
                  An Analysis of Alpha-Beta Pruning
                    ( - 祭)
                  Donald E. Knuth and Ronald W. Moor
              Artificial Intelligence, 6 (1975), 293-326

      0. 

       ᪥ 室  ⠪ ,  嬠, ணࠬ    
室 ந ᬮ 讣 ॢ  த.
⮡ ᪮          ଠ,  筮  -
 ⮤, 뢠 - 祭ﬨ ( ᮢ᪮ -
  뢠 ⮤ ࠭  業  --  ਬ.  ॢ稪).  
  뢠 १  ⮩ 楤, ஢-
  ⮣, ⮡   - ᫥ 業  ⢠
 ࠡ.

       ࠧ.  1 । ᭮ , 易  ॢ  -
.  ࠧ. 2 뢠 楤, 뢠 -  祭-
 (alpha-beta pruning),  ᭮ 易   ⮤, , -
,    ⮫  䥪⨢,  ᪮       ந    "
祭";    ४⭮  ⮤. ਬ  -
 ⬮   ஡  ।⠢ ᮤঠ   ࠧ.
3.  ࠧ. 4   ᯮᮡ ࠪ᪮ ਬ -
⮤,  ࠧ. 5 饭 ਨ - 祭.

       ࠧ. 6 稭  ࠡ  ⮤  --  祭  
業 㤮 ᪠, 室 楤    㣮
,  蠥   .    業  㤮
祭  ࠧ. 7  ᭮  ࠡ    砩묨  ॢﬨ,
  ந  祭. ,      -
   祭 ࠧ㬭 १.   ࠧ.  8    
祭  祭,   ࠧ.  9  ,    䥪⨢
⠥, ᫨  ᫥⥫묨 室  .  -
  ⥫쭮 ⥯ ,   ᫥ ࠧ  -
  ⥬⨪.

      1.   業 権

        ,  ⮫쪮          ,  筮
뢠 ⢮ "権"  ᮢ㯭 ࠢ 室   -
 樨  , 祬 ।,  ப 室   -
. 㤥 ,  ࠢ ࠧ襭      ᫥-
⥫쭮 権     樨    筮  ᫮
ࠧ襭 室.    樨 p  ᫮ N(p)  -
,   , 砢  p,      த  
N(p) 室 --  ᫥  "    ᪮筮"  (.  [11,  .
2.3.4.3]).

      ନ묨 뢠 樨,    ࠧ襭 -
.     । 楫᫥ 㭪 f(p),  
먣 ⮣  ப, ஬  ਭ  室    ⮩  樨;
먣 ண ப ⠥ ࠢ -f(p).

      ᫨  樨 p   d  ࠧ襭  室  p[1],...,p[d],
 ஡ 롮 襣  . 㤥 뢠  室  -
訬, ᫨  砭   ਭ 訩    먣
 ᫮,  ⨢ 롨ࠥ 室, 訥   ( ⮬ 
᫥).  F(p)  訩 먣, ⨦    樨  p
ப, ஬ ਭ । 室, ⨢ ⨬쭮 .
.. ᫥ 室   p[i] 먣 ⮣  ப  ࠢ  -F(p[i]),


                  /  f(p)                             d = 0,
   (1)  F(p) =   |
                  \  max {-F(p[1]),...,-F(p[d]), ᫨ d > 0.

  樨 p  㫠  ⨢ । F(p).

       設⢥ ࠡ  ⥮ਨ  ᯮ   -
஢;   ப 뢠 Max  Min     "-
 ७" ப Max.  ࠧ, ᫨ p   ନ쭠  -
  室 Max,  먣 ࠢ,   ࠭ f(p), ᫨    -
쭮 樨 p 室  Min,   먣 ࠢ

   (2)   g(p) = -f(p).

Max ⠥ ᨬ஢ ᢮  먣,    Min  ࠥ
஢ . ⭮襭 (1)  ⮬  ᮮ⢥    -
樨, ,

                   /  f(p)                           d = 0,
   (1)  F(p)V =   |
                   \  max {G(p[1]),...,G(p[d]), ᫨ d > 0,

  ᨬ ࠭஢ 먣 ப Max  -
樨 p, 

                  /  g(p)                           d = 0,
   (1)  G(p) =   |
                  \  min {F(p[1]),...,F(p[d]), ᫨ d > 0,

 ࠢ ⨬, ⨦   ப  Min.      ࠭,
 ।,  p[1],...,p[d]  ࠧ襭   樨  p
室. 樥   室  ,  㭪樨, ।-
 ᮮ⭮襭ﬨ (1)  (3), ᮢ     p

   (5)  G(p) = -F(p).

 ࠧ,  室 .

      ..  筮  業 樨   窨  ७  
- ப,  㤮 ᯮ짮 "" 室 (3)
 (4),   㦤  "--ᨬ" ନ  ᮮ⭮-
 (1).  㣮 ஭, ᮮ⭮襭 (1) ।⥫쭥,   
⨬  -, --  ⮬    室  ᫥
 (  --   ப) ࠧ .

      㭪 F(p) ࠢ ⮬ ᨬ,    ࠭஢,  ᫨
 ப  ⨬쭮. , ,  ,    
ࠦ १ 쬠 ஦  ⥣,      易-
⥫쭮  ⨢  ப  ப,   ᮣ᭮
㣮 ਭ樯 ⨬쭮. , ਬ,     室  
樨 p1  p2, 祬 p1 ࠭  (먣  0)      
 먣,   ६,  p2      먣,
᫨ ⨢ ᬮ 祭 ⮭ 먣뢠騩 室.   ⠪  -
樨   । ᪮ 室  p2, ᫨  ⮫쪮    
㢥७  ⮬,   ⨢ ᥬ  ᥧ. 祭  ,
  먣뢠  嬠 ணࠬ ⠪  ࠧ.

      2. ࠡ⪠ ⬠

      ᫥騩  ( ᯥ樠쭮  ਤ㬠  -
 몥), 祢,  F(p)  ᮮ⢥⢨  । (1):

      integer procedure F(position p):
         begin integer m,i,t,d;
         । 樨 p[1],...,p[d], 稭 p;
         if d = 0 then F := f(p) else
         begin  m := -INFINITY;
            for i:= 1 step 1 until d do
               begin t := -F(p[i]);
                  if( t > m then m:= t;
               end;
            F := m;
         end;
      end.

 +INFINITY 砥 ᫮, ஥   abs(f(p))   -
 ନ쭮 樨 p; ⮬ -INFINITY   F(p)    -F(p)
  p.   ॠ ᫥ F(p)  ᭮  "㡮
ᨫ" --   樨  業      த;
"  ᪮筮" ࠭ 砭 ᫥    筮
᫮ 室.

      ॡ,  ⢫   ,    㬥,
ᯮ ⮤ "⢥  ࠭" [14].  ⮨  ⮬,   
 ᪠  業 室,   ⠫ ⭮,      
  , 祬   室, ᬮ७ ࠭. ᫨,  -
ਬ, F(p1) = -10,  F(p) >= 10,    㦭  筮  -
 F(p2), ᫨ - ࠧ  ⠫ ⭮,  F(p2) >= -1
( ⠪ ࠧ,  -F(p2) <= 10). ⠪, ᫨ p21 -- ⨬  室
 p2  F(p21) <= 10,    ᫥ 㣨  室    p2.  
ନ ⥮ਨ  室   p2 "஢࣠" (⭮⥫쭮 -
  p1), ᫨  ⨢  樨 p2  ⢥,   ࠩ  
⮫  訩,   訩 ⢥  樨 p1. ᭮,   -
 室  ஢࣭,    ᪠ 襥 ஢থ.

       㦤 ਢ  , ࠧ   ,
祬  F. । F1  楤     ࠬࠬ  p  
bound;  楫 -- 㤮⢮ ᫥騬 ᫮:

          F1(p, bound) = F(p),    ᫨ F(p) < bound,
   (6)
          F1(p, bound) >= bound,  ᫨ F(p) >= bound.

 ᫮  । F1 ,  ,    ᫨
F(p)   砫쭮 樨 p, ᪮   ᫥, 

   (7)    F1(p,+INFINITY) = F(p).

 ⮤ ⢥  ࠭ ॠ ᫥騩 :

      integer procedure F1(position p, integer bound):
         begin integer m,i,t,d;
         । 樨 p[1],...,p[d], 稭 p;
         if d = 0 then F1 := f(p) else
         begin  m := -INFINITY;
            for i:= 1 step 1 until d do
               begin t := -F1(p[i], -m);
                  if( t > m then m := t;
                  if m >= bound then goto done;
               end;
      done: F1 := m;
         end;
      end.

ࠢ쭮 ࠡ ⮩ 楤  ,   १  㤮⢮
ᮮ⭮襭 (6),   ᫥騬  㦤ﬨ.    砫
i- ⪠ for- 믮 "ਠ⭮" ᫮

   (8)    m = max { -F(p[1]), ..., -F(p[i-1])}.

(।,  ᨬ  ⮬      -INFINITY.)
⮬, ᫨ -F(p[i]) > m, , ᯮ ᫮ (6)      
 , 稭饩  p, 砥 F1(p[i],-m) = F(p[i]). -
⥫쭮,  砫 ᫥饣 ⪠ (8) ⠪ 㤥 믮.  ᫨  
max {-F(p[1],...,-F(p[i])} >= bound    i,    F(p)  >=  bound.
 ᫥,  (6) 믮   p.

       楤   , ᫨   ⮫쪮 ,
   ࠭.   --   뢠  ᭮  -
楤ன   - 祭ﬨ --   ⥫
த  ࠢ  ஭ ⮤ ⢥  ࠭.  (
ᮦ,  ਬ    ,    ᯮ  ⮤
⢥  ࠭;  ࠡ⠥  ⮫쪮    ᫥  ஢  -
ॢ쥢.) । 楤 F2  ६ ࠬࠬ p,  alpha    beta
(祬 ᥣ 㤥 믮 alpha <  beta),    㤮⢮
᫥騬 ᫮,  (6):

          F2(p,alpha,beta) <= alpha,    ᫨ F(p) < alpha,
   (9)    F2(p,alpha,beta) = F(p),      ᫨ alpha < F(p) < beta,
          F2(p,alpha,beta) >= beta,     ᫨ F(p) >= beta.

 ᭮  ᫮  । F2 , ,      ᫥-
, 

   (10)   F2(p,-INFINITY,+INFINITY) = F(p).

뢠, ।⠢ ⮣ 襭 ⬠  몥  ண-
ࠬ஢  - ⫨砥  ⬮ F  F1:

      integer procedure F2(position p, integer alpha, integer beta):
         begin integer m,i,t,d;
         । 樨 p[1],...,p[d], 稭 p;
         if d = 0 then F2 := f(p) else
         begin  m := alpha;
            for i:= 1 step 1 until d do
               begin t := -F2(p[i], -beta, -m);
                  if( t > m then m := t;
                  if m >= beta then goto done;
               end;
      done: F2 := m;
         end;
      end.

      3. ਬ  筥

        楤 ந㥬  ਬ ॢ (. 1),
 ୥  樨  ६ 稭묨 ﬨ,      -
 ⠪   稭 樨,  ..   ,   
 稬 3**4 = 81 権, ⨦  4 室. ⨬  ନ
 ਯᠭ "砩" 業,  ࠢ  81  ࠬ
筮 ࠧ ᫠ pi.  ࠧ, ७ ॢ  (ᯮ-
 )  業 2, ࠢ 먣 ࢮ ப,  ᫨
 ப  ⨬ 室.

                                        2
             ͻ
                                                                 
                                                                 
                                                                 
                                                                 
            -2                         -2                         -1
    ͻ        ͻ        ͻ
                                                                    
                                                                    
                                                                    
                                                                    
    2        7        4        2        3        3        4        1        8
 ͻ  ͻ  ͻ  ͻ  ͻ  ͻ  ͻ  ͻ  ͻ
                                                     
                                                     
                                                     
-1 -1 -2 -3 -7 -2 -4 -2 -3 -2 -0 -2 -1 -1 -3 -3 -0 -2 -0 -4 -4 -0 -1 -0 -2 -0 -8



314159265358979323846264338327950288419716939937510582097494459230781640628620899

   .1.  業 ॢ .


       .  2 ।⠢ , ᮮ⢥ ॡ,  -
 ந 楤 F1. ⨬,   81  ନ  -
権  "頥" ⮫쪮 36. ஬ ⮣,   設  2-  ஢
 "ਡ" 業 3  ⨭ 祭 7;  ਡ-
⥫쭮, 筮,     業 .

                                        2
             ͻ
                                                                 
                                                                 
                                                                 
                                                                 
            -2                         -2                         -1
    ͻ        Ŀ        Ŀ
                                                                    
                                                                    
                                                                    
                                                                    
    2        3        4        2                        4        1        
 ͻ  Ŀ  Ŀ  ͻ  Ŀ  Ŀ  ͻ  ͻ  Ŀ
                                                     
                                                     
                                                     
-1 -1 -2 -3     -4     -2 -0 -2             -0 -4 -4 -0 -1 -0      
ſſſſſſſſſſſſſ


3141  265358      846      3279502                    0974944  230781640

   .2. ॢ , ।⠢  . 1, ᫥ 業
          楤ன F1 (⮤ ⢥  ࠭).


       . 3 ।⠢   , 蠥 -  -
ன. F2(p,-INFINITY,+INFINITY) ᥣ ᫥    㧫,    
F1(p,+INFINITY),   ⨣ 設 ⢥⮣ ஢;   
᫥⢨ ࠧ  ⥮ਨ.  ஢ 4, 5, ..., ,  -
楤 F2 ᯮᮡ ந " 祭",     ᯮ-
ᮡ F1. ࠢ . 3  . 2 뢠,      ⮬  ਬ
ᤥ 5  祭.

                                        2
             ͻ
                                                                 
                                                                 
                                                                 
                                                                 
            -2                         -2                         -1
    ͻ        Ŀ        Ŀ
                                                                    
                                                                    
                                                                    
                                                                    
    2        3        4        2                        4        1        
 ͻ  Ŀ  Ŀ  ͻ  Ŀ  Ŀ  ͻ  ͻ  Ŀ
                                                     
                                                     
                                                     
-1 -1 -2 -3     -4     -2 -0 -2             -0 -4 -4 -0 -1 -0      
ſſſſſſſſſſſſſ


3141  265358      846      32 9502                    0  4944  2  781640

   3. ॢ , ।⠢  . 1, ᫥ 業
         楤ன F2 (- ⥣).

       樨  ।⠢    ନ  "⥫쭮--
쭮" , ᬮ७  ࠧ. 1; ⥬,  ।⠥ "-
" ନ, 筮 ஢    .  1-3  .
楤 ࠧ. 2  ।⠢  ᭮ , ,  -
ਬ, F2 ᫥騬  楤ࠬ:

      integer procedure F2(position p, integer alpha, integer beta):
         begin integer m,i,t,d;
         । 樨 p[1],...,p[d], 稭 p;
         if d = 0 then F2 := f(p) else
         begin  m := alpha;
            for i:= 1 step 1 until d do
               begin t := G2(p[i], m, beta);
                  if( t > m then m := t;
                  if m >= beta then goto done;
               end;
      done: F2 := m;
         end;
      end;

      integer procedure G2(position p, integer alpha, integer beta):
         begin integer m,i,t,d;
         । 樨 p[1],...,p[d], 稭 p;
         if d = 0 then G2 := g(p) else
         begin  m := alpha;
            for i:= 1 step 1 until d do
               begin t := F2(p[i], alpha, m);
                  if( t > m then m := t;
                  if m >= beta then goto done;
               end;
      done: F2 := m;
         end;
      end.

 ⢥ ,  ⥫쭮 ࠦ । ,
 G2(p,alpha,beta) ᥣ ࠢ -F2(p,-beta,-alhpa).

       ਢ 楤  ᯮ  襡  ணࠬ,
 । 樨 p[1],...,p[d], 稭 樨 p.  ᫨
 ⨬  筮  ᯮᮡ ।⠢ 権,    -
⢥ ᯮ짮 ᯨ᪮ .  p  뫪  -
, ᮮ⢥ 室 樨, ⮣ first(p)  뫪  
 稭   NULL ( 뫪),  ᫨    p
ନ쭠. 筮, ᫨ q  뫪  ,  ᮮ⢥-
 p[i],  next(q)   뫪    ᫥  稭  
p[i+1]  NULL, ᫨ i = d. , १ generate(p) 稬 -
楤,       ᮧ    ,    ᮮ⢥騥    
p[1],...,p[d], ⠭     next      ⠪,
 first(p) 㪠뢠  p[1]  ࠢ NULL, ᫨ d =  0.  
- 楤  ।⠢  ᫥饩  筮 ଥ:

      integer procedure F2(ref(position) p,integer alpha,integer beta):
         begin integer m,t; ref(position) q;
         generate(p);
         q := first(p);
         if q = NULL then F2 := f(p) else
         begin  m := alpha;
            while q <> NULL and m < beta do
               begin t := -F2(q, -beta, -m);
                  if t > m then m := t;
                  q := next(q);
               end;
            F2 := m;
         end;
      end.

      ᭮ ।⠢  ४७ 楤    ⨢
(४७⭮)   ஡  ⨬騥  ८ࠧ-
,  ࠭ ४⭮ ணࠬ (. [13]).  -
 楤  㤨⥫쭮  ,    ⥫⢮    -
쭮  ⮫ 祢,   ४७⭮ ।⠢:

   integer procedure alphabeta(ref(position p);
      begin integer l; comment level of recursion;
         integer array a[-2:L];
            comment stack for recursion where a[l-2], a[l-1], a[l]
                denote respectively alpha, beta, m, -t in procedure F2;
         ref (position) array r[0:L+1];
            comment another stack for recursion where r[l] and r[l+1]
                denote respectively p and q in F2;
         l := 0; a[-2] := a[-1] := -INFINITY; r[0] := p;
     F2: generate(r[l]);
         r[l+1] := first(r[l]);
         if r[l+1] = NIL then a[l] := f(r[l]) else
         begin a[l] := a[l-2];
          loop: l := l+1; goto F2;
          resume: if -a[l+1] > a[l] then
            begin a[l] := -a[l+1];
              if a[l+1] <= a[l-1] then go to done;
            end;
         r[l+1] := next(r[l+1]);
         if r[l+1] <> NIL then goto loop;
      end;
   done: l := l-1; if l >= 0 then goto resume;
      alphabeta := a[0];
   end.


 楤    祭,   F2(p,-INFINITY,INFINITY);
祭 L 㦭  ⮫쪮 訬, ⮡ ஢ ४ᨨ -
  ॢ室 L.


      4. ਫ

      ࠡ뢠 ஢ ணࠬ,  ।    ,  
 業 । 室  ᬮ    ନ  -
 --  - 祭 筮 , ⮡    
"⨬" 嬠!    ,  ᠭ  楤  ਬ
, ᫨ ணࠬ,  । 樨 (室)  -
 ⠪, ⮡ 筮 㡮 樨  﫠 ନ-
. , ਬ,  ⨬  ॡ  㡨   6  室
( 3   ப).  ⮬ 砥   ⢮,    
, 室  6- ஢  業,  ⨬ -
 (..  ନ).  ᫥ 業 ⠪  "⢥-
 ନ" 権  㦭, 筮, ᯮ짮   -
 ,   ,    ⮬,    筮
㡮   ᫥⢨ 饭 訡.  (  
६ ࠡ ணࠬ 㤥 祭  ᫥ ⠪  ,
⮬ 室 ᯮ짮  ᫨ 業. 㣠  -
 ⮨  ࠧࠡ⪥  ⨬ 室, , 
 १砩 㤭.)

       ᪠  䨪஢ 㡨  ᫥ -
 ⢨ (᫥⥫쭮 -室) ॢ ஡, ᪠, -
᫥   ࠧ.   室  ।    1965  .
. [6],  -   ᫥ 筮 ப.  -
 室  奬  ᢠ  "ࠢ"  ᮣ᭮
᫥饬 饬 . "ࠢ" 㦤  室  ࠢ
1,   ६   室 (⠪,  ⢠    -
) ᢠ "ࠢ", ࠢ, ᪠, 0.01    
⮣.  嬠 ⢥⭮ ⨥  ࠢ, 襥  0.5,  
訥 室 -- , ᪠, 0.02.  ந  ࠢ-
 室,    , ⠭  ࠭ -
 (ਬ,  10**(-8)),   ᬠਢ  ନ-
쭠,        業.  ⮩  -
 " "  ॢ 㤥 襥 .

        ⮤  ᯮ짮  祭 ॢ  ਥ
ࠧ஢, ᠬ - 楤 ᪠ 襭, ᫨  ।-
⠢塞 ᥡ, 祬   ࠢ 業 室 樨. ᫨  
㬠,  業 樨   a      b,      F2(p,
-INFINITY, +INFINITY)   ஡ F2(p, a, b). ,  -
  . 3   F2(p,-10,+10)  ᯮ짮  F2(p,0,4)  --
 ⮬ "-4"  ஢ 3  稭 樨   祭.  ᫨
 । ୮,  祬    ॢ;    㣮
஭, ᫨ 祭 業  ᫨誮 , ᪠
F2(p,a,b) =  v,    v  <=  a,                 
F2(p,-INFINITY,v), ⮡   業.    ᯮ짮-
    嬠⭮ ணࠬ ਭ [8].



      5. 

      ० 祬  㯨  ⢥   䥪⨢
- 楤, ⪮ ᬮਬ ᭮ ⠯  ࠧ. 
࠭  筮 㬠, ⮬   ᭮뢠   -
㬥஢ ᯮ,  ⠪  ⮬,      
楤 F1   魮 楤ன F2. ⮬ ᫥饥 -
 ᯮ   ଠ, 㯭 ࠬ.

      - 㬠   ⮤  1965 .    ६  -
᪮ ⭥ ᫥⥫᪮ ७樨  ⢥  ⥫-
,  ⠩ ᪠    ᠬ ࠭  嬠  ண-
ࠬ [3],  ன   ᯮ짮    祭.  -
"⨪   ணࠬ,   㡥 ⠩.   ६ ᯥ-
䨪樨 ⬠ ⮢  뫨". 祭 ⭮,  -
 -, ᤥ  ⮩ ७樨, ਢ   ⮬,    
 50- . - 祭  ⠫  ᯮ짮    嬠
ணࠬ.

      ࢮ 㡫樥,  ன ᮤঠ 㦤 祭 
ॢ , 뫠  ,    [16]    ᠭ  
࠭ 嬠⭮ ணࠬ. ,  ਢ  ਬ  ࠡ  
"஭" ⮤, ॠ  楤 F1, ⠪   -
, ᯮ짮    "" 祭.

      -  䨪 alpha  beta  ᢮ ࢮ  ண-
ࠬ  LISP', ॠ饩  ⮤.  ணࠬ  ࠡ⠫  
 ७, 祬 襮ᠭ ⮤, ᪮  । -
⢮  㭪権 "⨬᪠ 業 (p)"    "ᨬ-
᪠ 業 (p)",  ⠢﫨    ࠭  業-
 樨. ணࠬ -, 믮   ⢨,   -
  楤 F2,  ।⠢  ᫥饬 :

      if optimistic value(p) <= alpha then F2 := alpha
      else if pessimistic value(p) >= beta then F2 := beta
      else begin <the above body of procedure F2> end.

-   - ⠫ - 祭 (,
騬 訡) ᪨ ਥ,  ᮧ,  筮  祭
業 砥,    p

      optimistic value(p) = +INFINITY

      pessimistic value(p) = -INFINITY.

 ਫ ⨥ ⮣ 䠪   ,   1961  
⮢  ⮬   [10].  ⮬ 㡫  -
  ਢ ਬ ࠡ 饣 ⮤,  ⮬ ᫥,   -
祭.  (  뫮 ਭ  1961 .),     ᭮-
 ⮤,  㪠 ᫮,    ࠡ⠥.

      ࢠ 㡫  ᠭ - 祭   
᪮ 몥  1963 . ᮢ襭 ᨬ    ࠡ  ਪ楢.
㤭 [4]   ࠧࠡ稪  ᨩ ᪮ 嬠⭮ ண-
ࠬ (   ணࠬ,      ஬᪨
     嬠 -- ਬ.  ॢ.),  ।  -
, ᮢ騩  - ⮤,   --  쭮-⠪  ᫮-
 -- ⥫⢮  ࠢ쭮.

       ᠭ - 祭   ""  -
  ᫥  ࠬ  1968 .        
᪮ [24]  ⥣ ⥫⢠ ⥮६, ,    -
 ࠤ ⪮   ᮤঠ 㦤    祭.
 ࠧ,  ⢥ত,  ࢮ ⪮  ⮤ 
᪮ 몥   1969 .     ᮭ  [25]
  [22];      㯮    
祭  㦤  砥 室 ⠫.

      ࠪ⨪ 뢠,  祭 㤭 ᭨  ⮤  ஥-
 - 祭 " ᫮"      筮  ⥬᪮
몥, ⮬  ஢  ࠡ  뫨  㦤  ਡ-
    ᫮ ᠭ.  ⮣,  ࢮ ⢥ 
⮤ 祭 㤭 ⠢ ᥡ    ,      ⢨-
⥫쭮 ࠡ⠥, ᮡ ⮣,  ⮤ 뢠    "筮"
몥   ᭮   祭.   ,
 ⮬ ᠭ ⮤     ᫥  ⮣,
   ⥭. ,  ࠧ. 2  ,  ⮤  -
  ᭮, ᫨ ࠧ   ⬨᪮ 몥;  -
 訬 ਬ஬ ,  ஬ "᪨" 室  -
ᠭ  뢠  ⥫쭮    ।⥫,  祬
 ⥬᪨.

      祭   ⮤  ᠭ        ᮭ  [18,
ࠧ.4]   [23, pp.16-24], ,   ।⠢  ⮤  "
஧",             ⬨᪮  ଥ.
- 祭 ⠫ "ப 묨", , ᪮쪮  -
⭮ ࠬ,    㡫 楤 ᠭ   -
᪮ 몥.  ᠬ ,  ࢮ   [27, ࠧ. 4.3.3], 
ਢ   - 楤,        ᫠,
祬 楤 F1. (   ⮫쪮  ந  -
,   孨 祭 ந     믮  ண
ࠢ⢠.) 㣠  ⬠, ਭ     ᭨
[5, ࠧ. 8.1],    襤襬  ࢥ᪮ 몥 祡-
  ࠬ ; , - ⮤ ।⠢   
 ᯮ짮 ࠬ஢-⮪, ⠪  ⥫⢮  ࠢ쭮-
 ⠭ 筮 㤭.    㣮    祡  [17,
ࠧ. 3.3.1] ᮤন ଠ쭮 ᠭ ⮣,    ⠬  
"- 祭ﬨ",  ᭮ ਢ    ⮤,  ॠ㥬
楤ன F1; -,   ,  -  楤
ᯮᮡ ந  祭. (    ஢  
(..)  5   த  ᫥,  ᠭ-
  ࠧ. 7,  ,   祭 . ,  -
 F1  ਭ  "- 祭",    -
 , ⠪    F2. -- ਬ. ஢.) -  ᥣ  ⮣
   㢥७  ⮬,    ⮤ -
 , --   ᬮ  ,    -  祭  -
 㦥  15 !
                                                                                                                                                                                                     
