΛ#/'ͷΑ͏ͳܗͰදݱ͢Δ

  47. generic programming ܕΛ#/'ͷΑ͏ͳܗͰදݱ͢Δ

  48. generic programming ܕΛ#/'ͷΑ͏ͳܗͰදݱ͢Δ

  49. generic programming ܕΛ#/'ͷΑ͏ͳܗͰදݱ͢Δ ͜ΕͰɺܕͷදݱ͕Ͱ͖ͨ

  50. generic programming ۩ମతͳܕͷදݱ

  51. generic programming දݱͷҙຯΛ༩͍͑ͯΔ

  52. generic programming ࣗ෼ࣗ਎Λ༩͑Δ͜ͱͰ࠶ؼ͢Δ

  53. generic programming ࠶ؼΛด͡ΔͨΊͷܕ

  54. generic programming ࠶ؼΛด͡ΔͨΊͷܕ

  55. generic programming ࠶ؼΛด͡ΔͨΊͷܕ

  56. generic programming ೚ҙʹେ͖͍ܕΛදݱͰ͖Δ

  57. ෆ౳੍ࣜ໿ ܕਪ࿦Λ࣮૷͢ΔࡍɺҎલ͸N``Λݫີʹܕʹ૊Έ ࠐΜͰ͍ͨ

  58. ෆ౳੍ࣜ໿ ͨͱ͑͹-BNTΛܕਪ࿦͢Δͱ͖ N``/BU N``DPVOUT  ϵ`$YUN`` ϵʜ

  59. ෆ౳੍ࣜ໿ ͨͱ͑͹-BNTΛܕਪ࿦͢Δͱ͖ N``JOU N``DPVOUT  ϵ`$YUN`` ϵʜ ܕͷ஋ϨϕϧͰܕΛ߹Θͤͯূ໌͠ͳ͍ͱ͍͚ͳ͘ͳΔ

  60. ෆ౳੍ࣜ໿ N``͸N≤N`Λຬ͍ͨͯ͠Ε͹Α͘ɺ۩ମతͳ਺ ͸Ҏޙ࣋ͪา͘ඞཁ͕ͳ͍ ୅ΘΓʹN≤N``ͱ͍͏ূ໌Λ࣋ͪา͘

  61. 5 ܕਪ࿦ • ؔ਺ఆٛ • ܕਪ࿦ͷྫ - ؔ਺ద༻ -

  62. ؔ਺ఆٛ ೖྗ
 TNݸͷܕม਺Λ΋ͭXFMMTDPQFEUFSN
 ϵܕ؀ڥ ग़ྗ N⒌⒌ N⒌ࣗવ਺ N≤N⒌⒌N≤N⒌⒌Ͱ͋Δ͜ͱͷূ໌ МN⒌⒌ݸͷܕม਺Λ࣋ͭܕΛN⒌ݸͷܕม਺Λ࣋ͭܕʹ͢Δ୅ೖ Нਪ࿦݁Ռͷܕ

    Xਪ࿦͞ΕͨXFMMUZQFEUFSNɻܕม਺ΛN⒌ݸ΋ͭ FSBTFX㲇Tฦ͞ΕͨXFMMUZQFEUFSNT͔Βܕ৘ใΛ࡟আͨ͠΋ͷ͕ɺ
 ΋ͱͷXFMMTDPQFEUFSNͱ౳͍͠ͱ͍͏ূ໌ JOGFS

  63. ؔ਺ఆٛ ೖྗ
 TNݸͷܕม਺Λ΋ͭXFMMTDPQFEUFSN
 ϵܕ؀ڥ .BZCF ग़ྗ N⒌⒌ N⒌ࣗવ਺ N≤N⒌⒌N≤N⒌⒌Ͱ͋Δ͜ͱͷূ໌ МN⒌⒌ݸͷܕม਺Λ࣋ͭܕΛN⒌ݸͷܕม਺Λ࣋ͭܕʹ͢Δ୅ೖ

    Нਪ࿦݁Ռͷܕ Xਪ࿦͞ΕͨXFMMUZQFEUFSNɻܕม਺ΛN⒌ݸ΋ͭ FSBTFX㲇Tฦ͞ΕͨXFMMUZQFEUFSNT͔Βܕ৘ใΛ࡟আͨ͠΋ͷ͕ɺ
 ΋ͱͷXFMMTDPQFEUFSNͱ౳͍͠ͱ͍͏ূ໌ JOGFS

  64. ؔ਺ఆٛ ೖྗ
 TNݸͷܕม਺Λ΋ͭXFMMTDPQFEUFSN
 ϵܕ؀ڥ .BZCF ग़ྗ N⒌⒌ N⒌ࣗવ਺ N≤N⒌⒌N≤N⒌⒌Ͱ͋Δ͜ͱͷূ໌ МN⒌⒌ݸͷܕม਺Λ࣋ͭܕΛN⒌ݸͷܕม਺Λ࣋ͭܕʹ͢Δ୅ೖ

    Нਪ࿦݁Ռͷܕ Xਪ࿦͞ΕͨXFMMUZQFEUFSNɻܕม਺ΛN⒌ݸ΋ͭ FSBTFX㲇Tฦ͞ΕͨXFMMUZQFEUFSNT͔Βܕ৘ใΛ࡟আͨ͠΋ͷ͕ɺ
 ΋ͱͷXFMMTDPQFEUFSNͱ౳͍͠ͱ͍͏ূ໌ JOGFS OPUIJOH͕ฦΔܕΤϥʔͰ͋Δ

  65. ؔ਺ఆٛ ίʔυͰ͸͜ͷΑ͏ʹͳΔʢࢀߟʣ

  66. ܕਪ࿦ͷྫ (App) "QQTTͷܕਪ࿦͸ TΛܕਪ࿦͠ɺМͱНΛಘΔ ϵʹМΛࢪ্ͨ͠ͰTΛܕਪ࿦͠ɺМͱНΛಘΔ ННˠЌ Ќ͸৽͍͠ܕม਺ ͱͳΔΑ͏VOJGZ͠  ੒ޭͨ͠ΒМΛಘΔ

    ฦ͞ΕΔܕ͸ЌͱͳΔ

  67. ܕਪ࿦ͷྫ (App) "QQTTͷܕਪ࿦͸ TΛܕਪ࿦͠ɺМͱНΛಘΔ ϵʹМΛࢪ্ͨ͠ͰTΛܕਪ࿦͠ɺМͱНΛಘΔ ННˠЌ Ќ͸৽͍͠ܕม਺ ͱͳΔΑ͏VOJGZ͠  ੒ޭͨ͠ΒМΛಘΔ

    ฦ͞ΕΔܕ͸ЌͱͳΔ

  68. ܕਪ࿦ͷྫ (App) "QQTTͷܕਪ࿦͸ TΛܕਪ࿦͠ɺМͱНΛಘΔ ϵʹМΛࢪ্ͨ͠ͰTΛܕਪ࿦͠ɺМͱНΛಘΔ ННˠЌ Ќ͸৽͍͠ܕม਺ ͱͳΔΑ͏VOJGZ͠  ੒ޭͨ͠ΒМΛಘΔ

    ฦ͞ΕΔܕ͸ЌͱͳΔ

  69. ܕਪ࿦ͷྫ (App) "QQTTͷܕਪ࿦͸ TΛܕਪ࿦͠ɺМͱНΛಘΔ ϵʹМΛࢪ্ͨ͠ͰTΛܕਪ࿦͠ɺМͱНΛಘΔ ННˠЌ Ќ͸৽͍͠ܕม਺ ͱͳΔΑ͏VOJGZ͠  ੒ޭͨ͠ΒМΛಘΔ

    ฦ͞ΕΔܕ͸ЌͱͳΔ

  70. ܕਪ࿦ͷྫ (App) TΛܕਪ࿦͠ɺМͱНΛಘΔɻ ϵʹМΛࢪ্ͨ͠ͰTΛܕਪ࿦͠ɺМͱНΛಘΔɻ

  71. ܕਪ࿦ͷྫ (App) TΛܕਪ࿦͠ɺМͱНΛಘΔɻ ϵʹМΛࢪ্ͨ͠ͰTΛܕਪ࿦͠ɺМͱНΛಘΔɻ

  72. ܕਪ࿦ͷྫ (App) TΛܕਪ࿦͠ɺМͱНΛಘΔɻ ϵʹМΛࢪ্ͨ͠ͰTΛܕਪ࿦͠ɺМͱНΛಘΔɻ

  73. ܕਪ࿦ͷྫ (App) ϵʹМΛࢪ্ͨ͠ͰTΛܕਪ࿦͠ɺМͱНΛಘΔɻ ННˠЌ Ќ͸৽͍͠ܕม਺ ͱͳΔΑ͏VOJGZ͠ 

  74. ܕਪ࿦ͷྫ (App) ϵʹМΛࢪ্ͨ͠ͰTΛܕਪ࿦͠ɺМͱНΛಘΔɻ ННˠЌɹɹɹɹͱͳΔΑ͏VOJGZ͠  N`൪໨ͷܕม਺Λ৽͘͠औΓग़͢

  75. ܕਪ࿦ͷྫ (App) ННˠЌɹɹɹɹͱͳΔΑ͏VOJGZ͠  ੒ޭͨ͠ΒМΛಘΔ

  76. ܕਪ࿦ͷྫ (App) ННˠЌɹɹɹɹͱͳΔΑ͏VOJGZ͠  ੒ޭͨ͠ΒМΛಘΔ

  77. ܕਪ࿦ͷྫ (App) "QQͷܕنଇΛద༻͢Δɻ

  78. "QQͷܕنଇΛద༻͢Δɻ

  79. ܕਪ࿦ͷྫ (App) "QQͷܕنଇΛద༻͢Δɻ ඞཁͳ୅ೖ

  80. ܕਪ࿦ͷྫ (App) "QQͷܕنଇΛద༻͢Δɻ ͳΔ୅ೖМΛฦ͍ͨ͠

  81. ܕਪ࿦ͷྫ (App) "QQͷܕنଇΛద༻͢Δɻ ͳΔ୅ೖМΛฦ͍ͨ͠

  82. ܕਪ࿦ͷྫ (App)

  83. ܕਪ࿦ͷྫ (App) ଞͷߏจ 'TU 4OEͳͲ ʹؔͯ͠΋ಉ༷ʹূ໌Ͱ͖Δ

  84. ·ͱΊ wܕม਺Λ਺Ͱ͸ͳ͘ෆ౳ࣜΛ༻͍ͯදݱ͢Δํࣜʹ վྑ͠ɺ·ͨܕఆٛΛHFOFSJDQSPHSBNNJOHͷߟ ͑ํΛऔΓೖΕͯΑΓҰൠతʹվྑ͢Δ͜ͱͰɺܕ ਪ࿦ثʹ͓͚ΔTZOUBYͷ֦ுͱূ໌ͷ؆ུԽΛ࣮ ݱͨ͠ wιʔείʔυHJUIVCDPNLEYVJOGFSBHEB
 "HEB߹ܭߦ

  85. ࠓޙͷ՝୊ wߏจͷ֦ு ଟ૬ͷ-FUจͳͲ 
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