PPL2016-9-3

Agda による定式化された型推論器の拡張と改良

Transcript

1. Agda ʹΑΔఆࣜԽ͞Εͨܕਪ࿦ ثͷ֦ுͱվྑ ͓஡ͷਫঁࢠେֶɹ໳࿬߳ࢠɹઙҪ݈Ұ
   
   
   
   
2. 1͸͡Ίʹ •ݚڀ֓ཁ •ݚڀഎܠ •ຊݚڀͷߩݙ •ൃදͷྲྀΕ
3. ݚڀ֓ཁ w"HEB
   Λ༻͍ͯɺఆࣜԽ͞Εͨܕਪ࿦ثʹ
   HFOFSJD
   QSPHSBNNJOH
   ΍ෆ౳੍ࣜ໿ͳͲͷςΫχοΫΛ༻ ͍ͯɺ࣮૷Λ؆໌ʹ͠ɺߏจͷ֦ுΛ؆୯ʹͨ͠
   wHFOFSJD
   QSPHSBNNJOH
   wෆ౳੍ࣜ໿
4. Agda ʹ͍ͭͯ wґଘܕΛ༻͍Δ͜ͱͷͰ͖Δɺఆཧূ໌ࢧԉγες Ϝ͓Αͼϓϩάϥϛϯάݴޠ
   w)BTLFMM
   ͳͲʹจ๏͕ࣅ͍ͯΔ
5. ґଘܕ(Dependent Types) w஋ʹґଘ͢ΔܕΛͭ͘Δ͜ͱ͕Ͱ͖Δܕ
   wྫ
   7FD
   O
   ܕ
   O
   ͸
   JOU
   ܕͷ஋ 
 ௕͕͞
   O
   ͷϕΫτϧΛ 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͸
    VOJpDBUJPO
    ͷػߏ͕ඞཁ͕ͩɺ ී௨ʹ࣮૷͢Δͱม਺ͷॻ͖׵͑ͳͲ͕ඞཁͰɺ
    "HEBͰ͸Ͱ͖ͳ͍
    wͦ͜Ͱ
    .D#SJEF
    ͸ܕม਺ͷ਺Λ޼ົʹ؅ཧ͢Δܗ ͰVOJpDBUJPOͷػߏΛ
    QVSF
    GVODUJPOBM
    ʹ࣮ݱͨ͠
    w͜ΕʹΑΓɺ"HEBͰܕਪ࿦ثΛͭ͘Δ౔୆͕ग़དྷ ͨ
11. ຊݚڀͷߩݙ w.D#SJEF
    ͕ࣔͨ͠
    VOJpDBUJPO
    ͷػߏ
    <.D#SJEF
    >
    Λ֦ு͠ɺ୯ ҰԽͰ͖͍ͯΔূ໌΋ฦ͢ܗͰఆࣜԽͨ͠ʢ11-ʣ
    w͜ͷ
    VOJpDBUJPO
    ͷػߏΛ΋ͱʹͯ͠ɺ
    "HEB
    ͰఆࣜԽ͞Εͨܕਪ࿦ث Λ࣮૷ͨ͠ʢ11-ʣ
    w͜ͷܕਪ࿦ثʹෆ౳੍ࣜ໿Λಋೖ͢Δ͜ͱͰܕม਺Λࢦఆ͢Δͷ Έͷ࣮૷ΑΓ؆୯ʹ࣮૷͢Δํ๏Λࣔ͠ɺͦΕΛܕنଇͷܗͰఆ ࣜԽͨ͠
    wVOJpDBUJPO
    ͷػߏΛHFOFSJD
    QSPHSBNNJOHʹΑΓ೚ҙͷܕʹ ֦ு͠ɺ
    ܕͷ֦ுΛ؆୯ʹͨ͠
    wܕਪ࿦ͷ݁ՌɺಘΒΕͨܕ෇͖߲͔ΒܕΛऔΓআ͘ͱݩͷ߲͕ಘ ΒΕΔ͜ͱ΋ূ໌ͨ͠
    wܕਪ࿦ͷ֦ுͷྫͱͯ͠ɺ૊Λಋೖͨ͠
    

12. ࠓ೔ͷίϯςϯπ ͸͡Ίʹ
    VOJpDBUJPO
    ܕఆٛ
    HFOFSJD
    QSPHSBNNJOH
    ͱෆ౳੍ࣜ໿
    ܕਪ࿦
    ·ͱΊ
13. 2 unification •McBrideʹΑΔख๏ •unificationͷఆࣜԽ
14. McBride ʹΑΔ unification ͷػߏ McBride͸ɺܕม਺Λ༗ݶू߹Ͱදݱ͢Δख๏Ͱఀࢭੑ͕໌ Β͔ͳ unificationΛ࣮ݱͨ͠ ܕ
    Н B
    B
    
    
    B N
    
     ܕม਺ͷू߹ ࠷େͰ
    N
    ݸ ༗ݶू߹
    pOJUF
    TFU N
15. McBride ʹΑΔ unification ͷػߏ ܕ
    Н B
    JOU
    
    
    B N
    
     N ܕม਺͸୅ೖʹΑΓ۩ମԽʢ۩ମతͳܕʹͳΔʣՄೳੑ͕͋Δ
16. ܕ
    U
    
    B
    ˠ
    B
    ˠ
    B
    
    B
    B
    B ܕ
    U
    
    B
    ˠ
    B
    ˠ
    JOU
    
    B
    JOU
    B
17. ܕ
    U
    
    B
    ˠ
    B
    ˠ
    B
    
    B
    B
    B ܕ
    U
    
    B
    ˠ
    B
    ˠ
    JOU
    
    B
    JOU
    B VOJpDBUJPO B
    
    JOU B
    JOU
    B
18. ܕ
    U
    
    B
    ˠ
    B
    ˠ
    B
    
    B
    B
    B ܕ
    U
    
    B
    ˠ
    B
    ˠ
    JOU
    
    B
    JOU
    B VOJpDBUJPO B
    
    JOU B
    JOU
    B ❌ ۩ମԽ͞ΕͨܕΛू߹͔Β࡟আ
19. ܕ
    U
    
    B
    ˠ
    B
    ˠ
    B
    
    B
    B
    B ܕ
    U
    
    B
    ˠ
    B
    ˠ
    JOU
    
    B
    JOU
    B VOJpDBUJPO B
    
    JOU B
    B
    B
    ˠ
    B
    ˠ
    JOU
    ܕม਺ͷ਺͕͔֬ʹݮ͍ͬͯΔ
    
    
20. ܕ
    U
    
    B
    ˠ
    B
    ˠ
    B
    
    B
    B
    B ܕ
    U
    
    B
    ˠ
    B
    ˠ
    JOU
    
    B
    JOU
    B VOJpDBUJPO B
    
    JOU B
    B
    B
    ˠ
    B
    ˠ
    JOU
    ܕม਺ͷ਺͕͔֬ʹݮ͍ͬͯΔ
    
    ܕม਺͕۩ମԽ͞ΕΔͨͼʹɺ۩ମԽ͞Ε͍ͯͳ͍ܕม਺ͷ਺ ͕ͻͱͭݮΔ
21. ܕ
    U
    
    B
    ˠ
    B
    ˠ
    B
    
    B
    B
    B ܕ
    U
    
    B
    ˠ
    B
    ˠ
    JOU
    
    B
    JOU
    B VOJpDBUJPO B
    
    JOU B
    B
    B
    ˠ
    B
    ˠ
    JOU
    ܕม਺ͷ਺͕͔֬ʹݮ͍ͬͯΔ
    
    ܕม਺͕۩ମԽ͞ΕΔͨͼʹɺ۩ମԽ͞Ε͍ͯͳ͍ܕม਺ͷ਺ ͕ͻͱͭݮΔ ఀࢭੑ͕໌Β͔
22. unificationͷఆࣜԽ wઌ΄Ͳࣔͨ͠ख๏ͰɺVOJpDBUJPOΛ"HEB্Ͱ࣮૷Ͱ͖ͨ
    wܕਪ࿦ثΛఆࣜԽ͢Δʹ͸z͔֬ʹVOJGZ͞Ε͍ͯΔzূ໌ Λฦ͢͜ͱ΋ඞཁ
 ɹࣜʹ͢Δͱ
    TVCTU
    М
    U
    
    TVCTU
    М
    U
 ʮ୅ೖͨ͠ޙͷܕ͕͔֬ʹಉҰʯ
    wVOJpDBUJPO
    ͷؔ਺Λ֦ுͯ͠ɺ͜ͷূ໌Λฦ͢Α͏ʹ͠ ͨʢ11-ʣ

23. 3 ܕఆٛ •ܕɺܕ൑அɺ߲ •ຊݚڀʹ͓͚Δಛघͳදݱ

24. ܕఆٛ ߲
25. ܕఆٛ ߲
    N
    ൪໨ͷܕม਺ ؔ਺ܕ
26. ܕఆٛ ߲
    ม਺
    ϥϜμந৅
    ؔ਺ద༻
27. ܕ൑அ ਪ࿦نଇ
28. ܕ൑அ ਪ࿦نଇ ʮܕ؀ڥ
    ϵ
    ͷ΋ͱͰɺ߲
    .
    ͷܕ͸
    Н
    ʹͳΔʯ
29. ܕ൑அ ਪ࿦نଇ
30. ܕਪ࿦ͷಛघͳදݱ ܕΛܭࢉͯ͠ग़ྗ͢ΔͨΊͷنଇ
31. ܕਪ࿦ͷಛघͳදݱ ೖྗ ग़ྗ
32. ܕਪ࿦ͷಛघͳදݱ ᶃ
    ߲
    .
    ͱ
    ܕม਺Λ࠷େͰ
    N
    ݸ΋ͭܕ؀ڥ
    ϵ͕ೖྗ͞ΕΔ
33. ܕਪ࿦ͷಛघͳදݱ ᶄ৽͍͠ܕม਺͕ܕਪ࿦தʹ
    N``
    
    N
    ݸׂΓ౰ͯΒΕΔ
34. ܕਪ࿦ͷಛघͳදݱ ᶅܕม਺ͷ਺ΛN`ݸ·ͰݮΒ͢୅ೖ͕ಘΒΕΔ
35. ܕਪ࿦ͷಛघͳදݱ ᶇܕม਺ͷ਺͕
    N`
    ·Ͱݮͬͨܕ
    Н
    ͕ฦͬͯ͘Δ
36. unification ͷಛघͳදݱ ܕ
    Н
    ͱ
    Н
    Λಉ͡ʹ͢Δ VOJGZ͢Δ ୅ೖМΛಘΔ
37. unification ͷಛघͳදݱ ೖྗ ग़ྗ
38. unification ͷಛघͳදݱ ܕม਺Λ࠷େͰ
    N``
    ݸ࣋ͭܕ
    Н
    ͱ
    Н
    ͸
 ܕม਺Λ
    N``
    ݸ͔Β
    N`
    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
    wܕΛ
    HFOFSJD
    ʹੜ੒͢Δ͜ͱͰɺύλʔϯϚονͷ࠶࣮૷ Λආ͚ɺϢʔβ͕ࢦఆͨ͠೚ҙͷܕΛ࠶࣮૷ͳ͘ਪ࿦Ͱ͖ ΔΑ͏ʹͳΔ
    wຊݚڀͰ͸ɺ
    3FHVMBS</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͹
    -BN
    T
    Λܕਪ࿦͢Δͱ͖
    N``
    
    /BU
    N``
    
    DPVOU
    T
    
    
    ϵ`
    
    $YU
    N``
    
    ϵ
    
    ʜ
59. ෆ౳੍ࣜ໿ ͨͱ͑͹
    -BN
    T
    Λܕਪ࿦͢Δͱ͖
    N``
    
    JOU
    N``
    
    DPVOU
    T
    
    
    ϵ`
    
    $YU
    N``
    
    ϵ
    
    ʜ ܕͷ஋ϨϕϧͰܕΛ߹Θͤͯূ໌͠ͳ͍ͱ͍͚ͳ͘ͳΔ
60. ෆ౳੍ࣜ໿ N``
    ͸
    N
    ≤
    
    N`
    Λຬ͍ͨͯ͠Ε͹Α͘ɺ۩ମతͳ਺ ͸Ҏޙ࣋ͪา͘ඞཁ͕ͳ͍
    ୅ΘΓʹ
    N
    ≤
    
    N``ͱ͍͏ূ໌Λ࣋ͪา͘
61. 5 ܕਪ࿦ • ؔ਺ఆٛ • ܕਪ࿦ͷྫ - ؔ਺ద༻ -
62. ؔ਺ఆٛ ೖྗ
 T
    
    N
    ݸͷܕม਺Λ΋ͭ
    XFMMTDPQFE
    UFSN
    
 ϵ
    
    ܕ؀ڥ
    
    ग़ྗ
    N⒌⒌
    
    N⒌
    
    ࣗવ਺
    N≤N⒌⒌
    N≤N⒌⒌Ͱ͋Δ͜ͱͷূ໌
    
    М
    
    N⒌⒌
    ݸͷܕม਺Λ࣋ͭܕΛ
    N⒌
    ݸͷܕม਺Λ࣋ͭܕʹ͢Δ
    ୅ೖ
    
    Н
    
    ਪ࿦݁Ռͷܕ
    X
    
    ਪ࿦͞Εͨ
    XFMMUZQFE
    UFSNɻܕม਺ΛN⒌ݸ΋ͭ
    FSBTF
    X
    㲇
    T
    
    ฦ͞Εͨ
    XFMMUZQFE
    UFSN
    T
    ͔Βܕ৘ใΛ࡟আͨ͠΋ͷ͕ɺ
 ΋ͱͷ
    XFMMTDPQFE
    UFSN
    ͱ౳͍͠ͱ͍͏ূ໌
    
    JOGFS
63. ؔ਺ఆٛ ೖྗ
 T
    N
    ݸͷܕม਺Λ΋ͭ
    XFMMTDPQFE
    UFSN
    
 ϵ
    
    ܕ؀ڥ
    
    .BZCF ग़ྗ
    N⒌⒌
    
    N⒌
    
    ࣗવ਺
    N≤N⒌⒌
    N≤N⒌⒌Ͱ͋Δ͜ͱͷূ໌
    
    М
    
    N⒌⒌
    ݸͷܕม਺Λ࣋ͭܕΛ
    N⒌
    ݸͷܕม਺Λ࣋ͭܕʹ͢Δ
    ୅ೖ
    
    Н
    
    ਪ࿦݁Ռͷܕ
    X
    
    ਪ࿦͞Εͨ
    XFMMUZQFE
    UFSNɻܕม਺ΛN⒌ݸ΋ͭ
    FSBTF
    X
    㲇
    T
    
    ฦ͞Εͨ
    XFMMUZQFE
    UFSN
    T
    ͔Βܕ৘ใΛ࡟আͨ͠΋ͷ͕ɺ
 ΋ͱͷ
    XFMMTDPQFE
    UFSN
    ͱ౳͍͠ͱ͍͏ূ໌
    
    JOGFS
64. ؔ਺ఆٛ ೖྗ
 T
    N
    ݸͷܕม਺Λ΋ͭ
    XFMMTDPQFE
    UFSN
    
 ϵ
    
    ܕ؀ڥ
    
    .BZCF ग़ྗ
    N⒌⒌
    
    N⒌
    
    ࣗવ਺
    N≤N⒌⒌
    N≤N⒌⒌Ͱ͋Δ͜ͱͷূ໌
    
    М
    
    N⒌⒌
    ݸͷܕม਺Λ࣋ͭܕΛ
    N⒌
    ݸͷܕม਺Λ࣋ͭܕʹ͢Δ
    ୅ೖ
    
    Н
    
    ਪ࿦݁Ռͷܕ
    X
    
    ਪ࿦͞Εͨ
    XFMMUZQFE
    UFSNɻܕม਺ΛN⒌ݸ΋ͭ
    FSBTF
    X
    㲇
    T
    
    ฦ͞Εͨ
    XFMMUZQFE
    UFSN
    T
    ͔Βܕ৘ใΛ࡟আͨ͠΋ͷ͕ɺ
 ΋ͱͷ
    XFMMTDPQFE
    UFSN
    ͱ౳͍͠ͱ͍͏ূ໌
    
    JOGFS OPUIJOH͕ฦΔ
    ܕΤϥʔͰ͋Δ
65. ؔ਺ఆٛ ίʔυͰ͸͜ͷΑ͏ʹͳΔʢࢀߟʣ
    
66. ܕਪ࿦ͷྫ (App) "QQ
    T
    T
    ͷܕਪ࿦͸
    
    T
    Λܕਪ࿦͠ɺМ
    ͱ
    НΛಘΔ
    
    ϵʹМ
    Λࢪ্ͨ͠ͰTΛܕਪ࿦͠ɺМͱНΛಘΔ
    
    Н
    
    Н
    ˠ
    Ќ
    Ќ͸৽͍͠ܕม਺ ͱͳΔΑ͏
    VOJGZ͠
    ੒ޭͨ͠Β
    МΛಘΔ
    
    ฦ͞ΕΔܕ͸ЌͱͳΔ
67. ܕਪ࿦ͷྫ (App) "QQ
    T
    T
    ͷܕਪ࿦͸
    
    T
    Λܕਪ࿦͠ɺМ
    ͱ
    НΛಘΔ
    
    ϵʹМ
    Λࢪ্ͨ͠ͰTΛܕਪ࿦͠ɺМͱНΛಘΔ
    
    Н
    
    Н
    ˠ
    Ќ
    Ќ͸৽͍͠ܕม਺ ͱͳΔΑ͏
    VOJGZ͠
    ੒ޭͨ͠Β
    МΛಘΔ
    
    ฦ͞ΕΔܕ͸ЌͱͳΔ
68. ܕਪ࿦ͷྫ (App) "QQ
    T
    T
    ͷܕਪ࿦͸
    
    T
    Λܕਪ࿦͠ɺМ
    ͱ
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