Playing Tic-Tac-Toe Using Genetic Neural Network with Double Transfer Functions

Transcription

1 Journa of Integent Learnng Sytem an Aaton, 0, 3, o:0.436/a Pubhe Onne February 0 (htt:// 37 Payng T-Ta-Toe Ung Genet Neura Network wth Doube Tranfer Funton Sa Ho Lng, Hak Keung Lam Fauty of Engneerng an Informaton Engneerng, Unverty of Tehnoogy Syney, New South Wae, Autraa; Dvon of Engneerng, Kng Coege Lonon, Lonon, UK. Ema: Steve.Lng@ut.eu.au, hak-keung.am@k.a.uk Reeve Otober 4 th, 00; reve November 3 r, 00; aete January 4 th, 0 ABSTRACT Comutatona ntegene a owerfu too for game eveoment. In th aer, an agorthm of ayng the game T-Ta-Toe wth omutatona ntegene eveoe. Th agorthm earne by a Neura Network wth Doube Tranfer funton (NNDTF), whh trane by genet agorthm (GA). In the NNDTF, the neuron ha two tranfer funton an exhbt a noe-to-noe reatonh n the hen ayer that enhane the earnng abty of the network. A T-Ta-Toe game ue to how that the NNDTF rove a better erformane than the tratona neura network oe. Keywor: Game, Genet Agorthm, Neura Network. Introuton Game uh a Bakgammon, Che, Cheker, Go, Otheo an T-Ta-Toe are wey ue atform for tuyng the earnng abty of an eveong earnng agorthm for mahne. By ayng game, the mahne ntegene an be reveae. Some tehnque of artfa ntegene, uh a the brute-fore metho an knowege-bae metho [], were reorte. Brutefore metho, e.g. retrograe anay [] an enhane tranoton-tabe metho, ove the game robem by ontrutng atabae for the game. For ntane, the atabae forme by a termna oton []. The bet move then etermne by workng bakwar on the ontrute atabae. For knowege-bae metho, the bet move etermne by earhng a game tree. For game uh a Cheker, the tree annng very arge. Tree earhng w be tme onumng even for a few e. Hene, an effent earhng agorthm an mortant ue. Some earhng agorthm, whh are afe a knowege-bae metho, are threat-ae earh an -earh, roof-number earh [3], eth-frt roof-number earh an attern earh. It an be een that the above game-ovng metho e many on the atabae ontruton an earhng. The robem are ove by formng a obe et of outon bae on the game onton, or earhng for the et of outon bae on the urrent game onton. The mahne annot earn to ay the game by tef. Unke an evoutonary aroah n [], neura network (NN) wa emoye to evove an to earn for ayng T-Ta-Toe wthout the nee of a atabae. Evoutonary rogrammng wa ue to egn the NN an nk weght. A mar ea ha been ae n a Cheker game [4-7]. Other game uh a Bakgammon [8], Otheo [9] an Cheker [0] ayng NN or omutatona ntegene tehnque an ao be foun. In th aer, a neura network wth oube tranfer funton (NNDTF) rooe to earn the rue of T-ta-toe. Eah obe move evauate by a rooe agorthm wth a ore. By maxmzng the tota ore (evauate vaue), the rue of T-Ta-Toe an be extrate by the NNDTF. Dfferent from the tratona fee-forwar mute-ereton NN, ome mofe tranfer funton wth a noe-to-noe reatonh are ntroue to the rooe NN. The mofe tranfer funton are aowe to hange the hae urng oeraton. Hene, the workng oman arger than that of the tratona one. By ntroung the noe-to-noe reatonh between hen noe, nformaton an be exhange between hen ayer. A a reut, the earnng abty enhane. A genet agorthm (GA) [] nvetgate to tran the NNDTF. The trane NNDTF w then be emoye to ay T-Ta-Toe wth a human ayer a an exame. Coyrght 0 SRe.

2 38 Payng T-Ta-Toe Ung Genet Neura Network wth Doube Tranfer Funton Th aer organze a foow. An agorthm to evauate eah move on ayng the T-Ta-Toe w be rooe n eton. NNDTF w be reente n eton 3. Genet Agorthm w be reente n eton 4. Tranng of the NNDTF ung GA to earn the rue of T-Ta-Toe w be reente n eton 5. An exame on ayng T-Ta-Toe w be gven n eton 6. A onuon w be rawn n eton 7.. Agorthm for Payng T-Ta-Toe The game T-ta-toe, ao known a naught an roe, a two-ayer game. Eah ayer w ae a marker, X for the frt ayer an O for the eon ayer, n turn n a three-by-three gr area. The frt ayer take the frt move. The goa to ae three marker n a ne of any reton on the gr area. An agorthm rooe n th eton to evauate the move on eah gr. An X an an O on a gr are enote by an reetvey. An emty gr enote by 0.5. The foowng roeure ue to evauate eah obe move. ) Pae an X on an emty gr. ) Correonng to te, um u a the gr vaue for eah ne n any reton, e.g., for a gr n the orner, we have three evauate vaue beaue there are three ne to wn or oe. 3) Remove the X ae n te an ae an X on another emty gr. Evauate th gr ung the agorthm n te. Reeat th roe t a emty gr are evauate. 4) After evauaton, eah gr w have been agne at eat evauate vaue for a obe ne, e.g. the enter gr w have 4 evauate vaue, orner gr w have 3 evauate vaue an other gr w have evauate vaue. There are totay 6 obe evauate vaue: 3 ( + + ),.5 ( ), ( ), ( + + ), 0.5 ( ) an ( + ). The mot mortant evauate vaue of a gr 3, whh nt a wnnng of the game (3 X n a ne) f you ut an X on that gr. The rorty of takng that move the hghet. The eon mortant evauate vaue of a gr ( O an X n a ne), whh nt that the oonent hou be revente from wnnng the game. The rorty of takng that move the eon hghet. Ung th ratonae, the t of rorty n a eng orer : 3,,.5,,, 0.5. Bae on thee agne evauate vaue, a ore w be agne to eah obe move. Frt, eah evauate vaue agne a ore: , 5 6, , 3 4, 3 3 an 05., The hoen ore have the foowng roerte, 4 () () (3) 3 4 (4) 4 (5) The um of the ore of a gr the fna ore. The fna ore w be ue to etermne the rorte of the obe move. A hgher fna ore of a gr nate a hgher rorty of that move. The reaon for hoong the ore n th way wth the roerte of () to (5) are a foow. A the evauate vaue of 3 nate a wnnng of the game (3 X n a ne), the ore of 6 mut be the hghet. There are at mot four evauate vaue for a gr. Hene, 6 mut be greater than 4 tme the eon arget evauate vaue,.e Conequenty, the rorty of a gr havng an evauate ore wth a hgher rorty w not be affete by other ower evauate ore. For ntane, oner a gr havng evauate vaue of 3 an 0.5, an another gr havng evauate vaue of,.5, an 0.5. The fna ore of the former gr (7 7 + ) bgger than an atter gr (7 7 + an ). Thu, the X hou be ae at the gr havng an evauate vaue of 3 to wn the game. Take the game a hown n Fgure a an exame, we have 3 X an 3 O. The next move w be to ae an X. After agnng an emty gr to be 0.5, an X to be, an O to be, Fgure (b) obtane. Foowng Ste to Ste 3, we obtan the evauate vaue a hown n Fgure (). Bae on Ste 4, Fgure () how the fna ore for the emty gr. A the hghet ore 87334, the mot arorate move to ut an X on the bottom rght orner. Th move not (a) ,.5,.5 (b).5, () () Fgure. Evauaton roe. 3, -,.5 Coyrght 0 SRe.

3 Payng T-Ta-Toe Ung Genet Neura Network wth Doube Tranfer Funton 39 ony ne u 3 X to wn a game, but ao revent the oonent to ne u 3 O. The eon arorate move, nte by the fna ore of 5906, an gan a hane to wn by nng u X, an revent the oonent to wn. 3. Neura Network wth Doube Tranfer Funton (NNDTF) NN wa rove to be a unvera aroxmator []. A 3-ayer fee-forwar NN an aroxmate any nonnear ontnuou funton to an arbtrary auray. NN are wey ae n area uh a reton, ytem moeng an ontro []. Owng to t artuar truture, a NN goo n earnng [] ung ome earnng agorthm uh a GA [] an bak roagaton []. In genera, the roeng of a tratona fee-forwar NN one n a ayer-by-ayer manner. In th aer, by ntroung a noe-to-noe reatonh n the hen ayer of the NN, a better erformane an be obtane. Fgure how the rooe neuron. It ha two atvaton tranfer funton to govern the nut-outut reatonh of the neuron: tat tranfer funton (STF) an ynam tranfer funton (DTF). For the STF, the arameter are fxe an t outut e on the nut of the neuron. For the DTF, the arameter of the atvaton tranfer funton e on the outut of other neuron an t STF. Wth th rooe neuron, the onneton of the rooe NN hown n Fgure 3, whh a three-ayer NN. A noe-to-noe reatonh ntroue n the hen ayer. Comarng wth the tratona fee-forwar NN [], t wa reorte n [3] that the rooe NN an offer a better erformane an nee fewer hen noe. The eta of the NNDTF are reente a foow. 3.. The Neuron Moe We oner the STF frt. Let v be the ynat onneton weght from the -th nut omonent x to the -th neuron. The outut of the -th neuron STF efne a, x v v x v nn x nn, z STF net () net () DTF th neuron z, Fgure. Moe of the rooe neuron. z x x x nn v nh v nn v v v nn v n n n h n h nh z z n h nh nh z nh nh n h w w w nh w nout w n h n out w nout Fgure 3. Conneton of the NNDTF. n n net xv (6) where,,, n n,,,, n h ; nn enote the number of nut an net a tat atvaton tranfer funton. The atvaton tranfer funton efne a, n n xv m nn xv n n xv m e f xv m net e otherwe (7) where m an are the tat mean an tat tanar evaton for the -th STF reetvey. The arameter ( m an ) are fxe after the tranng roeng. Thu, the atvaton tranfer funton tat. The outut of the STF e on the nut of the neuron ony. From (7), the outut vaue range from to. The hae of the rooe atvaton tranfer funton hown n Fgure 4 an Fgure 5. It an be oberve from thee fgure that net f a f. nn y y nout net f a f an Conerng the DTF, the neuron outut z of the -th neuron efne a, z net,m, (8) where net the DTF efne a foow, m e f m net,m, m e otherwe (9) Coyrght 0 SRe.

4 40 Payng T-Ta-Toe Ung Genet Neura Network wth Doube Tranfer Funton net m=0. m=0 m=-0. m=-0.4 m= f Fgure 4. Same tranfer funton of the rooe neuron ( = 0.). net f Fgure 5. Same tranfer funton of the rooe neuron (m = 0). where m z (0),, z () m an are the ynam mean an ynam tanar evaton for the -th DTF. z an z rereent the outut of the -th an -th neuron reetvey., enote the weght of the nk between the -th noe an the -th noe an, enote the weght of the nk between the -th noe an the -th noe. It hou be note that f,, equa to n h,an f nh,, equa to,. In th DTF, unke the STF, the atvaton tranfer funton ynam a the arameter of t atvaton tranfer funton e on the outut of the -th an -th neuron. Referrng to (4), the nut-outut reatonh of the rooe neuron a foow, n n z net net xv,m, () 3.. Conneton of the NNDTF A hown n Fgure 3, the NNDTF ha three ayer wth n n noe n the nut ayer, n h noe n the hen ayer, an n out noe n the outut ayer. In the hen ayer, the neuron moe reente n the revou eton emoye. The outut vaue of the hen noe e on the neghborng noe an nut noe. In the outut ayer, a tat atvaton tranfer funton emoye. Conerng an nut-outut ar ( x, y), the outut of the -th noe of the hen ayer gven by n n z net net xv (3) The outut of the NNDTF efne a, n out y neto zw,,,, n out (4) nout n n neto net net xv w (5) where w enote the weght of the nk between the -th hen an the -th outut noe; neto enote the atvaton tranfer funton of the outut neuron. The tranfer funton of the outut noe efne a foow, zk mo o e f zk mo netoz (6) zk mo o e otherwe where mo an o are the mean an the tanar evaton of the outut noe atvaton tranfer funton reetvey. The arameter of the NNDTF an be trane by GA []. 4. Genet Agorthm Genet agorthm (GA) are owerfu earhng agorthm. The tratona GA roe [4-6] hown n Fgure 6. Frt, a ouaton of hromoome reate. Seon, the hromoome are evauate by a efne ftne funton. Thr, ome of the hromoome are eete for erformng genet oeraton. Forth, genet oeraton of roover an mutaton are erforme. The roue offrng reae ther arent n the nta ouaton. Th GA roe reeat unt a uer-efne rteron reahe. In th aer, the tratona GA mofe an new genet oerator [] are ntroue to mrove t erformane. The mofe GA roe hown n Fgure 7. It eta w be gven a foow. 4.. Inta Pouaton The nta ouaton a otenta outon et P. The Coyrght 0 SRe.

5 Payng T-Ta-Toe Ung Genet Neura Network wth Doube Tranfer Funton 4 Proeure of the me GA begn 0 // : teraton generaton ntaze P() //P(): ouaton for teraton t evauate f(p()) // f(p()):ftne funton whe (not termnaton onton) o begn + eet arent an from P(-) erform genet oeraton (roover an mutaton) reroue a new P() evauate f(p()) Fgure 6. Proeure of me GA. Proeure of the mrove GA begn 0 // : teraton ntaze P() //P(): ouaton for teraton t evauate f(p()) // f(p()):ftne funton whe (not termnaton onton) o begn + eet arent an from P(-) erform roover oeraton aorng (3) to (8) erform mutaton oeraton aorng to (30) to three offrng no, no an no 3 // reroue a new P() f ranom number < a The one among no, no an no 3 wth the arget ftne vaue reae the hromoome wth the maet ftne vaue n the ouaton ee f f(no ) > maet ftne vaue n the P(-) no reae the hromoome wth the maet ftne vaue f f(no ) > maet ftne vaue n the uate P(-) no reae the hromoome wth the maet ftne vaue f f(no 3 ) > maet ftne vaue n the uate P(-) no 3 reae the hromoome wth the maet ftne vaue evauate P() Fgure 7. Proeure of the mofe GA. frt et of ouaton uuay generate ranomy. P,,, (7) o _ ze no_var,,,, o _ ze;,,,no _ var (8) ara ara (9) mn max where o_ze enote the ouaton ze; no_var enote the number of varabe to be tune;,,,, o_ze;,,, no_var, are the arameter to be tune; ara mn an ara max are the mnmum an maxmum vaue of the arameter for a. It an be een from (7) to (9) that the otenta outon et P ontan ome anate outon (hromoome). The hromoome ontan ome varabe (gene). 4.. Evauaton Eah hromoome n the ouat on w be evauate by a efne ftne funton. The better hromoome w return hgher vaue n th roe. The ftne funton to evauate a hromoome n the ouaton an be wrtten a, ftne f (0) The form of the ftne funton e on the aaton Seeton Two hromoome n the ouaton w be eete to unergo genet oeraton for rerouton by the metho of nnng the rouette whee [6]. It beeve that hgh otenta arent w roue better offrng (urvva of the bet one). The hromoome havng a hgher ftne vaue hou therefore have a hgher hane to be eete. The eeton an be one by agnng a robabty q to the hromoome : q f o _ ze f,,,, o _ ze () The umuatve robabty ˆq for the hromoome efne a, ˆq q () The eeton roe tart by ranomy generatng a nonzero foatng-ont number, 0. Then, the hromoome hoen f qˆ ˆ q,,,, o_ze, an q ˆ0 0. It an be oberve from th eeton roe that a hromoome havng a arger f w have a hgher hane to be eete. Conequenty, the bet hromoome w get more offrng, the average w tay an the wort w e off. In the eeton roe, ony two hromoome w be eete to unergo the genet oeraton Genet Oeraton The genet oeraton are to generate ome new hromoome (offrng) from ther arent after the eeton roe. They nue the roover an the mutaton oeraton. Coyrght 0 SRe.

6 4 Payng T-Ta-Toe Ung Genet Neura Network wth Doube Tranfer Funton Croover The roover oeraton many for exhangng nformaton from the two arent, hromoome an, obtane n the eeton roe. The two arent w roue one offrng. Frt, four hromoome w be generate aorng to the foowng mehanm, o o o o no_var (3) o o o o no_var (4) w max, w o max o o o w, w no_var mn mn (5) o o o o no_var max mn w w (6) no _ var max aramax aramax ara max (7) no _ var mn aramn aramn ara mn (8) where w 0 enote a weght to be etermne by uer, max, enote a vetor wth eah eement obtane by takng the maxmum among the orreonng eement of an. For ntane, max 3, Smary, mn, gve a vetor by takng the mnmum vaue. For ntane, mn 3, 3. Among 4 o to o, the one wth the arget ftne vaue ue a the offrng of the roover oeraton. The offrng efne a, o o o o o no _ var o (9) where o enote the nex whh gve a maxmum vaue of f o, 34,,,. If the roover oeraton an rove a goo offrng, a hgher ftne vaue an be reahe n e teraton. A een from (3) to (6), the offrng rea over the oman: (3) an (6) w move the offrng near entre regon of the onerne oman (a w n (6) a- 4 roahe, o aroahe ), an (4) an (5) w move the offrng near the oman bounary (a w 3 n (4) an (5) aroahe, o an o aroahe max an mn reetvey). The hane of gettng a goo offrng thu enhane Mutaton The offrng (30) w then unergo the mutaton oeraton. The mutaton oeraton to hange the gene of the hromoome. Conequenty, the feature of the hromoome nherte from ther arent an be hange. Three new offrng w be generate by the mutaton oeraton: no o o o no_var (30) bno bno bno _ var no no _ var where b,,,, no _ var, an ony take the vaue of 0 or ; no,,,, no _ var, are ranomy generate number uh that aramn o no ara max. The frt new offrng ( ) obtane aorng to (30) wth that ony one b ( beng ranomy generate wthn the range) aowe to be an a the other are zero. The eon new offrng obtane aorng to (30) wth that ome ranomy hoen b are et to be an other are zero. The thr new offrng obtane aorng to (30) wth a b. Thee three new offrng w then be evauate ung the ftne funton of (). A rea number w be generate ranomy an omare wth a uer-efne number a 0. If the rea number maer than a, the one wth the arget ftne vaue f among the three new offrng w reae the hromoome wth the maet ftne f n the ouaton (even when f f.) If the rea number arger than a, the frt offrng w reae the hromoome wth the maet ftne vaue f n the ouaton f f f ; the eon an the thr offrng w o the ame. a effetvey the robabty of aetng a ba offrng n orer to reue the hane of onvergng to a oa otmum. Hene, the obty of reahng the goba otmum ket. We have three offrng generate n the mutaton roe. From (30), the frt mutaton ( ) n fat a unform mutaton. The eon mutaton aow ome ranomy eete gene to hange mutaneouy. The thr mutaton hange a gene mutaneouy. The eon an the thr mutaton aow mute gene to be hange. Hene, the oman to be earhe arger a omare wth a oman haraterze by hangng a nge gene. A three offrng are roue n eah generaton, the gene w have a arger ae for mrovng the ftne vaue when the ftne vaue ma. When the ftne vaue are arge an neary teay, hangng the vaue of a nge gene (the frt mutaton) may be enough a ome gene may have reahe the otma vaue. After the oeraton of eeton, roover, an mutaton, a new ouaton generate. Th new ouaton w reeat the ame roe. Suh an teratve roe an be termnate when the reut reahe a efne onton, e.g. a efne number of teraton have been Coyrght 0 SRe.

7 Payng T-Ta-Toe Ung Genet Neura Network wth Doube Tranfer Funton 43 reahe. 5. Tranng of the NNDTF In th eton, the GA w be emoye to tran the arameter of the NNDTF to ay T-Ta-Toe bae on the gamng agorthm n Seton. The NNDTF wth 9 nut an outut emoye. The gr are numbere from to 9 from rght to eft an from to to bottom. An X on the gr enote by, an O enote by, an an emty gr enote by 0.5. The gr attern rereente by numera vaue w be ue a the nut of the NNDTF. The outut of the NNDTF ( y t whh a foatng ont number range from to 9) rereent the oton of the marker that hou be ae on. In orer to have a ega move (ae a marker on an emty gr), the marker ae on an emty gr that ha t gr number oet to the outut of the network. To erform the tranng, we have to etermne the arameter to be trane an the ftne funton erbng the robem obetve. The arameter of the mofe network to be turne v m,, w mo o for a,,, whh w be hoen a the hromoome for the GA. 00 fferent tranng attern (obtane bae on the rooe gamng agorthm tate n Seton ) are ue to fee nto the NNDTF for tranng. The ftne funton egne a foow, ftne 00 m 00 max S y t, x t S x t (3) where y t enote the outut of the NNDTF wth the t-th tranng attern x t a the nut, Sm yt, x t enote the fna ore for gr y t an the t-th tranng attern x t bae on the gamng agorthm. Smax x t enote the maxmum fna ore vaue among a the emty gr for the t-th tranng attern x t. The GA to maxmze the ftne vaue (range from 0 to ) o a to fore the outut of the NNDTF to the gr number havng the arget fna ore to enure the bet move. 6. Exame In th eon, a 9-nut--outut NNDTF ue for tranng. The number of hen noe hoen to be tranng attern are ue for tranng wth teraton. The ouaton ze, robabty of aetane, an w are hoen to be 0, 0.5 an 0., reetvey. After tranng, the ftne vaue obtane The uer an ower boun of eah arameter are an, Tabe. Reut of the rooe NN ayng T-Ta-Toe agant wth the tratona NN for 50 game. Prooe aroah move frt Tratona aroah move frt Prooe NN Wn: Draw: Loe 8: 3: 4 3: 4: 8 reetvey. The nta vaue of the arameter are generate ranomy. For omaron uroe, a tratona 3-ayer feeforwar NN [7] trane by GA wth arthmet roover an non-unform mutaton [7] ao ae uner the ame onton to earn the gamng trategy n Seton. The robabte of roover an mutaton are eete to be 0.8 an 0., reetvey. The hae arameter of the tratona GA for non-unform mutaton [7] eete to be 5. Thee arameter are eete by tra an error for the bet erformane. After tranng for teraton, the ftne vaue obtane To tet the erformane of our rooe metho, our trane NN ay T-Ta-Toe wth the trane tratona NN for 50 game arre out. The frt 5 gr attern, whh are generate ranomy wth O an X, are the ame a the next 5 gr attern. For the frt 5 game, the rooe aroah move frt. For the eon 5 game, the tratona aroah move frt. The reut are tabuate n Tabe. It an be een that the rooe aroah erform better. The number of wn 8 by ung NNDTF whe ony 3 by ung the traton NN. 7. Conuon A neura network wth oube tranfer funton an trane wth genet agorthm ha been rooe. An agorthm of ayng T-Ta-Toe ha been reente. A new tranfer funton of the neuron wth a noe-to-noe reatonh ha been rooe. The rooe neura network trane by genet agorthm to earn the agorthm of ayng T-ta-toe. A a omaron, the trane NN ha aye agant the tratona NN trane by the tratona GA. The reut ha hown that the rooe aroah erform better. REFERENCES [] H. J. V. D. Herk, J. W. H. M. Uterwk an J. V. Rwk, Game Sove: Now an n the Future, Artfa Integene, Vo. 34, No. -, 00, o:0. 06/S (0)005-7 [] H. J. V. D. Herk an I. S. Herhberg, The Contruton of an Omnent Engame Data Bae, ICCA Journa, Vo. 8, No., 985, Coyrght 0 SRe.

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