==============================================================================
                          Υ쥸
  Copyright (C) 1996  Tokuyasu Kakuta, Makoto Haraguchi, Yoshiaki Okubo

            ֥˰¸ݲѤˡŪθ
                     ɽ :   ()
                            1996. 7.5
==============================================================================

 Ϥ
-----------

ˡŪˤơͿλ㤫ʷ뤿ˡ͡μȴ
ι⼡ʿԤäƤȹͤ롣Τ褦͡μΰ
˳ǰؤμͤ졢ºݤѤƤ롣[Oho],[Har]   γǰ
μϡϤȤꡢˤƤͭѤμǤ롣̤˳ǰ
ȤΤ¿ζ̤ǤȻפɸŪμǤ롣ʤ顢
ˡŪˤƤϡͿλŬɬɸŪǤʤɬפȤʤ롣
ǡ桹ˡŪκݤ˻Ѥ복ǰؤGDA(˰¸ݲ)
ˤäƵʸ̮˱ơΰ뤿ȤƤƥ
μ¸ԤäܸǤϤΥץ򼨤

ǰؤȤϳǰˡ˽äɽȸ뤳ȤǤ롣[Nebel]
ȤС֥ȥåפ礭ʲʪѺܤǽǤּ֡פȤȤ
ּ֡פ̤β̳ǰǤ֥ספȡ֥ȥåפˤϡּ֡פäƤ
°Ѿͭ뤿ᡢ¸ߤƤȹͤ롣Τ褦ˤƳ
ǰؤŪˤäͿȤߤʤȡŪ
Ƥ֤ȤǤ롣

ŪȤϡ嵭Τ褦˵ʸ̮˰ͤ餺Ū줿
Ǥ뤬ˡŪˤƤϡ¿ͤ갷ɬפ⤢롣षˡŪ
١[Ashley]θ餫ˤ줿褦ˡΤ褦¿ͤ
椫ʸ̮˱ƤŬڤФ뤳ȤפˤʤäƤ롣
äơ桹ŪȤΩˡʸ̮˱ưŪʳǰؤ
䤤뤿ܸΥץƤƤ롣ܥץ
ϡŪ˿ޣΤ褦ʼˡǤ롣


  [ޣ]

                      +----------+
                      | ΰ |    ʸ̮()
                      +----------+        |
                                        |
                        << GDA >> --------+ 
                           
       +--------+    +--------------+
       |ǰ|  |ʳǰ|
       +--------+    +--------------+

            |            |
            +------------+
                  |
              Ѥ


ǡΰʹߤǤϡ桹ΥץɬפˡŪˤʸ̮
ϤɤΤ褦ʤΤͤ˼ϰʹߤǤʸ̮˱ǰؤκˡβ
ɬפȤʤԤ


1.1 ˡŪˤ롼Ŭ

ˡŪȤơˡŪ롼ФŬѤԤˡŪȽǤԤ
롣̤ˡΧƤΥѥˡŪ롼ɽ뤳ȤԲǽ
롣ǡ롼ľܵҤƤʤäˤ͡ʻФˡΧ
ͽꤷƤŬѤԤˤϡˡŪŬѤѤ롣

ȤСּ֤ϤθäƤʤפȤˡŪ롼ˤĤƹͤ롣
ΤȤ֥ȥåפ֥ספϼ֤β̳ǰʤΤǡǰμľ
ѤǤ롣ּž֡פֿפŬѤϲǽǤ  ǰؤ˱
äơּ֡פּž֡פʪǤ롢Ȥ˾̳ǰ򸫤Ȥ
äơּž֡פؤŬѤβǽϹͤ롣ֿפϡưʪפʬ
ƤΤŪǤȡñ˳ؤ˱äԤˡǤϥ롼
ŬѤϺǤ롣ˡΧȤοǤŬѤǽ⤤Ȥ
СˡΥ롼뤬줿аޤȤơֿͤˤ֤ĤǽΤ
ʤΤफӽפȤäȤС¿ˡΧȤϡֿ
ˤ⤳Υ롼ŬѤ뤳ȤȹͤǤʤ롼Ωˡ
ݤˡ(ˡΧҲˤ⤿餹)Ȥä롼ŬѺˡŪ
˽äơּ֡פȡֿפ򸫽ФƤ뤳Ȥˤʤ롣ϡʸ̮˱
Ūʳؤȹͤ뤳ȤǤ롣

κݡˡŪοΤˤϡˡפΩˡݤ˴ؤˡŪΰ
ԲķǤ롣ǡ˼褦ˡ桹ΥץǤΰμѤ
뤳Ȥˤ


1.2 ˤθ

ϰ̤ˤϡĤγǰζ̣ͭƤ롣äơΰ
鶦ͭˤäƸФ뤳ȼΤϸŪ˲ǽǤ롣
ΰ褫ǤϡϤ¿Τͭפ
ӽФˡˤĤƤϲͿƤʤ桹Υץΰ
ѤƤ뤬ˡͭפ֤˵ʸ̮˱Ŭڤ
򸡽ФǤȤȤʤäƤ롣ޤǡؽñĴʬǤ
(Integrity Constraint ICά)ФлѤƤIC
ԴꡢIC򤢤餫ˡΧȤΰμ¾ˤ뤳Ȥ
Ǥ롢Ȥ⤢롣桹ƤˡϤФͿƤ
ΰ顢СưŪŬڤڤʬƤԤäƤ롣


1.3 ʸ̮˴Ϣݲȳǰ

˽Ҥ٤褦ˡ롼ŬѤˡŪ(ˡ)˴Ϣ뤳Ȥˤäơ
桹ΰŬڤʤФʤʤ桹ƤȤ
ϡݡGDA[Okubo]Ѥ뤳Ȥˤäơפʵʸ̮ȤʤŬ
Ѥκ˴Ϣǰؤ롣

ǡݲ(ǰ)δطˤĤƽҤ٤롣̤ˡݲȤϽͿ
γǰΤ¦̤˾򤢤ơŪʵҤ뤿Ǥ
ͤ롣κݡ̤¦̤ϼξݤ롣


    [ޣ]
                   ----------------------------------
                   |     ȳ <- Ȥ     |
                   ----------------------------------
                                                
                              ݲ            
                                                
      ----------------------------    ------------------------------
      |̳ <- 򤹤|    |ȳʧ  <-  ®ٰȿ|
      ----------------------------    ------------------------------


ޣΤ褦ݲԤ줿ȤΥ٥Ǥб복ǰƱΤϸߤ
ƤΤȤߤʤ롣ʤ򤹤פȡ®ٰȿסַ̳
פȡȳʧפϤ줾Ƥ롣ޤξ硢ȤС
Ȥפβ̳ǰȤơ򤹤פȡ®ٰȿפƤ褦
ʳǰؤͤ뤳ȤǽǤ롣Τ褦ˤơݲ뤤
ǰؤɽˤʤäƤ롣ǡʸ塢ݲǰؤƱ
ǻѤ롣

ʸǹͤݲJ.D.Tenenberg[Ten]ˤ󾧤줿ݲ˴Ť
롣TenenbergˤݲȤϡηTŪʷT'
ѴǤ롣٤ˤĤƤϡTT'򵭽ҤLL'ȤL
L'ؤͦդ¸ߤȤL'LŪǤȤŪ롣
ǡդݲȸƤ֡ʤTenenbergݲˤƤϡޣ˼
褦TΰΤߤݲ졢T'ȤϿ롣ʤ
դˤäơTݲȤʤʬ뤳ȤǤ롣


    [ޣ]

                              ݲ٥
                             --- T' ----
                             |   a<-b  |
                             -----------
                                
                                
                                
        ------- T --------------------      ---  ---
        |             ---------------  |      |   b    |
        |             | a<-b1       |  |      |      |
        |  c<-b1      |    a<-b2    |  |      | -----  |
        |             ---------------  |      | |   |  |
        --------------------------------      | b1  b2 |
                                Υ٥    ---------- 


դݲT'ϿϡľŪ˸Сݥ٥Ƕβ˰¸
ΩפǤꡢդ˸СֶΥ٥ǤбβƤΥ
᤬TƳΡפǤ롣

ܸǴäȤƤGDAǤϤT'ؤݲˤڤʬѤơ
ŬڤƳƤ롣ݡGDAǤϡξϢΤ᤬դ
ݲƤˤΦդŬڤȤƤ롣椨˰¸Ŭڤ
ݲ(ǰ)򤵤뤿ᡢ֥˰¸ݲפȸƤФ롣

ܸǤϡ¤˴ϢŬڤǤդ뤿ˡڤ˸
Ƥζβ᤬T'˼̤褦ʦդŬڤȤƳ롣ʤ
Ŭڤδ
  ֥ξ¤ݲˤä¸褦ʳؤΤߤŬڤʳ
    Ȥߤʤ
ȤˤäݲդŬڡŬڤȽǤԤͻҤޣ˼


    [ޣ]
                              ݲ٥
                             =====T'=====
                            ||          ||
                             ============
                                
                                
                                
        ------- T --------------------
        |             ---------------  |
        |          ------           |  |
        |   Pr2 |   |  |          |  |
        |   Pr3    ------   -----   |  |
        |           |    |     |  |  |
        |   ------    |    |     |  |  |
        |   |    |    |    |     |  |  |
        |   |    |    |     -----   |  |
        |   ------    -------------  |
        |                    Pr1       |
        --------------------------------
                                Υ٥


˻Ȥ줿ὸ礬Pr1ξϦդŬڤȤʤꡢPr2Pr3ǤϦդŬڤȤ
롣

Τ褦ˤơ桹ϥξѤ줿(¤)˴ϢΤ
Ŭڤݲ()Ȥ뤳ȤǤ롣

ϵʸ̮˱ǰؤĤμˡˤʤäƤ롣
ơ롼ŬѤεʸ̮ȤʤפʴϡŬѤˡŪʺǤ
뤫顢κ򥴡Ȥ¤Ф嵭ǼݲȤŬڤ
ŬѤС桹˾Ǥ褦Ŭڤʳǰؤ롣


1.4 ŬѺȤƤΥ

롼ŬѤκȤƹͤΤϡ˽Ҥ٤褦ˡΥ롼
Ωˡݤˡ˾Ȥ餷ƹͤ롼εǽŪŪǤ롣äơ
롼ʤŬѤΤȤ르򸫤ĤʤФʤʤ
ľŪ˥롼ŬͳνҸ줬ΰ˴ޤޤƤȤϤۤȤɴ
ԤǤʤΤǡŪŬͳȹͤΤϡΥ롼¸ߤˤä
ˡפǤ롣ǡ桹ʸꤹ롼ϡ׷򽼤Τ
Ф餫ζػߤɽΤȤ롣ȡ롼Τ⤿餹ˡפȤϡ
Ǥζػߤ򤤤硢ɤΤ褦ˡ׿뤫뤳
ǡˡ׿򤽤Υ롼뤬ɤȤǽŪŪФ줳롼
ŬѤˤˡפȤʤ롣ʤˡ׿뤿ˤϡɽ
Ҹ췲ɬפȤʤ롣㤨СǤ롢ǡĶץ饤Х
͡ˡ׿νҸ줬ͤ롣ΥץǤϡΤ褦ʽҸν
餫桼뤤ˡΧȤͿΤȲꤹ롣


---------------
 γ
---------------

ϤǼ褦ˡʸ̮˱򸫤Ĥ뤿ˡˡŪ롼Ŭ
Ѻ¤¸򸡽Ф롣ΤᡢޤŬѥ롼
롣ˡŬѺȤʤˡ׿¤ФǸˡGDAˤ
ꤽ¤¸褦ʳ()򸡽Ф롣

ͻҤޣ˼


  [ޣ]

      ΰT0          ǰئ          ꥸʥΥG0
                                                  
      +---------------------------------------------------------------+
      |  Υȥꥬ  :  Ŭѥ롼оݥȷ  |
      +---------------------------------------------------------------+
                                                  
                 ԤȷSG    Ŭо(Ԥ)롼R
                                                  
      +---------------------------------------------------------------+
      |        :      롼ŬѤκ¤      |
      +---------------------------------------------------------------+
                                              
          롼Ѵ줿T   ΥG  (GCF)
                                              
      +===============================================================+
      ||   G D A         :            Ŭڤη׻             ||
      +===============================================================+
                                   
                                Ŭڤʦ


IFSΥ꡼󶡤ʬϿΣǰϤGDAʬǤ롣
                     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
äơǤϤʬؤϤȤʤǡˡ⤹롣

ܼˡϤȤʤΤϿޤΤ褦˰ʲ̤Ǥ롣

  ΰ: T0  (ؿͳʥȳὸǤꡢоݤΥ롼R
               ޤ)
  ȳ: = <S,>  (SǤ򥽡ȤȸƤ)
  η: CS  (ɤΥȤ°뤫Ҥ)
  ˡ׿νҸ콸: CF
  ȿФνҸФν: NP
  ڹۤΥ: G0

ޤܼˡǵΤ Ŭڤʦ Ǥ롣

ޣǡSUG0ڤμԤθȤʤäȽ²TΤ
T0ѴΤǤꡢGϺΥǤ롣T0ѴˤNPѤơR
ꤷT0RƤդäΤTǤ롣

ǡˤĤƴñ롣
ϴؿͳ¤ƤΤǡΥ󥿥åϡƱ
ѿ˰٤ȤդȤǤ¾ϰ쳬ҸƱͤǤ롣
ѿˤ

    x:s

ηǥȤź뤳ȤǤ롣̣ϡxˤϥṣΥ
ǤʤФʤʤȤ(׻)ˤʤäƤ롣ʤ
ȤդƤʤѿϥȥåץȤդƤȤߤʤ
ȤС

   groundmother(X:human,Y:famale) :- parent(X,Z:famale), parent(Z,Y).

ȤhumanǤXfamaleǤYǤꡢXοƤfamaleǤZ
Y(YfamaleȻꤵƤ뤳Ȥ)Ǥ롣դȹͤ뤳ȤǤ
롣ܤ[Oho],[Har],[Ait],[Wal]򻲾ȤΤȡ

ʤIFS桼Helic-IIȤ䤹Ȼפ롣


------------
 GDA
------------

󶡤ƤGDAʬѤ뤿ɬפϤˤĤƵҤ롣
ɬפϤϡ

    ѷ줿ΰ: T
    Υ: G
    Gξ(ὸ): Proof
    ԸΥȷ: SG
    ǰ: 
    ݲ˴ؤõ: TH

Ǥ롣¸ߤ릲ʲϥ桼ѤƤ륪Ȥο󥸥
ƼƤɬפ롣ˡʲ˼

  ѷ줿ΰTȼԸΥȷSG

    ΥGT0ѤƿȤ롼RǼԤƤȤ롣
    ΤȤʲΤ褦ʼTSG롣

    (1) TμԸRꤹ롣
    (2) RǼԸȤʤäƤѿˤĤơΥȤȥåפȤߤʤ
        RʳѿˤɤΤ褦ʥȤղä뤫Ĵ٤롣
    (3) ѿȤˡ
          ([롼RǤΥ],[RʳǤΥ,.....])
        ȤƤޤȤᡢƤѿˤĤƺ顢ꥹȲƤ
        줬SGǤ롣
    (4) R롣ʤΤǥإåϤ餫ᡢNPȤͿ
        ƤGDAѤ桼ϡȤʤҸ֤Ƥ
        㤨С allow<-->prohibit  in<-->not_in ʤɤꤷƤ硢
        ~prohibit (prohibit)allowǤꡢ~innot_inǤ롣
        ѿϥಽɬפ롣äơץ˸ʤ
        ƳդƤϡˤinstance-ofط
        ä롣㤨С
               not_in(X:vehicle):-exist(X,Y:park).
        ʤС
               in(c1).
               exist(c1,c2).
        Ȥʤ롣ΤȤ
               c1  vehicle
               c2  park
        򦲤˲ä롣
    (5) T0ФR嵭Τ褦ˤƺä롣
        줬TǤ롣

            [꼰] T=T0/{R}{~R}

  ΥGȾProof

    T桼ο󥸥Gξߤ롣consequence-finding 
    ˾ޤT0ο˻Ѥ󥸥Ѥ뤳Ȥˤʤ
    Ǥ顢CFҸȤƤ餫ϿƤƽҸˤĤƽT
    ƳФΤߤ롣ʤT0ƳФΤϽ
    ƳФ줿⥴ϢȤäGȤ롣
    κݤξڤProofǤ롣

        [꼰]  T |- Gi  (GiCF)
                  G= i Gi


  ݲ˴ؤõTH

    ꡼ΤGDAʬΤߤǤΤǡGDAȤο󥸥
    Ǥο̤򻲾ȤФơ餫ᡢƤʤФ
    ʤɬפTHǤ롣

    Ƥ٤ϡProof˸줿ΥȤƥȥåץȤѴ
    줬TƳвǽǤ롣ΥȤ˴ؤݴط
    ΤΤΤߤǤ褤


------------------
 GDAεǽ(What)
------------------

ʳ줿RκΥG¸ݲŬڤݲȤƵ뤳
ȤGDAŪǤ롣ݲˤäŬڤʳǰ()ľ
롣

ʸǤݲϥȤ˸¤äƤꡢGDAǵ٤ȼդϰʲͿ
롣

[](SortAbs)
  :L | L' 򥽡ȼT  L Ȥ롣
  ΤȤT   ΤȤǤὸ SortAbs_(T) ʲ롣

        SortAbs(T)= {C'| F^{-1}(C') T|- F }     

ǡCT(C) = C'Ǥꡢ^{-1}(C')={Ci|(Ci)=C'}Ǥ롣ݥ٥
CϡǤդζβ T ǽǤȤSortAbs_(T) ˴ޤޤ
뤳Ȥˤʤ롣ΤȤ(C) = C' ʤ C C'ζβȸƤ֡SortAbsľ
Ū˼ȿޣ==ǰϤʬˤ롣

SortAbsȺΥGŬڤ뤿ˡʲˤäơŬڤ
ݲդ̤Ѥ롣

[](˴ؤŬڤ)
   ȼ  ϲξ򤹤٤Ȥ
   롼RκG˴ؤŬڤǤȸƤ֡
    դϥ롼κڤ¸:
        (Proof(G))  SortAbs_(T)
         Proof(G) GƳФ뤿˻Ȥ줿ὸ򤵤
    Ѿ:
        s1  s2  (s2)=(s)  (s1)=(s) 

ʤºݤˤϦդϥ롼Rκ˴ؤŬڤǤ뤳Ȥ˲äơΥ
ꥬǵ᤿SUƱФߺѤˤʤäƤʤƤϤʤʤäơ嵭
᤿ŬڤѤơ
    Rκ˴ؤŬڤʦդǤꡢ
      G0ڤμԵߺѤȤʤդܥƥŬڤݲǤ
Ȥ롣

ηѾȤϡʬǰϾ̳ǰ¾γǰ˷Ѿ뤳Ȥ
̣ƤꡢܸǤϡǣģ르ꥺबФ٤Ф
Ȥꤹ롣ޣΤ褦ˡľŪˤ vehicle  horse ƤȽ
ʤСvehicle β̳ǰǤ car  horse ƤϤǤ롢
ȤȤ̣롣


  [ޣ]

        vehicle    horse              vehicle    horse
           |                               |        /
           |                   ===>        |    
           |                               |  / 
          car                             car    


-------------------
 GDA(How)
-------------------

ܥ르ꥺǤνоݤΥǡϥȽʬǤ롣ϡդб
ƷޤƤƱ򽸤᤿ΤǤ롣ʬϦդդ˷ޤ뤿ᡢ
Ϳ줿Ƚ S ʬª뤳ȤǤ롣ΦդбʬФ
ŬڤȽԤŬڤȽǤ줿ʬĤȤܽŪǤ롣

Ǥ, ʬ򥽡Ƚ²ɽ, γ(Ƚ)  ޤ 
Ʊ ȸƤ֡

ޡȽ {a,b,c,d} ʬ {{a,b},{c},{d}} ͤƤߤ褦ʬϡ

   (a)=(b)=abs1(c)=abs2(d)=abs3 

ʤ륽ȼդ롣ä, ʣȤʤ륻(Ǥ{a,b})
ȷɽƤ롣

ǡGDAΨ褯¸뤿ɬפʬνطƳ롣

[](ʬνط)
  ǤʤʬPP'ͤ롣
  PǤդΥ c Фc  c' ʤP'Υc'¸ߤ
  P  P'Ȥ롣 

ΤȤPP' ե ȸƤ. 

嵭{a,b,c,d}ˤСäȤ⾮ʬϡ{{a},{b},{c},{d}}Ǥꡢ
Ǥ礭ʬ{{a,b,c,d}}Ǥ롣ȤСνطǤ
   {{a,b},{c},{d}}  {{a,b,c},{d}}
   {{a,b},{c},{d}}  {{a,b,d},{c}}
   {{a,b},{c},{d}}  {{a,b},{c,d}}
  
   {{a,b},{c},{d}}  {{a},{b,c,d}}
  Ǥ롣

ΡƳˤꡢʲΩġͭʻ޴구§ȵ§Ȥʤ롣

[](ˤ޴구§)
  ȼդCProof(G)ˤĤơ(C)SortAbs_(T)˴ޤޤʤ
  ա'ʤǤդΥȼ'ˤĤơ'(C)SortAbs_'(T)˴ޤޤʤ
    //// [Okubo]Ʊ͡

ŬڤǼƾθϰʲΤ褦˼¸ΨפäƤ롣

¤¸ˤĤƤϡ嵭ꡢ޴§ѤǤ뤿ᡢ˼
  եȤΥˤäơΤñ-ˡ
  ˤäŬڤʤΤ򸫤Ĥ롣

ѾˤĤƤϡ餫꡼եȤΤߤǾ嵭ԤŬڤȽ
  Ǥ줿ʬФƥˤơѾξ̥
  Ȥβ̥Ȥ򤢤餫ʬΥƤ꡼եȷФ
  ղä롣(ʬȤʤ褦˥Хꥨθ롣)

ñ첽ԤεߺѾϡ˥եƤƤ륻뤷Ĺʤ
  褦ˤ뤳ȤǼ¸õ֤ʤǤ롣


 ʸ
-----------

[Ait]
H.Ait-Kaci and A.Podelski
"Towards meaning of LIFE",
Digital Equipment Corporation, Paris Research Laboratory,
PRL Presearch Report 11, 1991.

[Ashley]
K.D.Ashley (1990)
"Modelling Legal Argument: Reasoning with Cases and Hypotheticals",
MIT Press, 329 pages.

[Har]
Haraguchi, M. (1996) 
"A Reasoning System for Legal Analogy",
Machine Intelligence, Vol.14, Oxford University Press.

[Nebel]
B.Nebel
"Reasoning and Revision in Hybrid Representation Systems",
Springer LNAI Vol. 422, 270 pages , 1990.

[Oho]
Ohtake, Y., Nitta, K., Maeda, S., Ono, M., Osaki, H., Sakane, K. (1994)
"Legal reasoning system HELIC-II",
Trans. Inf. Process. Soc. Jpn. (Japan), vol.35, no.6, 986--96.

[Okubo]
Y.Okubo, M.Haraguchi (1994)
"Construction of Predicate Mappings for Goal-Dependent Abstraction",
Algorithmic Learning Theory '94, Springer LNAI 872, 516--531.

[Ten]
J.D.Tenenberg (1989)
"Abstracting First-Order Theories",
Change of Representation and Inductive Bias, 
67--79, Kluwer Academic Publishers

[Wal]
C.Walter (1988)
"Many-Sorted Unification",
JACM 35, 1, 1--17.


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                                                      kaku@int.titech.ac.jp
