utomti Strtegy Verifition for Hex Ryn B. Hywrd, Broderik rneson, nd Philip Henderson Deprtment of Computing Siene, University of lert, Edmonton, Cnd {hywrd,roderi,ph}@s.ulert. strt. We present onise nd/or-tree nottion for desriing Hex strtegies together with n esily implemented lgorithm for verifying strtegy orretness. To illustrte our lgorithm, we use it to verify Jing Yng s 7 7 entre-opening strtegy. Introdution Hex is the lssi two-plyer ord gme invented y Piet Hein in 9 nd independently y John Nsh round 98 [,,6,7,8,9]. The gme is nmed fter the ord, whih onsists of prllelogrm-shped m n rry of hexgons, lso lled ells. Eh plyer is ssigned set of stones nd two opposing ord sides; plyers lterntely ple stone on n unoupied ell; the first plyer to form pth onneting her two sides with her stones wins the gme. For exmple, Fig. shows the strt nd end of gme on ord. White sueeds in joining her two sides, so White wins this gme. For more on Hex, see the reent survey y Hywrd nd Vn Rijswijk [] or the we pge y Thoms Mrup [7]. 6 5 7 Fig.. The strt (left) nd finish (right) of Hex gme on ord n intriguing spet of the gme of Hex is tht for ll n n ords, lthough winning first-plyer strtegy is known to exist [,,9], expliit suh strtegies hve een found only for smll ords. While finding suh strtegies is routine on very smll ords, the tsk quikly eomes hllenging s ord size inreses. This is not surprising sine, s Stefn Reish hs shown, determining the winner of ritrry Hex positions is PSPCE-omplete []. uthors grtefully knowledge the support of NSERC nd the University of lert Gmes Group. For revity we use she nd her whenever she or he nd his or her re ment. H.J. vn den Herik et l. (Eds.): CG 006, LNCS 60, pp., 007. Springer-Verlg Berlin Heidelerg 007
utomti Strtegy Verifition for Hex Fig.. winning first-plyer Hex strtegy. Fig. shows one line of this strtegy. For 7 7, 8 8, nd 9 9ords, Jing Yng found strtegies y hnd [,,5,6]. Lter,Hywrdet l. found other 7 7 strtegies y omputer [,5], while Noshit found 7 7 strtegies nd one 8 8 strtegy similr to Yng s y hnd [0]. For ords 0 0 or lrger, no winning strtegies re known. s the serh for winning strtegies on lrger ords ontinues, it is of interest to provide lgorithms for verifying strtegy orretness. Reently, Noshit desried strtegies in mnner tht rguly filittes humn verifition [0]. By ontrst, in this pper we present system tht llows for omputer verifition. To demonstrte the utility of our system, we use it to onfirm the orretness of Yng s originl 7 7 strtegy []. Exised Trees nd utotrees The underlying feture of our verifition system is the ondensed tree nottion we use to represent strtegies. Our nottion llows the stndrd tree desription of strtegy to e ondensed in three wys. First, it permits the use of n nd opertion orresponding to the omintoril sum of independent sustrtegies. Seond, it permits the use of mro desriptor for representing repetedly ourring sustrtegies. Third, it llows ll opponent moves to e exised from the tree y repling eh set of opponent responses with single nonymous met-response. The first two of these three ides re well known; for exmple, they were used y Yng in his desription of his proofs [,,5,6]. The third ide, nmely using exised trees, is new. In the rest of this setion we illustrte the exision proess nd show tht it does not hmper verifition. To egin, onsider the first-plyer strtegy tree in Fig.. The nodes t even depth indite first-plyer moves; the nodes t odd depth indite seond-plyer moves; the gme in Fig. follows one root-to-lef pth through the tree. Notie tht the first-plyer strtegy desried y the tree is omplete: fter eh seondplyer move, there is unique first-plyer response; fter eh first-plyer move, This nottion ould lso e used for other two-plyer ord gmes in whih gme piees re fixed one they hve een pled.
R.B. Hywrd, B. nderson, nd P. Henderson Fig.. The tree otined from the strtegy tree in Fig. y repling eh set of opponent response nodes with single met-node (top), nd the exised tree otined y then repetedly merging identil sutrees (ottom) there is every possile seond-plyer response. lso, eh lef node estlishes first-plyer win, so this is winning strtegy for the first plyer. Next, onsider the two trees shown in Fig.. The top tree is otined from the tree in Fig. y exising nodes orresponding to seond-plyer moves; eh set of seond-plyer moves is repled with single met-node, indited in our digrms y dot (). The ottom tree is otined from the top tree y repetedly merging identil sutrees into single sutree until, for eh node, ll sutrees re distint. We refer to the ottom tree s n exised tree. More generlly, given ny omplete (ut not neessrily winning) strtegy tree, the following proess, whih we ll exision, reples the tree with n exised tree. For eh non-lef first-plyer node, merge the hildren into single metnode. Next, s long s some seond-plyer node hs two identil sutrees, remove one of these sutrees. Exised trees represent equivlene lsses of strtegies, so some informtion is lost when strtegy tree is repled with its exised tree. However, exision n e reversed in the following sense: for ny exised tree E for plyer, there is set S of strtegy trees suh tht E is the exised tree of every tree in S. Furthermore, it is esy to onstrut ll elements of S from E vi the following proess, whih we ll restortion: t eh met-node m, for eh possile opponent move to ell, selet for the plyer s responding move ny ell r tht is the root of sutree of m in whih does not pper.
utomti Strtegy Verifition for Hex 5 For exmple, onsider the restortion proess for the exised tree shown t the ottom of Fig.. Strt with the top-most met-node m, nmely the hild of. For this ord position, the ell set of possile opponent moves is {,,,,,,, }. Consider the first suh ell,. The ell sets of the sutrees of m re {,,}, {,,,,}, nd{,,}. Sine is not in the first or third of these three ell sets, we n selet the root of either the first or third sutree of m. Let us ssume in this exmple tht we lwys selet the root of the first ville sutree. Thus, s the response to we selet the root of the first sutree, nmely. Continuing in this fshion, we selet s the response for opponent moves to,,, nd, nd we selet sthe response for opponent moves to,, nd. Hving seleted ll responses to m, we ontinue in top-down order to proess met-nodes until ll suh nodes hve een delt with nd the exised tree hs een repled with omplete strtegy tree S of S. Notie tht S is different from the strtegy tree S of Fig. from whih E ws derived; for exmple, in the restortion proess we never seleted the root of the third sutree of m s response to n opponent move. However, y repeting the restortion proess one for eh of the possile permuttions of hoies for r, we would onstrut ll possile strtegy trees ssoited with E, inluding S. In the restortion proess it will lwys e possile to find t lest one vlue of r t eh met-node s long s the exised tree eing restored ws otined from omplete strtegy tree. This follows from Oservtion, whih in turn follows from the ft tht in Hex, stones never move one plyed. With respet to strtegy, π-move is move mde y plyer π. With respet to strtegy tree, π-node is node ssoited with π-move, nd π-node is node ssoited with π s opponent. Oservtion. Let T e omplete Hex strtegy tree for plyer π, letp e π-node of T tht is not lef, let S,...,S k e the sutrees of T rooted t the hildren of p, ndforehs j let P j e the set of ells ssoited with the π-nodes of S j. Then the omined intersetion I = P... P k is empty. Proof. For eh index j, letq j e the ell ssoited with the root of S j. T is omplete, so Q = {q,...,q t } orresponds to ll possile opponent responses to p, nmely ll the unoupied ells fter the move p. lso, for eh index j, q j is oupied y n opponent s stone nd so is not in P j, nd so is not in I. ThusI is empty. The following is orollry of the preeding oservtion. Oservtion. Let E(T ) e the exised tree otined from omplete Hex strtegy tree T for plyer π, letm e met-node of E(T ) tht is not lef, let S,...,S k e the sutrees of E(T ) rooted t the hildren of m, ndforeh S j let P j e the set of ells ssoited with the π-nodes of S j. Then the omined intersetion I = P... P k is empty. We refer to the lss of trees tht we use in our verifition system s utotrees ; we use this term sine suh trees mke expliit mention only of plyer s own
6 R.B. Hywrd, B. nderson, nd P. Henderson moves. utotrees hve the sme form nd funtion s exised trees; however, they my not hve risen vi exision, nd so we do not define them with respet to exision. n utotree is defined s follows: eh node t one set of lternting levels is speil node lled met-node; eh node t the other set of lternting levels is leled with ord ell. We ll n utotree elusive if it stisfies the onditions of Oservtion. Notie tht restortion genertes omplete strtegy tree from n utotree if nd only if the utotree is elusive. s n initil step in our verifition lgorithm, we hek whether the input utotree is elusive. The seond nd finl step in our verifition lgorithm is to determine whether the strtegies ssoited with the input utotree re winning. We ll n utotree of plyer stisfying if, for every lef, the ells of the rootto-lef pth stisfy the onditions of win, nmely join the plyer s two sides on the Hex ord. n elusive utotree represents winning strtegy if nd only if the utotree is stisfying. This follows from the following theorems, whih in turn follow y strightforwrd rguments from our definitions nd the disussion to this point; we omit the detils of the proofs. Theorem. For Hex, for ny omplete strtegy tree there is unique ssoited elusive exised tree, nd for ny elusive utotree there is unique set of ssoited omplete strtegy trees. Furthermore, for ny omplete strtegy tree S nd the exised tree E(S) derived from S, S is winning if nd only if ll strtegy trees S reted vi restortion from E(S) re winning. Theorem. n utotree represents winning strtegy if nd only if the utotree is elusive nd stisfying. nd/or utotrees with Lef Ptterns To omplete the desription of our nottion, we need only to desrie how we dd two fetures to utotrees: nd-nodes nd lef ptterns. Notie tht the hildren of met-node in n utotree orrespond to n or deision in strtegy; depending on the opponent s move t the met-node, the plyer will ply the strtegy orresponding to the first sutree, or the next sutree, or the next sutree, nd so on; see the exised tree in Fig.. By ontrst, in Hex s in mny other gmes, prtiulr strtegy often deomposes into two or more independent sustrtegies tht eh need to e followed. Suh nd opertions re esily inorported into our nottion y llowing eh leled node (nmely, not met-node) of modified utotree to hve ny numer of hildren. We refer to utotrees tht re modified in this wy s nd/or utotrees sine, when interpreting them s strtegies, the odd depth nodes (the met-nodes) re or-nodes while the even depth nodes (with ell lels) re ndnodes. Consider for exmple Fig., whih shows n nd/or utotree for winning strtegy.therootisnnd-node, so we hve to ply ll sustrtegies simultneously; in this se, there is only one sutree so there is only one sustrtegy
utomti Strtegy Verifition for Hex 7 d d d d d d d d Fig.. n nd/or utotree for winning first-plyer Hex strtegy. Odd depth nodes () re or -nodes; even depth nodes (ell lels) re nd -nodes. Fig. 5 shows one line of this strtegy. to follow. Suppose tht the opponent s response to the plyer s initil move d is. Then the plyer n selet ny sutree not ontining, sy the first sutree; thus the plyer moves to, the root of the first sutree. This root is n nd-node with two sutrees, so now the plyer hs to follow these two sustrtegies simultneously; the plyer must ensure tht she rehes lef node in eh of the sutrees of every nd-node. For exmple, if the opponent s next move is t one of {,}, the plyer must immeditely reply with the other of these two ells or risk not rehing lef of the {,} sutree. Similrly, if the opponent s next move is t one of {,d}, the plyer must immeditely reply with the other of these two ells. If the opponent s next move is not in {,} or {,d}, the plyer n move nywhere. Fig. 5 illustrtes nother line of ply of this strtegy. Finlly, sutrees of nd/or utotrees tht orrespond to isomorphi sustrtegies n e repled with speil node orresponding to suh sustrtegies. This is illustrted in Fig. 6, where two sustrtegy mros re used to simplify the tree of Fig.. Modifying our verifition lgorithms to hndle nd- nd or-nodes is strightforwrd. For or-nodes, the test for the elusive property is the sme s with unmodified utotrees: hek whether the omined intersetion of ll hild nodes is d d 9 5 6 5 0 8 d 7 d Fig. 5. The strt (left) nd finish (right) of one line of the strtegy of Fig.
8 R.B. Hywrd, B. nderson, nd P. Henderson d d B d d B d Fig. 6. n nd/or utotree with two mro pttern nodes. This tree is equivlent to the tree in Fig. ; pttern prmeters hve een omitted. the empty set. For nd-nodes, it is neessry to hek whether the intersetion of eh pir of hild nodes is empty. nother lgorithmi pproh one might tke here is to expnd the nd/or utotree into the orresponding equivlent utotree; however, the resulting trees n e lrge, so this pproh would require signifintly more spe thn our pproh. Testing the stisfying property on nd/or utotrees involves heking every root-to-lef pth in the ssoited expnded utotree. For resons of effiieny we do not wnt to generte the expnded utotree; we thus rry out this tsk in n impliit fshion. By using simple indexing sheme for eh root-to-lef pth in the nd/or utotree, we n reonstrut the ell sets for eh possile root-to-lef pth in the ssoited utotree. Eh node stores the numer of root-to-lef pths it ontins. We onsider ll suh pths nd verify tht eh stisfies the winning ondition. We implement the isomorphi sustrtegy feture in the simplest possile wy, nmely using mro sustitution to generte the equivlent nd/or utotree. Verifying Yng s Proof s enhmrk for testing our system, we used it to verify the first known winning 7 7 Hex strtegy, nmely Yng s originl 7 7 enter-opening strtegy [,]. Yng desried his strtegy in n esily understood nottion similr to tht used in the C progrmming lnguge; n pplet tht follows this strtegy is ville on his homepge[]. The version of the strtegy tht we tested is from preprint lso ville from his we pge []. In Yng s nottion, his strtegy uses out 0 ptterns (not ounting pttern vritions) omprising out six pges of text. reursion tree inditing the hierrhy of his ptterns is shown in Fig. 7. For exmple, n nd-node with k sutrees of two nodes eh orresponds in the expnded utotree to node with k sutrees.
utomti Strtegy Verifition for Hex 9 + + + + + + + + 9 56578 9055+ + + + + 0 5 90+ 5657890 8 596 9799 677996666 + 9 5+ + + + 9+ 0 56 7 8 5+ 66779866 665+ + 666 0 + 7 795796768 5 8 9 Fig. 7. Prt of the reursion tree for Yng s proof. Referenes to frequently ourring smll ptterns hve een omitted. Lels indite pttern numers. Nodes leled + re nd-nodes; ll other nodes re or-nodes. 6 ( pttern8 // lled y: ((6 BR) (d BR)) (d6 e e e5 e6 f f f f5 f6 g g g g g5 g6) (6 d BR) [(f [(pttern (e e) (d f))] [(pttern (g g) (f BR))]) (e5 [(d6) (e)] [(pttern (e6 f f5 f6 g g g5 g6) (e5 BR))]) (f [(pttern (g g) (f BR))] [(pttern9 (g5 g f5 f f e5 e e) (BR f d))]) (e [(pttern7 (d6 e5 e6 f f f f5 g g g g g5) (6 d e BR))]) ]) Fig. 8. Yng s Pttern 8 in our nottion We trnslted Yng s proof into our nottion y hnd, following his pttern nming onvention. s n exmple of our nottion, see Fig. 8. The first line gives the nme of the pttern. The seond line is omment noting tht the only pttern lling this pttern is Pttern. The third line gives the onnetions tht re hieved y the pttern; in this se t lest one of two onnetions is hieved, either etween 6 nd the ottom right side of the ord, or etween d nd the ottom right side; this informtion is given only to id in humn deugging purposes nd is not used y our lgorithm. The fourth line lists the ells tht
0 R.B. Hywrd, B. nderson, nd P. Henderson pttern onnet: (TL BR) empty: ( 5 6 7 5 6 7 5 6 7 d d d d5 d6 d7 e e e e e5 e6 e7 f f f f f5 f6 f7 g g g g g5 g6 g7) plyed: (TL d BR) stts: ND = 80, OR = 9, Lefs = 5 pths: 557/557 VLID pttern. Fig. 9. Dignostis returned fter verifying Yng s proof must e unoupied t this point; the fifth line lists the ells tht the plyer must lredy oupy. The susequent lines desrie the nd/or utotree, where prentheses surround the sutrees of n or-node nd squre rkets surround the sutrees of n nd-node. In the proess of verifying the desription of Yng s proof, we found only one typogrphil error: in the desription of Pttern there is ll to Pttern 7 tht should insted e ll to Pttern 9. Our nottion represents Yng s strtegy in out 700 lines of text. The dignosti messge returned y our progrm fter reursively verifying Yng s proof is shown in Fig. 9. The resulting tree hd,80 nd-nodes,,9 or-nodes,,5 leves, nd 5,57 impliit root-to-lef pths. The verifition took less thn one seond to exeute on our omputer, single-proessor thlon6 00+ with gigyte of memory. 5 Conlusions We hve introdued the notion of n exised tree s ompressed representtion of omplete strtegy tree from whih ll expliit opponent moves hve een exised. We used exised trees in simple lgorithm tht verified the orretness of Yng s originl winning 7 7 Hex strtegy. One wy in whih our system ould e improved would e to utomte the proess of trnslting strtegies from other nottions into our nottion. nother improvement onerns the numer of pths tht our lgorithm heks in verifying the orretness of strtegy. Currently our system expliitly verifies tht every possile ell set tht plyermightendupwithontinswinning pth. For exmple, for Yng s strtegy this ws totl of 5,57 ell sets tht were heked. The prolem with this pproh is tht the numer of suh ell sets, orresponding to the numer of root-to-lef pths in the omplete strtegy tree, inreses exponentilly in the ord size. Consider for exmple Mrtin Grdner s winning seond-plyer strtegy for the plyer with the longer sides on n n n ord []. The strtegy onsists of the nd of f(n) =n (n )/ sustrtegies eh onsisting of the or of two moves. The ssoited exised tree thus hs f(n) root-to-lef pths. Even for n s smll s, f(n) = 9, nd heking this mny pths individully is urrently omputtionlly infesile.
utomti Strtegy Verifition for Hex Thus, s ord size inreses, verifition lgorithms will e required tht do not expliitly hek the winning ondition for eh root-to-lef pth. Referenes. Grdner, M.: Mthemtil Gmes. Sientifi merin 97, July, pp. 5 50, ugust, pp. 0 7, Otoer, pp. 0 8 (957). Grdner, M.: The Sientifi merin Book of Mthemtil Puzzles nd Diversions, pp. 7 8. Simon nd Shuster, New York (959). Hywrd, R.B., vn Rijswijk, J.: Hex nd Comintoris (formerly Notes on Hex). Disrete Mthemtis 06, 55 58 (006). Hywrd, R.B., Björnsson, Y., Johnson, M., Kn, M., Po, N., vn Rijswijk, J.: Solving 7 7 Hex: Virtul Connetions nd Gme-stte Redution. In: vn den Herik, H.J., Iid, H., Heinz, E.. (eds.) dvnes in Computer Gmes (CG0), Mny Gmes, Mny Chllenges, pp. 6 78. Kluwer demi Pulishers, Boston (00) 5. Hywrd, R.B., Björnsson, Y., Johnson, M., Kn, M., Po, N., vn Rijswijk, J.: Solving 7 7 Hex with Domintion, Fill-in, nd Virtul Connetions. Theoretil Computer Siene 9, 9 (005) 6. Mrup, Th.: Hex Everything You lwys Wnted to Know out Hex ut Were frid to sk. Mster s thesis, Deprtment of Mthemtis nd Computer Siene, University of Southern Denmrk, Odense, Denmrk (005) 7. Mrup, Th.: Hex Wepge (005), http://mrup.net/thoms/hex/ 8. Nsr, S.: Beutiful Mind. Touhstone, New York (998) 9. Nsh, J.: Some Gmes nd Mhines for Plying Them. Tehnil Report D-6, Rnd Corp. (95) 0. Noshit, K.: Union-Connetions nd Strightforwrd Winning Strtegies in Hex. ICG Journl 8(), (005). Reish, S.: Hex ist PSPCE-vollständig. t Informti 5, 67 9 (98). Yng, J.: Jing Yng s We Site (00), http://www.ee.umnito./ jingyng. Yng, J., Lio, S., Pwlk, M.: Deomposition Method for Finding Solution in Gme Hex 7x7. In: Interntionl Conferene on pplition nd Development of Computer Gmes in the st Century, pp. 96 (Novemer 00). Yng, J., Lio, S., Pwlk, M.: On Deomposition Method for Finding Winning Strtegy in Hex Gme (00), http://www.ee.umnito./ jingyng/hexsol.pdf 5. Yng, J., Lio, S., Pwlk, M.: nother Solution for Hex 7x7. Tehnil report, University of Mnito, Winnipeg, Cnd (00), http://www.ee.umnito./ jingyng/tr.pdf 6. Yng, J., Lio, S., Pwlk, M.: New Winning nd Losing Positions for 7x7 Hex. In: Sheffer, J., Müller, M., Björnsson, Y. (eds.) CG 00. LNCS, vol. 88, pp. 0 8. Springer, Heidelerg (00)