Solving Problems with Uncertainty: A case study using Tsuitate-Tsume-Shogi

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1 olvn Prolms wt nrtnty: A s stuy usn sutt-sum-o Mkoto kut n Hroyuk I Dprtmnt o Computr n, zuok nvrsty {skut, }@s.n.szuok..jp Astrt s ppr xplors t posslty o omputn mprt-normton ms wt ous on ts nm, nmly, mtn prolms wt unrtnty. v osn t omn o sutt-o, w s Krspl-lk o vrnt. Our prsnt ol s to mk prorm tt n solv mtn prolms o sutt- o (sutt-sum-o). v propos n so-ll mtposton, y w m-tr sr or mprt-normton ms turns out to ntl to nr AND/OR-tr sr o mtpostons. In orr to solv sutt-sum- o, w ntrou sr lortm tt w ll oul-nst trtv-pnn sr. lortm s two trtons. outr trton s tt o t sr pt n t nnr trton s tt o t llow ount o ouls n prolm. mplmnt solvr o sutt-sum-o s on t propos s, n xprmnts usn tst st o mtn prolms wr prorm. All prolms wt rouly lss tn 17 stps n t tst st wr solv. 1 Introuton In t omn o omputr-m rsr, most orts v n vot to prormmn m wt prt normton su s Css. r r lttl ltrturs on omputr-m rsr usn mprt-normton ss-lk ms su s Krspl. know tt t s xtrmly r to mk stron prorm o m wt mprt normton, sn tr r som unrtn tors. Our lon-trm lln s to mk worl-mpon-lvl prorm o ss-lk ms wt mprt normton. urrnt trt s to mk prorm w n solv t nm (mtn prolm) o su ms. A mtn prolm o ss-lk ms wt mprt normton ontns mu unrtnty tt os not ppr n omputn ms wt prt normton. v osn t omn o sutt-o, w s Krspl-lk o vrnt. Our prsnt ol n ts ppr s to mk prorm w n solv sutt- o mtn prolms so-ll sutt-sum-o. In t omn o Krspl, Cnrn t l. rport on tr works on som smpl nms o Krspl wl sown t prormn o tr prorm w n ply KPK nms y t knowl-s ppro[cnrn97. soul not tt t r-us rul o o mks sutt-sum-o mu mor omplt tn t nms o Krspl s wll s n t s o ortoox ms. 2 sutt-o sutt-o s o vrnt w s on o t st known n most populr mon ll vrnts s wll s Krspl n n Css[Prtr94. r sutt (n Jpns) srn, tror w my ll sutt-o (srn o). 2.1 Ruls o sutt-o Lt us v sort summry o t ruls. Not tt t ruls o sutt-o sm mostly smlr wt Krspl. sutt-o rqurs two plyrs, n umpr, two o ors wt st o ps, n srn. Bot plyrs r unl to s t otr s us o t srn. prnpl o t m s tt plyr movs normlly ut s not tol t opponnt's movs w ttmpts to sovr trou juous ply. Not tt mov o o nlus not only ptur ut lso rop. Atr on plys mov on s own or, t umpr pprovs t mov to prov normton to ot plyrs s rqur y t ruls. Most mportnt normton s vn y t umpr syn k, lll, n Blk/t s ply. n t umpr nnot tll otr normton lou,, nst o plyrs, prorms pturn on t or nvolv.

2 E plyr s llow to mk lll movs wtn rtn numr o tms, typlly t tms. 1 I on s m lll movs mor tn t llow numr o tms, s to los t m on ouls. A m lso ns up wn t kns o ny two plyrs r n mt. 2.2 sutt-sum-o sutt-sum-o () s mtn prolm o sutt-o[kto95, w s lso vrnt o sum- o, mtn prolm o o[grmrn98. Hr w v summry o ruls o. Not tt t ruls o sm lmost smlr wt sum-o xpt w ss. stnus two plyrs n sutt-sum-o, s t ttkr n t nr. In, t ttkr s unl to s t opponnt's ps n rspons xpt t ntl poston o mtn prolm onsr. ol o solvn prolm s to mt t opponnt's kn tr t squn o k n ts rspons s wll s n sum-o. Howvr, n, t ttkr s unl to s t nt normton on t nr's rsponss, wr t nr s t prt normton on ot ss. s orrspons to t st ns mol o m wt mprt normton[frnk98. In otr wors, t oul rr tt t nr wo s unl to s t opponnt lwys mks t st mov t ny poston rrlss o t prolty. ttkr must l to kmt wtvr movs t nr my ply. ttkr s llow to try t lll movs n rtn numr o tms, typlly t tms or lss. ttkr soul try to uss t poston o t nr's ps s t mtn sr prorsss y tryn movs tt n tr ll or lll wt rspt to t ull poston. In ts sns, to mk som (ut lmt) lll movs w vn my not k, s usul to n nst nto t poston. Nmly, t ttkr s to ply n ontxt o unrtnty n not ull ut prtl normton. In sutt-o, t lll mov s k n nnoun y t rr. A plyr n vn mk n xplt nonsns mov s nt n orr to onus t opponnt. In, tryn t lll mov s lso llow. A quston rss: s t xplt nonsns mov llow n? Hr w v osn t mor strt nton. v n t oul mov s t sust o t lll mov. oul mov s t mov tt sms to ll on t or only wt t own ps ut s lll on t or wt ps o ot ss. In otr wors, t lll movs on onton tt only t own p n sn r not t oul movs. us, t oul movs r: 1. sln p (Rook, Bsop, Ln) jumps ovr t opponnt p 2. Drop p nto t squr wr tr s t opponnt p 3. pwn-rop mov tt uss t ropp pwn mt An or prolms wt oul kns: 4. kn rmns n k or s om to n k tr t mov r my oul mov n volton o t rul tt ors t 4 tms rptton o t poston y t ontnuton o t k mov n ts rspons, ut w v not yt mplmnt. 2.3 Clsston Hr w summrz t turs o ss-lk ms n puzzls n t vwpont o unrtnty (s l 1). ornry ss-lk ms r two-prson zro-sum ms wt prt normton, n or solvn t nm (mtn prolm) o su ms AND/OR-tr sr s us. A no n AND/OR-tr rprsnts poston. opposts r Krspl n sutt-o, n w tr s mu unrtnty sn on os not know t opponnt's rspons urn mtn sr pross. It s too r to sr t m sp o ts ms or omputr. r r mny prolms o Krspl s t ntrmt typ twn Css n Krspl. s prolms r t rly rstrt ss o Krspl, or w omputr my l to solv. prolms o r t ntrmt typ twn Krspl prolms n sutt-o. In, t ntl poston s nt n t prt soluton n trmn. A no o m tr or s mtposton (sr ltr) n tus m tr n ntl to nr AND/OR tr, n tn n solv y omputton. In m tr o, AND nos n OR nos o not nssrly ppr y turns us o oul movs n t splttn o mtpostons, wl AND nos n OR nos ppr y turns n n AND/OR tr or sum-o. 1 o, pnn on t numr o tms or lll movs, strty woul n. n mtn t stuton wr tr r no mor ns to ply lll movs, n you rop p or mov sln p to som stn?

3 ntl poston sum-o, Css nm prolm sutt- sum-o Krspl prolm sutt-o, Krspl ow to solv AND/OR m tr mor nr AND/OR tr AND/OR tr or Mnmx tr? - no us lu us prolty poston No No Dnt No mtposton Ys No Dnt No mtposton o ot ss mtposton o ot ss Ys Ys or No nrtn or Dnt spultv ply No omputton mu str tn umn Ys Ys Dnt Ys too r l 1: Clsston o ss-lk ms n puzzls n vw o unrtnty s unrtnty, so t normton sts[own95 o otn onsst o mny postons. Consquntly, t s nssry to ollt ny lus o t urrnt poston or solvn just t sm s t prolm o Krspl. s r t turs o t m wt mprt normton. ou n solv y t sr o AND/OR tr n ts s t tur o t m wt prt normton. ror, ts s our rmrk tt s t yr turs o prt normton n mprt normton. 3 Mtposton n ts splttn y osrvtons o l wt prolms wt unrtnty, w v ntrou t onpts o mtposton n mtmov. 3.1 Mtposton Mtposton s omposton o ll possl postons, not st o possl postons. In ts mnn, mtposton s smlr to t wv unton n quntum mns. In, t ntl poston s nt, ut tr w movs, t s us nto t st o som mtpostons. solvr must xmn ll t mtpostons. ount o postons n mtposton n rr s t nx o unrtnty. 3.2 Mtmov At rtn mtposton wr t ttkr s to mov, possl movs r t movs tt r tr k or oul movs or poston n t mtposton. On o t movs s t mtmov to t mtposton. At rtn mtposton wr t nr s to mov, possl movs r ll t ll movs o ll postons n t mtposton. omposton o ll possl movs s t mtmov. Mtposton s smlr to t normton st n m tory[own95. Howvr, w v nl ts s snl no n t m tr n tr s only on r twn two mtpostons y mtmov, wl tr r proly svrl rs twn two normton sts y svrl movs. s s n ssntl rn rom t onpt o t normton st. 3.3 Clus (osrvtons) ttkr n uss t urrnt poston n t opponnt movs y som lus (osrvtons): 1. tr t mov o t ttkr s k mov or oul mov 2. tr t poston s kmt 3. tr n wr pturn ours tr t opponnt mov 4. tr pturn ours n wt kn o p s ptur tr t ttk mov An or prolms wt oul kns: 5. tr t ountrk ours tr t opponnt mov By ts osrvtons, t mtposton s splt nto som mtpostons wt lss unrtnty. s s ANDsplttn, so ll t splt mtpostons r to solv. Fnlly prolm s rsolv nto t sr o t nr AND/OR tr y us o t mtposton n ts splttn.

4 3.4 H-lvl nrn R000 00PO0 0A00KO N000 0L0BP 000O0 PN0O0O 0R00 Fur 1: A Krspl prolm Mt n 2 (M); (G.R.Fostr). nnot tll xtly wtr t -lvl nrns tt n not rv rom t omnton o t ov prmtv lus xst or not. In Krspl, tr s unny prolm n w on n only s ps o t ttkr s s t t vn poston[fostr96 (s Fur 1). urprsnly, xplotn t Css-sp turs, ts prolm s solv qut sly. From t plmnts o t s pwns w n s tt tr v n t lst 14 pturs n Blk s n own to kn n my on pwn n l. In ton, t Blk kn n only on v squrs: 2, 5, 6, 3, n 6. s soul so-ll -lvl nrn. Howvr, ll t knowl tt n ottn rom t prolm poston woul lso l to ottn rom t omnton o t prmtv lus n t Krspl m rom t strtn poston, supposn tt w oul nor t normous tm n mmory sp. In our xptton, tr s no -lvl nrn n no n or tt n prolms. 3.5 A smpl prolm n soluton o A smpl prolm o n t ssntl prt o ts soluton tr r sown n Fur 2. poston t t uppr lt s t vn on o t prolm. rst mov s lvr 2-1 n tr r our movs or ts rspons. I t Kn movs nto t squr 1 n pturs t lvr, t ttkr knows t mov us t p t 1 spprs n so n l to kmt sly. I tr s no ptur, tr movs r possl n t mtposton s us to on wt tr postons. son ttkr s mov s roppn Rook t 2. I t s oul mov, t Kn squr s x t 2 n t poston n l to kmt. Otrws, t mtposton s ru to on wt two postons n tr r our movs or ts rspons. I t Kn movs nto t squr 2 n pturs t Rook, t ttkr knows t mov n t poston n n l to kmt. I t Kn movs nto t squr 1 n pturs t lvr, t ttkr lso knows t mov n t poston n n l to kmt. I tr s no ptur, svn movs r possl n t mtposton s om to v svn postons. In ts ur, t ntrposn movs r rprsnt y roppn Pwn ut tully tr r sx possl movs. r s no ommon k mov mon t postons n ts mtposton, ut utlzn oul mov, Rook 3-1 n mtmov. I t s oul mov, t ntrposn mov s onrm n t postons n l to kmt. Otrws, t mtposton s om to v only on poston n tr r sx ntrposn movs n on pturn mov. I tr s no ptur, t ntrposn mov s onrm n t postons n l to kmt. Otrws, t mtposton s only on poston n n l to kmt. Atr ll, t st squn s nn pls n no oul wrs t llow oul soul on or solvn. st squn o t soluton s: (-) 3. *2 4.(-) x (-) mt. (-) nts mtmov tr w no pturn s ourr. 4 Implmntton n Exprmnts o olvr v mplmnt t solvr o n omputr. 4.1 Ruton o postons In mtposton, tr rtn mtmov, tr ppn to som sm postons. In ts s, t runnt postons r sr n t poston ount s ru. s rutons usully sp up solvn, ut n som prolms t s str not to prorm t ruton o postons. o tr s n opton swt n our solvr.

5 Fur 2: oluton o t smpl prolm (ompos y Knku Koys n Ktsuro Km) 1 *1, 2, 1+ kmt. *2 I *2 s oul 3+, 1, 2 kmt. 3+, 1, 2 kmt. 2+ kmt. 1 I 1 s oul x2+ kmt. x1-1+ kmt. 2, 1(1), 2+ kmt.

6 4.2 Itrtv pnn v prsntly n usn t qut nnt trtv pnn sr. Itrton s ouly nst to n t soluton wt t sortst stps n t lst ouls. outr trton s tt o t sr pt n t nnr trton s tt o t llow ount o ouls. 4.3 Dnn t st squn o t soluton Atr n t stps n ouls, t sr wt t multpl trtv pnn t OR nos s prorm to n t st squn o t soluton. squn s ronz ttr wn numr o stps o on squn s lss tn tt o notr squn. I t s sm, t numr o ouls s lss tn tt o notr squn. I t s sm, t st o ps n n t t mt poston s sust o tt o notr squn. mtposton tt s t worst squn s slt n t nr s turn. 4.4 Exprmntl sn Our solvr ws o n C++ (sul C++ 6.0). Exprmnts wr on unr t ollown nvronmnt: Gtwy2000, G6/GP6 rs (B ) Pntum II 447MHz, RAM: 384MB nows prolms rom t sour[kto95, ut t rst w tr to solv t rst 15 prolms sown n Appnx. No F M P M P MP AP BM BP B E6 1.39E E E E6 7.29E E E7 1.72E8 1.27E7 6.52E E7 8.27E7 1.06E7 3.01E vlty o prolm tonl mt, lonr vrton rlr mt (ntn 27 stps) vlty o soluton n to omt uslss roppn ps? l 2: Rsults o solvn prolms :stps o t soluton, F:ouls, M:totl ount o t nrt mtpostons, P:totl ount o t nrt postons M:mtpostons, P:postons, MP:mxmum vlu o poston ount o mtposton, AP:vr vlu o poston ount o mtposton BM:tv rnn tor (s on mtposton), BP:tv rnn tor (s on poston) B:tm or nn t st squn, :totl tm 4.5 Rsults n Dsussons Rsults o t xprmnts r sown n l 2. solvr oul solv ll prolms n sow t propr st squns xpt #5, n w t s nssry to omt t uslss ntrposn mov. M (mtpostons) n P (postons) r t ount o mtpostons n postons tr nn t solvl sr pt n ouls rsptvly. MP (mxmum poston ount) s t mxmum ount o postons n t mtposton. vr ount o postons n t mtposton, tt s, t vr unrtnty nx rouly rn rom 1.4 to 6. An t mxmum ount o postons n t mtposton, tt s, t mxmum unrtnty nx vr wly rom 9 to tv rnn tors wr lult supposn t worst ss n turn out to rouly rn rom 1.9 to 2.7. For rrns t tv rnn tors s on t ounts o postons r lso sown.

7 y r lttl lrr tn tos o t ormr r. upposn t st uto ss, ts vlus woul tw s lr, rn rom 3.9 to 5.4. By llown t ttkr to try oul movs, t numr o t possl movs s not so smll. Consquntly, quntty o omputn s lso not so smll. r sms to v t tnny tt t rnn tors r lrr or t r prolms wt lon stps. In our urrnt mplmntton, t most o solvn tm s t prt nn t solvl pts n ouls y t ouly nst trtv pnn. By srn mor tvly, t totl ount o nrt mtpostons M n postons P woul nrly om own to M n P y tor o on, n onsquntly t totl solvn tm woul ru to t vlus o t tm or nn t st squn B. 5 Conluson n Futur works As t rst stp to lln omputn t ms wt mprt normton, w v osn t prolm o. v ntrou t mtposton s t omposton o t possl postons n t m tr o s turn out to om own to t nr AND/OR tr o mtpostons. rn s n prorm y ouly nst trtv pnn. outr trton s tt o t sr pt n t nnr trton s tt o t llow ount o ouls. Our solvr oul solv ll t prolms wt rouly lss tn 17 stps. Howvr, w v prsntly n usn t qut nnt trtv pnn sr. r r som motons to sp up solvn. Frst, t ny oul mu r to us t trnsposton tl. sn t trnsposton tl s t wll-known n tv mto n srn m tr n t orst mto, t st mto o on postons usn t psuo-rnom numrs s known or or ms. Howvr, or, t s not rtly ppll or on t mtposton. v n onsrn t us o t CRC (or ksum) o poston os s t o o mtposton. v su to solv mny prolms n t sour, ut tr stll r svrl r prolms tt v not yt n solv. s r prolms wt qut lon stps, or prolms tt v rly lr rnn tors. only, to solv ts n sort tm, w tnk w soul xmn t otr srs tt v st-rst mnnrs, pn-sr n PD (or PD*). s must t omn sujt. Aknowlmnts woul lk to tnk tsu Kto wo s plsntly prov us t prolms n xplntons o. In ton, w tnk Ktsuro Km, Knku Koys n Kto or prmttn us rryn tr ps n ts rtl. Rrns [Cnrn97 Cnrn, P., Lr, F.D., n Mrn F. (1997). Dson Mkn unr nrtnty: A Rtonl Appro to Krspl. Avns n Computr Css 8 (. H.J. Hrk n J..H.M. trwjk), pp , Drukkrj n pjk B.., lno, Ntrlns. IBN [Prtr94 Prtr, D.B. (1994). Enylop o Css rnts. Gms & Puzzls Pultons, Golmn, K. IBN [Own95 Own, G. (1995). Gm ory. Am Prss, Nw York, IBN [Fostr96 Fostr, G. R. (1996). rnt Css, ssu 20, ummr lso vll t A Krspl Prolm n Hns Bolnr s Css rnt Ps, ttp:// [Kto95 Kto.. (1995). Collton o Kptn Doumnts No.1, Explntons n Prolms o sutt- sum-o. (n Jpns). [Grmrn98 Grmrn R. (1999). A urvy o sum-o Prorms sn rl-dpt r. n Jp vn n Hrk n Hroyuk I (Es.) Pro. Intrnt. Con. on Computrs n Gms, CG'98, Ltur Nots n Computr n, vol.1558, prnr, Hlr, pp [Frnk98 Frnk, I., Bsn, D. (1998). r n ms wt nomplt normton: s stuy usn Br r ply. Artl Intlln, ol.100, pp

8 [ 5 Y 7 ;3 _ ?3 93 _ 5 9 Y 5 9 _ 7 [ 7 Y 9 _ 9 [ 5 5 _ 9 #1 * Y #4 * _ [ _ l #7 * #10 * [ [ 4 Appnx: st prolms o 9 ;3 5 _ 7 =3 _ _ _ Y 9 5 _ 5 [ 5 5 #2 * _ #5 * _ l Y Y #8 * l l l m l #11 * 5 Y 7 53 _ _ 7 53 _ Y 7 53 _ 5 [ 7 Y 9 #3 * l #6 * [ _ [ #9 * #12 * #13 * #14 * #15 * ompos y *:tsu Kto, *:Ktsuro Km, *:Knku Koys, *:Km, Koys, Kto _ [ 4 =3 _ 7 l [ 7 Y 5 Y Y

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5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees /1/018 W usully no strns y ssnn -lnt os to ll rtrs n t lpt (or mpl, 8-t on n ASCII). Howvr, rnt rtrs our wt rnt rquns, w n sv mmory n ru trnsmttl tm y usn vrl-lnt non. T s to ssn sortr os to rtrs tt our