Article信州大学工学部紀要 65: 17-24 (1989)departmental bulletin pape

FUWA,  Yasushi

KOSHISAKA,  Yoshihiro

NAKAMURA,  Yatsuka

English

FUWA  Yasushi

NAKAMURA  Yatsuka

KOSHISAKA  Yoshihiro

Shinshu University Institutional Repository

     Journal of the Faculty of Engineering, Shinshu University, No.65, 1989 17                       ･Se･[･lti({¥iXfl}kscS*ilee ee6s-"Ei'r        An Algorithm for Solving Alive Situation                     Puzzles in GO Game              Yasushi FUWA,* Yatsuka NAKAMURA,**                    and Yoshihiro KOSMSAKA"*"                       (Received October 31, 1988)    For the process of deciding whether the situation of stones in GO game is alive  or not, an improved algorithm was devised, and a program for solving alive sit-  uation puzzles in GO .crame was constructed with the algorithm and put into test  uses. Conventional algorithms employ metheds based on pattern matchlng and  require a lot of processing time and Iarge databases. A new algorithm first par-  titions stones into groups and then analyze the sltuation. Every group enclosed  by opposite stones is pruned, and if at !east one group remains, the situation is  consrdered to be alive. The program has succeeded in solving alive situation puzzles  reasonably rapidly.                         1. Introduction   In general, it is very dlthcuit to find efficient algorithms for playingstrategic games. A great progress in the art how to program a computerto play a strategic game has been made since l955 when the first computerchess program appeared2). In 1976 a Northwestern University chess pro-gram named CHESS 4. 5 defeated several Experts and Class "A" players. Thisprogram is based on the full width search by the MIN-MAX principle. Butthere have been no such successful programs for GO game. The main reasonfor it is that GO game has higher complexity than chess. Statistically, theaverage of possible moves is 40 in chess, but 200 in GO. Moreover, theanaiysis of static situatlons in GO game is more complicated and dithcultthan ik chess.i),3),4) Here, this paper presents a new simple algorithm forrecognizing alive groups of stones on the GO board, which has been succe-ssfully applied to solving alive situation puzzles of GO game. In section2, the notion of alive situation is explained. The algorithm is described insection 3. In section 4, a program for solving Tsume GO with applicationsof the algorithm and the possibility of speeding up the algoyithm are dis-cussed. 'S Assistant, Department of Information Engineering. :k'k Professor, Department of Information Englneering.;S** Student of master course.18 Y. FuwA Y. NAKAMuRA Y. KosmsAKA                    2. AIRye situation   In a GO game stones are placed on lntersections of horizontal and verticallines. Two stones are called steighbors if they are positioned on two neigh-boring intersections. A set of stones of the sapae color on the GO board iscalled a group if each stone X in the set has at least one neighbor in theset and all the neighbors of the stone X belong to the same set. Thus, thegroup ls a set of stones closed under the relation of neighborhood. A singlestone without neighbors is the minimal possible group. The boundary ofa group is composed of the stones of the group each of which has at leastone neighboring intersection unoccupied by stones of the same group. Theneighborhood of a stone in a boundary is all its vacant neighboring intersec-tions. The Reighborhood ef a group is the union of the neighborhoods ofall stones in the boundary. The neighborhood of a group consisting of asingle vacant intersection is called the single interseetioR neighborhood ofthe group. If a group has a single intersection neighborhood and the oppo-nent now has the turn to play, the opponent can capture this group by plac-ing his stone on this single vacant intersection. If this intersection is a singleintersection neighborhood of more than one group, all these groups arecaptured. Naturally at most four groups can share a single intersection nei-ghborhood. The group is said to be surrounded if there are no intersectionsin its neighborhood.   The black stones in Figs. 1 (A) (B) form one group, and in Fig. 1 <C)there are two black and two white groups. The basic form of capture is thecapture of one stone (Fig. 2). A capture of a group of three stones is shownin Fig. 3.   Consider the situation shown in Fig. 4. The white stones form one groupand its neighborhood consists of two intersections. Assume that a blackstone has been placed on one of them (triangular mark in Fig. 4). The whitegroup cannot be captured because there still remains one vacant intersectionin its neighborhood. The black stone has become a group with no neighbo-        tiglttL.I,        (A') (B)'v'F         (c)Fig. 1 Relation of stones.ttsiiiliilil,il  (A)- (B)Fig  2 Capture of????  (c)one st ne.         An Algorithm for Solving AHve Situation Puzzles in GO Game 19       Xsi9 eci ,[ ce      (A) (B) (C) (A) (B)      Fig.3 Capture of stones. Fig.4 AIive situation.rhood on the board. To avoid this strange situation, the rules of GO gamedo not allow this move, and thus such a group of white stones can neverbe captured. We will call such a group alive.   Now consider the situation shown in Fig. 5(A). The white stones form                             two groups. Assume that a blacl< stone hasza     HX         x- . fl   lt ew i' ,iri      (A)   Fig. 5 Not-alive   The' 'of the group lf itbe shared byeye if it isonly alive groups.eye. It is easy toperfect eyes.   The aim invacantregion ef GOrefore, the ''for the success '   The functionGO board. It canon the boardit has at Ieastall stones of a 'Then it makesl         rx            I         t ?Cii      -+ra          (B)        s tuatlon. Intersectlon ln         is     groups. The  surrounded by        The        GO gamelntersectlons as      board   ability to        in GO         of this      according to      two      given color       thesurrounded         stoRes      ey understand       is       possible completely    determine  game.   algorithm de ermine        our    perfect       intosequentially    been placed on intersection (1).' Although    it is placed on an intersection entirely sur-    rounded by white stones, it is alive because    the effect of this move leads to the capture    of the white group of three stones. These    white groups are not alive because they    can be captured as shown in Flg.5 (B)neighborhood of a group will be calied the eye    by stones of the group's color. Eyes caneye of an alive group wlll be cailed the perfect  of only this group or if it is shared withwhich is not perfect will be called the imperfeet that the alive group should have at least twoto secure territory and to surround as many   at the end of game. The territory is the    enclosed with groups which are alive The-   which groups are alive is of basic lmportance3. AIgoritkrrt     is to analyze the static situation on thewhether or not there is at least one group alive preceding criterion that a group is allve ifey s. The algoritlim (see Fig.6) first partitlons  gr ups and marks ail eyes of these groups.ana ses of all the groups witk respect to the20 Y. FuNrerA Y.N KAMURA Y.KOSHISAKAAlive(plate,color){     captured = TRUE;     set-ofmgroups = divide-into-groups<piate,stone>;     while< (captured==TRUE) && notdeempty(setmofwwgroups) ) {          markneali-eye(piate,color);          group = first-of(setmef-groups);          captured = FALSE;          whi-le{ not-empty(group) ) (               if( count-eye<plate,group}<2 ) {                    removewwfrom(setwnof-groups,group);                    remove.eyes(plate,group);                    captured = TRUE;               }               group = nextNof(group);          }     if( not-empty(setwwof-groups) )          return{ isnvaiive );     else          return( is.not-alive );Fig.6 Algorithm.number of their eyes. If a group does not have at ieast two eyes, it ispruned from the beard. If at least one group has been pruned from theboard, analysis of all the remaining groups is repeated. The algorithm endsif no groups can be pruned. If at least one group has remained on the board,it may be concluded that on the board there is at least one group with twoperfect eyes, and so the result of analysis is that the situation of the stonesof the glven color on the board is alive.   It will be explained how the algorithm recognizes alive groups in thesimple examples shown in Figs.7, 8, and 9. Suppose that the stones on theN O P Q E S T(A).ISle17]6I5la13NOPQRST?.,,il'VLg'tL',,,,,,.-......ntdi-          (B)Fig. 7 Process to solutioii.IIII-??  iI.IIIIww.rm.  I   fy??nvteei  (A>?  lt [!] set.of.groupsepm-".-T7TT.rfp171918i7Ifit514ISercElwwtsrrmN-wtr          (B>Fig. 8 Process to solution.'->LZP!rm5,.-ew       N,ign            cc>       N soL.of.areu[,s      -v      C:-urmhn,x,-".Ln'-1;'i-.'{ILI::"tiiZE!]--- -)-{?rlSnz]                (D)                      Fig. 9 Process to solution.an afarmative answer after the first iteration. Foralgorithm will find two groups, but the group ofhave only one eye. Therefore. in the first iteration,(see Fig.9 (B)) is pruned from the board and ingroup of seven stones is found to have only onetoo, ancl the algorithm ends with a negative answerhave remained on the board.                        4. Applieation   This aigorithm applies to a program forin GO game. This puzzle is cailed "Tsttme GO" inan element of GO game which appears anytimeboard in course of game. The program which usesdes other functions adopt the MIN-MAX principle.     An Algorithm for Solving AHve Situation Puzzles in GO Game 21                        figures are oRly the ones on the board and No y, Qns 'r             ig that we are to analyze the skuation of theitllliill.'Sll,tl'//'x･ts/･-/S'lisi/iij1'i'i'iILi.,h/iie/6,{,FSg:e{S,i/f:Fe,,S,,gi,:'i:h,W,meil,az,ij'/ksniien/tlln//,/li'11'S,E.      <,x)                        than two eyes. The group is alive and so  [/"] sct-of-sroups is the situation of wliite stones･ For the('i Lt'.tlaTii'r-:L'E/!-LzllLtti"'mi'L--- -.･ijFpT･ -!                        white stones in Fig. 8, the algorithm wiilC'.th" !uari,--.irwt' /･-{iim---.･ifpm , find two groups and thelr two eyes･ As in          (B)                        the preceding case, the analysis ends with    NOPQRS 'V1,k",Yts,ki ,T･esieei±ri,*iT-NiL   the   three   the   eye.solving live situati n puzzies  J panese. "Tsume GO" is  and nywhere oR the GO   this algorit m and inclu-     To anaiyze a game, the"tit"Meeitnvi3    <E)IZ] Set.Of.8reur,t,    <1:> stones in Fig. 9, the   tones is found to th s group and its eye   s cond iteration the  Thus, it is removedbecause no white groups22 Y. FuwA Y. NAKAMuRA Y. KosHIsAKAMIN-MAX principle makes a game tree. The game tree indicates the processof the game. Fig. Ie shows a sampie of game tree to analyze the puzzle. EachpoinS indicates a situation, and each line indicates a move. In this case, thebest move for white, i. e., the answer for the puzzle, corresponds to thebold line, because as a result of this move the white player may expect thatwhite stones are alive after all. Thls method is called the full width search bythe MIN-MAX principle, and it requires so much tlme as not to be feasible.We have tried to speed up the analysis. For this purpose, the authors usethree methods, i. e., the alpha-beta pruning method, the seeondary searchmethod, and the register meehod which registers all skuations, evaluations,aRd a part of game tree in memory. An example of alpha-beta pruning isshown in Fig. 11. The broken lines which indicate moves of situation are not    ag¥hkiX;x,",S//ebe,E ig.e6,l/31Ililli.eg,:･z.ie,Qg    ECt!CC!!!ECEE-C.=-.fEEE c te ccc-==-M=.-"- tr-i- -H-- = == ===    noooooonnnceoanaOeOO Oig Sttuation rr :: ;SX O@ Siiuation    se to tt tt c7 ta le caMVi ee Ld t" ca L} ts CI C7- ca mm nm :}    vpnm==pa i:pvp=a=avva un MOVe v =" vvp mu- Xove    --nnon.o--""noo.oHHHo - oo ---  l, i'xx z-g;igg ii g.' s'g". ".ag, g' z- g.' zs' i' g, z.'        Fig. le Game tree. Fig. 11 A!pha-beta pruning.searched actually. The number of moves to be searched actually is decreasedfrom 51 to 19 by this method. An example of secondary search is shown inFig.12. We decide the level of boundary, and this method searches such asituation as has an evaluation of depth less than the boundary. No resultsof thls searching gives the best move, and then a deeper situation issearched. The number of moves to be searched is decreased from 51 to 16by alpha-beta pruning combined with this method. An example of registermethod is shown in Fig.13. This method is such that all situations, andeva}uations and a part of game tree above some Ievel are registered andthat if the same situation as has been registered before is found, the asso-ciated evaluation and part of game tree are copied from memory. In thiscase, when the situa£ion of move (E:l)-(G:1)-(D:1) is searched, the samesituation (D:1)-(G:1)-(E:1) which has been searched before is found, andthus the evaluation and the part of game tree are copied.An Algorith m for Solving Alive Situation Puzzles in GO GamegN,Slw.ite/"eeig'i.s:e.cr ct. ct tt nt ct ctcccc========T=7r"rrxrr-,--.iiililllXXX oe sitoation====uts")--,..      g.klg. im ,,,,      ::; Fig. 12 Secondary search. ABCDEFfiHJK/13 reSllllijE     (A) p:1 /-C)×1 E:'{------･-. .-. ??     A/             XA    E:I F:l g:1 D:IF:la:1  s( b XQ gf c) xQ F:: C:1       E:] F:i F:I S:1    s Jx bll aj pt eq pt op @ @         F:1 E:'1 G:1 F:iO:I G:1 D:1 F:1             F:1Ob"b         P6QO              Qa:s Fx       O:I E= E:I F:1             'O:1 E:l-:         tt tt             aj igaj tw   op w pt ep t'        e:l B:: F':I E:1             S:1 D:1        "O ""             OO        D:I E:1 E:I F:1            O:1 E:        /l r!             Tl   tw @ ee tw     tw dyccccccccccccccr==== tr m= tr == tr ==-t -t "t -t Lt -t -t -  -t rt -t -t -t -trF r- -  r- r" - e r- e- - e- " -ny r-OOh fi no ,? eeonoo"-t -t -t -t -t ft -t -t "t -t -t -t -t -t" tt t) 47 Ce t- tt en ca 4" te SS Ct "cwnv""npan ts "ae: =, :=::g=::= =. goe{actaata n. ct==  n ea o n"    xv･              Copy         (B)   Fig. 13 Register method.I#ove of Sl:ck:Ieve of Vhitenovc of Blacknev,e of Vhite,Eeve ofBeacknove of Uhltcrlove ef glack23'(A)@4of(B)144--<D6fo4??)??o8 ?-o-ee                                     @ tw                     (D) (E)                    Fig. 14 Results of execution.   The authors have instalied this algorithm on a mini-super computer (CO-NVEX CI-XP) by the C language, and solved some problems of "Tsume GO"game. Fig. 14 shows results of this execution. To solve the problems, Fig. 14(A) has needed 108 minutes, Figs.I4 (B) and (C) two minutes, Fig.14 (D) 233minutes, and Fig. I4 (E) 24 minutes.24 Y. FuwA Y. NAKAMuRA Y. KOSHISAKA                           5. Conclusions   This algorithm makes it possible to evaluate the situation of stones andto solve alive situation puzzles within realistic time. It is possible to applythe method to solving the GO game. The authors are investigating a hard-ware of Paraliel Processings that can execute this algorithm.                           6. Acknewledgment   The authors would like to express our gratitude to Mr. Pavel Tvrdikfor reading the maRuscript                              References1) Reitman, Vgr. & Wilcox, B., `'Pattern recognition and pattern directed inference  in a program for playing GO", in D. Waterman and F. Hayes-Roth (Eds.), Pattern-  directed inference Systems, New York Academic Press, pp.503-523 (1978).2) Berliner, H., "A chronology of computer chess and its literature", Artificial Intel-  ligence le, 201-214 (1978).3) Sanechika, Ohigashi, Mano, Sugawara & Torii, "Notes on Modelling and Imple-  mentation of the Ruman PIayer's Decision Processes in the Garne of GO", Bul, of  ETL 45, 1-2 (1981).4) Mano, Y., "An approach to Conquer Dithculties in Developing a GO PIaying Pi'o-  gram", fournal of Infbr}nation Processing, Vol.7 (19S4).

An Algorithm for Solving Alive Situation Puzzles in GO Game

     Journal of the Faculty of Engineering, Shinshu University, No.65, 1989 17                       ･Se･[･lti({¥iXfl}kscS*ilee ee6s-"Ei'r        An Algorithm for Solving Alive Situation                     Puzzles in GO Game              Yasushi FUWA,* Yatsuka NAKAMURA,**                    and Yoshihiro KOSMSAKA"*"                       (Received October 31, 1988)    For the process of deciding whether the situation of stones in GO game is alive  or not, an improved algorithm was devised, and a program for solving alive sit-  uation puzzles in GO .crame was constructed with the algorithm and put into test  uses. Conventional algorithms employ metheds based on pattern matchlng and  require a lot of processing time and Iarge databases. A new algorithm first par-  titions stones into groups and then analyze the sltuation. Every group enclosed  by opposite stones is pruned, and if at !east one group remains, the situation is  consrdered to be alive. The program has succeeded in solving alive situation puzzles  reasonably rapidly.                         1. Introduction   In general, it is very dlthcuit to find efficient algorithms for playingstrategic games. A great progress in the art how to program a computerto play a strategic game has been made since l955 when the first computerchess program appeared2). In 1976 a Northwestern University chess pro-gram named CHESS 4. 5 defeated several Experts and Class "A" players. Thisprogram is based on the full width search by the MIN-MAX principle. Butthere have been no such successful programs for GO game. The main reasonfor it is that GO game has higher complexity than chess. Statistically, theaverage of possible moves is 40 in chess, but 200 in GO. Moreover, theanaiysis of static situatlons in GO game is more complicated and dithcultthan ik chess.i),3),4) Here, this paper presents a new simple algorithm forrecognizing alive groups of stones on the GO board, which has been succe-ssfully applied to solving alive situation puzzles of GO game. In section2, the notion of alive situation is explained. The algorithm is described insection 3. In section 4, a program for solving Tsume GO with applicationsof the algorithm and the possibility of speeding up the algoyithm are dis-cussed. 'S Assistant, Department of Information Engineering. :k'k Professor, Department of Information Englneering.;S** Student of master course.18 Y. FuwA Y. NAKAMuRA Y. KosmsAKA                    2. AIRye situation   In a GO game stones are placed on lntersections of horizontal and verticallines. Two stones are called steighbors if they are positioned on two neigh-boring intersections. A set of stones of the sapae color on the GO board iscalled a group if each stone X in the set has at least one neighbor in theset and all the neighbors of the stone X belong to the same set. Thus, thegroup ls a set of stones closed under the relation of neighborhood. A singlestone without neighbors is the minimal possible group. The boundary ofa group is composed of the stones of the group each of which has at leastone neighboring intersection unoccupied by stones of the same group. Theneighborhood of a stone in a boundary is all its vacant neighboring intersec-tions. The Reighborhood ef a group is the union of the neighborhoods ofall stones in the boundary. The neighborhood of a group consisting of asingle vacant intersection is called the single interseetioR neighborhood ofthe group. If a group has a single intersection neighborhood and the oppo-nent now has the turn to play, the opponent can capture this group by plac-ing his stone on this single vacant intersection. If this intersection is a singleintersection neighborhood of more than one group, all these groups arecaptured. Naturally at most four groups can share a single intersection nei-ghborhood. The group is said to be surrounded if there are no intersectionsin its neighborhood.   The black stones in Figs. 1 (A) (B) form one group, and in Fig. 1 <C)there are two black and two white groups. The basic form of capture is thecapture of one stone (Fig. 2). A capture of a group of three stones is shownin Fig. 3.   Consider the situation shown in Fig. 4. The white stones form one groupand its neighborhood consists of two intersections. Assume that a blackstone has been placed on one of them (triangular mark in Fig. 4). The whitegroup cannot be captured because there still remains one vacant intersectionin its neighborhood. The black stone has become a group with no neighbo-        tiglttL.I,        (A') (B)'v'F         (c)Fig. 1 Relation of stones.ttsiiiliilil,il  (A)- (B)Fig. 2 Capture ofililiEl   (c)one stone.         An Algorithm for Solving AHve Situation Puzzles in GO Game 19       Xsi9 eci ,[ ce      (A) (B) (C) (A) (B)      Fig.3 Capture of stones. Fig.4 AIive situation.rhood on the board. To avoid this strange situation, the rules of GO gamedo not allow this move, and thus such a group of white stones can neverbe captured. We will call such a group alive.   Now consider the situation shown in Fig. 5(A). The white stones form                             two groups. Assume that a blacl< stone hasza     HX         x  - . fl   lt  ew i' ,iri      (A)   Fig. 5 Not-alive   The' 'of the group lf itbe shared byeye if it isonly alive groups.eye. It is easy toperfect eyes.   The aim invacantregion ef GOrefore, the ''for the success '   The functionGO board. It canon the boardit has at Ieastall stones of a 'Then it makesl          rx            I       t ?Cii        -+ra            (B)         sltuatlon. Intersectlon ln        is     groups. The  surrounded by         The       GO gamelntersectlons as     board  ability to       in GO         of this      according to      two     given color       the   surrounded         stoRes       eye  understand         is       possible    completely    determine      game.      algorithm determine          our    perfect          intosequentially    been placed on intersection (1).' Although    it is placed on an intersection entirely sur-    rounded by white stones, it is alive because    the effect of this move leads to the capture    of the white group of three stones. These    white groups are not alive because they    can be captured as shown in Flg.5 (B)neighborhood of a group will be calied the eye     by stones of the group's color. Eyes caneye of an alive group wlll be cailed the perfect     of only this group or if it is shared withwhich is not perfect will be called the imperfeet  that the alive group should have at least two to secure territory and to surround as many     at the end of game. The territory is the    enclosed with groups which are alive The-   which groups are alive is of basic lmportance3. AIgoritkrrt       is to analyze the static situation on thewhether or not there is at least one group alive    preceding criterion that a group is allve ifeyes. The algoritlim (see Fig.6) first partitlons    groups and marks ail eyes of these groups.analyses of all the groups witk respect to the20 Y. FuNrerA Y. NAKAMURA Y. KOSHISAKAAlive(plate,color){     captured = TRUE;     set-ofmgroups = divide-into-groups<piate,stone>;     while< (captured==TRUE) && notdeempty(setmofwwgroups) ) {          markneali-eye(piate,color);          group = first-of(setmef-groups);          captured = FALSE;          whi-le{ not-empty(group) ) (               if( count-eye<plate,group}<2 ) {                    removewwfrom(setwnof-groups,group);                    remove.eyes(plate,group);                    captured = TRUE;               }               group = nextNof(group);          }     if( not-empty(setwwof-groups) )          return{ isnvaiive );     else          return( is.not-alive );Fig. 6 Algorithm.number of their eyes. If a group does not have at ieast two eyes, it ispruned from the beard. If at least one group has been pruned from theboard, analysis of all the remaining groups is repeated. The algorithm endsif no groups can be pruned. If at least one group has remained on the board,it may be concluded that on the board there is at least one group with twoperfect eyes, and so the result of analysis is that the situation of the stonesof the glven color on the board is alive.   It will be explained how the algorithm recognizes alive groups in thesimple examples shown in Figs.7, 8, and 9. Suppose that the stones on theN O P Q E S T(A).ISle17]6I5la13NOPQRST?.,,il'VLg'tL',,,,,,.-......ntdi-          (B)Fig. 7 Process to solutioii.IIII-Ieei  iI.IIIIww.rm.  I   fygpa.nvteei  (A>as  lt [!] set.of.groupsepm-".-T7TT.rfp171918i7Ifit514ISercElwwtsrrmN-wtr          (B>Fig. 8 Process to solution.'->LZP!rm5,.-ew       N,ign            cc>       N soL.of.areu[,s      -v      C:-urmhn,x,-".Ln'-1;'i-.'{ILI::"tiiZE!]--- -)-{?rlSnz]                (D)                      Fig. 9 Process to solution.an afarmative answer after the first iteration. Foralgorithm will find two groups, but the group ofhave only one eye. Therefore. in the first iteration,(see Fig.9 (B)) is pruned from the board and ingroup of seven stones is found to have only onetoo, ancl the algorithm ends with a negative answerhave remained on the board.                        4. Applieation   This aigorithm applies to a program forin GO game. This puzzle is cailed "Tsttme GO" inan element of GO game which appears anytimeboard in course of game. The program which usesdes other functions adopt the MIN-MAX principle.     An Algorithm for Solving AHve Situation Puzzles in GO Game 21                        figures are oRly the ones on the board and No y, Qns 'r             ig that we are to analyze the skuation of theitllliill.'Sll,tl'//'x･ts/･-/S'lisi/iij1'i'i'iILi.,h/iie/6,{,FSg:e{S,i/f:Fe,,S,,gi,:'i:h,W,meil,az,ij'/ksniien/tlln//,/li'11'S,E.      <,x)                        than two eyes. The group is alive and so  [/"] sct-of-sroups is the situation of wliite stones･ For the('i Lt'.tlaTii'r-:L'E/!-LzllLtti"'mi'L--- -.･ijFpT･ -!                        white stones in Fig. 8, the algorithm wiilC'.th" !uari,--.irwt' /･-{iim---.･ifpm , find two groups and thelr two eyes･ As in          (B)                        the preceding case, the analysis ends with    NOPQRS 'V1,k",Yts,ki ,T･esieei±ri,*iT-NiL    the    three    the    eye.solving alive situation puzzies    Japanese. "Tsume GO" is   and anywhere oR the GO    this algorithm and inclu-     To anaiyze a game, the"tit"Meeitnvi3    <E)IZ] Set.Of.8reur,t,    <1:>  stones in Fig. 9, the    stones is found to this group and its eye   second iteration the   Thus, it is removedbecause no white groups22 Y. FuwA Y. NAKAMuRA Y. KosHIsAKAMIN-MAX principle makes a game tree. The game tree indicates the processof the game. Fig. Ie shows a sampie of game tree to analyze the puzzle. EachpoinS indicates a situation, and each line indicates a move. In this case, thebest move for white, i. e., the answer for the puzzle, corresponds to thebold line, because as a result of this move the white player may expect thatwhite stones are alive after all. Thls method is called the full width search bythe MIN-MAX principle, and it requires so much tlme as not to be feasible.We have tried to speed up the analysis. For this purpose, the authors usethree methods, i. e., the alpha-beta pruning method, the seeondary searchmethod, and the register meehod which registers all skuations, evaluations,aRd a part of game tree in memory. An example of alpha-beta pruning isshown in Fig. 11. The broken lines which indicate moves of situation are not    ag¥hkiX;x,",S//ebe,E ig.e6,l/31Ililli.eg,:･z.ie,Qg    ECt!CC!!!ECEE-C.=-.fEEE c te ccc    -==-M=.-"- tr-i---H-- = == ===    noooooonnnceoanaOeOO Oig Sttuation rr :: ;SX O@ Siiuation    se to tt tt c7 ta le caMVi ee Ld t" ca L} ts CI C7- ca mm nm :}    vpnm==pa i:pvp=a=avva un MOVe v =" vvp mu- Xove    --nnon.o--""noo.oHHHo - oo ---  l, i'xx z-g;igg ii g.' s'g". ".ag, g' z- g.' zs' i' g, z.'        Fig. le Game tree. Fig. 11 A!pha-beta pruning.searched actually. The number of moves to be searched actually is decreasedfrom 51 to 19 by this method. An example of secondary search is shown inFig.12. We decide the level of boundary, and this method searches such asituation as has an evaluation of depth less than the boundary. No resultsof thls searching gives the best move, and then a deeper situation issearched. The number of moves to be searched is decreased from 51 to 16by alpha-beta pruning combined with this method. An example of registermethod is shown in Fig.13. This method is such that all situations, andeva}uations and a part of game tree above some Ievel are registered andthat if the same situation as has been registered before is found, the asso-ciated evaluation and part of game tree are copied from memory. In thiscase, when the situa£ion of move (E:l)-(G:1)-(D:1) is searched, the samesituation (D:1)-(G:1)-(E:1) which has been searched before is found, andthus the evaluation and the part of game tree are copied.An Algorith m for Solving Alive Situation Puzzles in GO GamegN,Slw.ite/"eeig'i.s:e.cr ct. ct tt nt ct ctcccc========T=7=r"rrxrr-,--.iiililllXXX oe sitoation====uts")--,..      g.klg. im ,,,,      ::; Fig. 12 Secondary search. ABCDEFfiHJK/13 reSllllijE     (A) p:1 /-C)×1 E:'{------･-. .-. ??     A/             XA    E:I F:l g:1 D:IF:la:1  s( b XQ gf c) xQ F:: C:1       E:] F:i F:I S:1           s Jx bll aj pt eq pt op @ @         F:1 E:'1 G:1 F:iO:I G:1 D:1 F:1                    F:1Ob"b         P6QO              Qa:s Fx       O:I E= E:I F:1           'O:1 E:l-:         tt tt             aj igaj tw    op w pt ep t'        e:l B:: F':I E:1             S:1 D:1        "O ""             OO        D:I E:1 E:I F:1            O:1 E:        /l r!             Tl   tw @ ee tw     tw dyccccccccccccccr==== tr m= tr == tr ==-t -t "t -t Lt -t -t -  -t rt -t -t -t -trF r- -- r- r" - e r- e- - e- " -ny r-OOh fi no ,? eeonoo"-t -t -t -t -t ft -t -t "t -t -t -t -t -t" tt t) 47 Ce t- tt en ca 4" te SS Ct "cwnv""npan ts "ae: =, :=::g=::= =. goe{actaata n. ct==  n ea o n"    xv･              Copy         (B)   Fig. 13 Register method.I#ove of Sl:ck:Ieve of Vhitenovc of Blacknev,e of Vhite,Eeve of Beacknove of Uhltcrlove ef glack23'(A)@4of(B)144--<D6fo46(c)op･o8i4-o-ee                                     @ tw                     (D) (E)                    Fig. 14 Results of execution.   The authors have instalied this algorithm on a mini-super computer (CO-NVEX CI-XP) by the C language, and solved some problems of "Tsume GO"game. Fig. 14 shows results of this execution. To solve the problems, Fig. 14(A) has needed 108 minutes, Figs.I4 (B) and (C) two minutes, Fig.14 (D) 233minutes, and Fig. I4 (E) 24 minutes.24 Y. FuwA Y. NAKAMuRA Y. KOSHISAKA                           5. Conclusions   This algorithm makes it possible to evaluate the situation of stones andto solve alive situation puzzles within realistic time. It is possible to applythe method to solving the GO game. The authors are investigating a hard-ware of Paraliel Processings that can execute this algorithm.                           6. Acknewledgment   The authors would like to express our gratitude to Mr. Pavel Tvrdikfor reading the maRuscript                              References1) Reitman, Vgr. & Wilcox, B., `'Pattern recognition and pattern directed inference  in a program for playing GO", in D. Waterman and F. Hayes-Roth (Eds.), Pattern-  directed inference Systems, New York Academic Press, pp.503-523 (1978).2) Berliner, H., "A chronology of computer chess and its literature", Artificial Intel-  ligence le, 201-214 (1978).3) Sanechika, Ohigashi, Mano, Sugawara & Torii, "Notes on Modelling and Imple-  mentation of the Ruman PIayer's Decision Processes in the Garne of GO", Bul, of  ETL 45, 1-2 (1981).4) Mano, Y., "An approach to Conquer Dithculties in Developing a GO PIaying Pi'o-  gram", fournal of Infbr}nation Processing, Vol.7 (19S4).

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An Algorithm for Solving Alive Situation Puzzles in GO Game

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