1. Constraint propagation

Transcription

1 6.034 Artificial Intelligence, Fall 2006 Prf. Patrick H.Winstn Prblem Set 3 This prblem set is due Wednesday, Octber 18th at 11:59 PM. If yu have questins abut it, ask the TA list. Yur respnse will prbably cme frm a TA. T wrk n this prblem set, yu will need t get the cde, much like yu did fr earlier prblem sets. The cde can be dwnladed frm the assignments sectin. Yur answers fr the prblem set belng in the main file ps3.scm. Things t remember Avid using DrScheme's graphical cmment bxes. If yu d, take them ut r use Save definitins as text... s that yu submit a Scheme file in plain text. If yu are ging t submit yur pset late, see the prblem set grading plicy. 1. Cnstraint prpagatin 1.1. Sudku In this prblem yu will write a prgram t slve Sudku puzzles. This kind f puzzle has 81 cells arranged in a square grid which is subdivided int nine smaller squares. Each rw, clumn, and small square must cntain the digits 1 9. There is a graphic that ges with this part f the prblem. T ensure that it runs crrectly, d the fllwing in DrScheme: Open the Language menu. Select Chse Language... Under PLT, chse the language Graphical (MrEd, includes MzScheme). Click OK. Run the tester and make sure that the Sudku windw appears. In general, Sudku puzzles may require search in additin t cnstraint prpagatin. In this prblem, hwever, yu will never encunter a Sudku bard which cannt be slved with cnstraint prpagatin alne. There are several methds defined in sudku.scm that yu may find useful. Here are the mst relevant nes: Prf. Patrick H. Winstn Page 1 f 9

2 (get-cells bard) - get all the cell data structures frm the bard (cells-in-same-rw cell bard), as well as cells-in-same-cl and cells-in-same-square - given a cell data structure, returns the cells in the same rw, clumn, r square (cell-values cell) -- a list f values this cell can have (set-values! cell new-values) -- set the pssible values fr a cell. newvalues shuld be a list cntaining integers frm 1 t 9. (get-cell-by-number bard n): lks up a cell by its unique index number, frm 0 t 80. Yu can als use get-rw-by-number, get-cl-by-number, and get-square-by-number t get at the rw, clumn, and square data structures. Lk in sudku.scm fr mre accessrs like this if yu need them. There is als a functin t draw the bard in the graphics windw: (draw-bard bard) - draw a Sudku bard n the screen. Yu'll be wrking with just ne bard in this prblem, and changing it as yu g. T get a fresh cpy f the bard, call the prcedure (testing-bard), which takes n arguments. Yu may find this helpful t visualize the bard as yu wrk n the prblem, because the scheme-list representatin f the bard is lng. The fllwing cde snippet, fr example, sets the first cell t value 7 and redraws the bard: (define thebard (testing-bard)) (set-values! (get-cell-by-number thebard 0) '(7)) (draw-bard thebard) 1.2. Identifying cnstraints Write (can-prpagate? cell). cell is a cell data structure. This functin shuld return whether a cell in the Sudku puzzle can prpagate cnstriants nt ther cells. (There are many tactics fr slving Sudku puzzles that invlve lking at cnstraints in multiple cells, but in this prblem, yu shuld use the nly cnstraint yu can prpagate by examining a single cell.) Write (prpagate-ne-cell cell thers). cell is a cell data sructure. Others is a list f cell data structures that are either in the same rw, clumn, r square as cell. cell des nt appear anywhere in thers. Yur functin shuld mdify each cell in thers which can be mdified (i.e. cnstraints can prpagate), and return #t if cells were mdified, #f if n cells were mdified Slving the puzzle Write (prpagate-ne-rund bard). bard is a bard data structure. Yur functin shuld apply (prpagate-ne-cell) t each f the cells in the bard, and the grups Prf. Patrick H. Winstn Page 2 f 9

3 f crrespnding cells in the same clumn, rw, and square. Usefull tip: use cellsin-same-rw, etc. These methds will return a list f cells sharing the same rw, clumn r square as a cell. The returned list will nt cntain the cell argument. Write (cnstraint-prpagate bard cunt). This will be called initially with cunt=0. It shuld return the number f steps (prpagate-ne-runds) that it tk t reslve all f the cells in the puzzle t single values. The return value f this functin ught t be small (less than 10) if yur cnstraint prpagatin is wrking prperly. 2. Game search In this sectin, yu'll be wrking with a game AI that uses minimax search, then alphabeta search, t play a simple game Multiple chice Answer the questins in ps3.scm. The questins are there t check yur understanding, s yu can verify yur answers with the tester The eight-pawn game This sectin f the prblem set invlves a chess-like game that uses nly pawns n a 4x4 bard. The trivially-slvable 3x3 versin f this game is called "Hexapawn"; it was invented by Martin Gardner t demnstrate AI learning n a very small game tree. A pawn in chess is a piece that can nly mve in ne directin acrss the bard: white pawns nly mve up, and black pawns nly mve dwn. Ordinarily, they mve in straight lines, remaining in the same clumn. A pawn can't mve frward in this way if there's already anther pawn, f either clr, ccupying the space in frnt f it. In additin t these straight mves, thugh, pawns can capture diagnally. If there is a piece f the ppsite clr n ne f the tw squares that are diagnally in frnt f a pawn, it can capture the ther pawn by mving nt its square, which remves the captured pawn frm the bard. Diagnal mves can nly be made when capturing, and this is the nly way that a pawn can change clumns. The pawns in this game are nt allwed t mve tw spaces frward n their first mve, like they can in chess. In the eight-pawn game, tw players begin with 4 pawns each n ppsite sides f the bard. There are tw ways t win: t mve ne f yur pawns t the ppsite side f the bard, r t leave yur ppnent in a situatin where he can't mve. Prf. Patrick H. Winstn Page 3 f 9

4 Here are sme examples. These bards lk the same as they will lk in the Scheme prgram. W represents a white pawn, B represents a black pawn, and - indicates an empty space. (B B B B) ( ) This is the starting psitin. ( ) (W W W W) (B B B B) ( ) White makes the first mve, advancing his pawn (- - W -) in the third clumn. (W W - W) (B - B B) (- B - -) Black advances her pawn in the secnd clumn. (- - W -) (W W - W) (B - B B) (- W - -) White captures by mving the pawn diagnally frward ( ) and t the left. Black has tw pieces that can capture (W W - W) this piece in respnse. (- - B -) (- B W B) This situatin might arise late in the game. Nne (B W - W) f these pawns can mve. Whever's turn it is lses. (W - - -) (B - - B) (- - B W) Anther situatin that can ccur after several mves. (B W - -) It's black's turn t mve. (- - W -) (B - - B) (- - B W) Black mves a pawn int the bttm rw and wins. (- W - -) Pawned! (B - W -) The cde in 8pawns.scm enfrces the rules f the game and lets yu knw what mves are valid frm a given psitin Playing the game Yu can get a feel fr hw the game wrks by playing it against the cmputer. Fr example, by uncmmenting this line in ps3.scm, yu can play white, while a cmputer player that des minimax search t depth 5 plays black. (play-game startps (human-player) (chatty-minimax-player 5 simpleevaluate)) Fr each mve, the prgram will prmpt yu t make a chice, by chsing what lcatin t mve a piece t. (In a few cases, this will be ambiguus. These cases aren't Prf. Patrick H. Winstn Page 4 f 9

5 cmmn enugh t matter, thugh. If it helps, the pieces yu've mved mre recently tend t be near the tp f the list.) Yur chices may lk like this: (B B - B) (- - B -) (- - - W) (W W W -) 1: (W 4 3) 2: (W 3 3) 3: (W 1 2) 4: (W 2 2) 5: (W 3 2) Which mve d yu want t make? Chice 1, fr example, represents mving a White pawn t clumn 4, rw 3 (the third frm the bttm f the bard). This means mving the advanced pawn frward again. Chice 2 -- mving t 3, 3 -- indicates mving that same pawn diagnally t capture. The ther three chices advance ne f the ther pawns frm the first rw t the secnd. Yu'll get the hang f it. The pint f this isn't t be a gd interface t a game, it's t let yu interact with the game at all withut large amunts f supprt cde. The cmputer, meanwhile, is making the best mve it can while lking ahead t depth 5 (three mves fr itself and tw fr yu). At each mve, it utputs the value it gets frm minimax. If this value is 0, it cnsiders the game even s far. A value f 1000 means it's certain it will win, and means it's certain it will lse (assuming yu play ptimally) The cde Here's an verview f the cde in 8pawns.scm that yu may need t wrk with Abstractins A psitin indicates the state f the bard and whse turn it is. It's represented as a list cntaining three things: 1. the tag 'psitin 2. the clr that mves next, expressed as 'W r 'B 3. the bard A bard is simply a list f all the pieces in play. Prf. Patrick H. Winstn Page 5 f 9

6 A piece represents the lcatin and clr f ne piece. It is represented as a list f three items: the piece's clr (W r B), the clumn the piece is in (1-4), and the rw the piece is in (1-4). This same structure is used t represent spaces n the bard, whether r nt there is actually a piece there. S the bttm left crner f the bard is passed t functins as (W 1 1), r maybe as (B 1 1). The piece's clr is disregarded if all that matters is the lcatin. Given these abstractins, the starting psitin f the game lks like this: '(psitin W ((W 1 1) (W 2 1) (W 3 1) (W 4 1) (B 1 4) (B 2 4) (B 3 4) (B 4 4))) Dn't cnfuse psitins with lcatins n the bard. A psitin represents the entire state f the game, while a lcatin is a small part f the game Prcedures This is an verview f the mst useful prcedures in 8pawns.scm. Yu may want t lk at the cde fr a better understanding. (valid-mves psitin): return a list f psitins, representing all psitins that the current player culd put the game in after mving. If the current psitin has White t mve, then all the returned psitins will have Black t mve, and vice versa. (get-clr psitin): which clr has the next mve in this psitin? (get-bard psitin): extract the bard (list f piece lcatins) frm a psitin. (minimax-search psitin depth player max? evaluatin-functin): the main functin that finds the minimax value f the game tree. psitin is the psitin it shuld analyze. depth is hw many mre levels it shuld descend in the game tree befre using a static evaluatin functin. If depth is 0, then it will nt explre any mre mves and simply run the static evaluatin functin n the given psitin. player is W r B, depending n whse pint f view the bard shuld be evaluated frm. A psitin that's wrth +3 fr White is wrth -3 fr Black. player is nt necessarily the player that the psitin says has the next mve, because yu need t explre yur ppnent's mves as well as yur wn. max? is #t r #f. If it's #t, then the prcedure will return the mve that maximizes the heuristic value; if #f, it will return the mve that minimizes the heuristic value. evaluatin-functin is a prcedure t use fr static evaluatin. Yu can change the behavir f minimax-search by giving it a different evaluatin- Prf. Patrick H. Winstn Page 6 f 9

7 functin. The prvided prcedure simple-evaluate is an example f an evaluatin-functin. minimax-search returns a list f tw things: the mve t make (represented as the psitin that results frm the mve), and the heuristic value it calculated. (simple-evaluate ps player depth): a simple static evaluatin functin. It takes in a psitin ps, the player whse pint f view t use, and the depth remaining t search (infrmatin which this functin desn't use). If the psitin is a lss fr the player t mve, the psitin is wrth fr that player. Otherwise, a psitin is cnsidered better fr the current player if that player has mre pssible mves. The psitin is wrth the number f pssible mves, minus 3 (s that it is rughly centered arund 0). If the psitin is fr the ppsite player, simple-evaluate will negate the value it returns. If it's evaluating psitins frm White's pint f view, fr example, then a lss fr Black shuld be wrth 1000, nt (maximize-psitin p1 p2): p1 and p2 shuld be the kind f tw-element (psitin value) lists that are returned frm minimax-search. This functin cmpares the tw results and returns the psitin with the higher heuristic value. (minimize-psitin p1 p2): Same as abve, except it returns the psitin with the lwer heuristic value. (check-lss psitin): is this psitin a lss fr the player t mve? (There is n check-win, s just use check-lss n the ther player's turn.) (minimax-player depth evaluatin-functin) (and related prcedures) Several f these "player factry" prcedure are prvided. These prcedures take in a depth t search t and an evaluatin functin t use, and return a "player" functin, which will take in a psitin and return the psitin it wants t mve t. (human-player): Anther "player factry", but here the "player" will stp and ask yu what mve t make. (play-game psitin player1 player2): takes in a starting psitin, and tw players that are returned frm player factries. play-game plays ut the eightpawn game until ne player wins, and returns the clr f the winning player Just win already! Okay, we're abut t ask yu t actually cde stuff. Yu shuld stp skimming nw. If yu end up in a lsing psitin when playing against the cmputer, it may seem that the cmputer is tying with yu: instead f making a single mve that lets it win, it may prlng yur misery by mving ther pieces in such a way that it can still be guaranteed t win. Prf. Patrick H. Winstn Page 7 f 9

8 Similarly, if the cmputer sees that yu are in a winning psitin, the cmputer may make an apparently stupid mve that lets yu win faster. It's nt stupid frm the cmputer's pint f view, thugh - all mves are equally dmed, s why des it matter which ne it makes? This isn't hw peple generally play games. Peple want t win as quickly as pssible, and lse slwly s that their ppnent has several pprtunities t mess up. A small change t the static evaluatin functin will make the cmputer play this way t. In ps3.scm, write a new evaluatin functin, fcused-evaluate, which prefers winning psitins when they happen sner and lsing psitins when they happen later. It will help t fllw these guidelines: Leave the "nrmal" values (the nes that are like 1 r -2, nt 1000 r -1000) alne. Yu dn't need t change hw the prcedure evaluates psitins that aren't guaranteed wins r lsses. Dn't return values less than r greater than we'll be using thse numbers t mean "infinity" in a bit. Indicate a certain win with a value that is greater than r equal t 1000, and a certain lss with a value that is less than r equal t The "chatty" versins f the cmputerized players will cmment when they see these kinds f values Alpha-beta search The cmputerized players yu've used s far wuld fit in well in the British Museum - they're evaluating all the psitins dwn t a certain depth, even the unhelpful nes. Yu can make them search much mre efficiently by using alpha-beta search. It may help t read Chapter 6 f the textbk: Winstn, Patrick H. Artificial Intelligence. 3rd ed. Reading, MA: Addisn-Wellsley, 1992, ISBN: if yu're unclear n the details f alpha-beta search. Write a prcedure alpha-beta-search, which wrks like minimax-search, except it takes tw additinal arguments (alpha and beta) and des alpha-beta pruning t reduce the search space. Yu shuld always search yur pssible mves frm left t right, in the rder they are returned frm valid-mves, and return a result fr a particular level as sn as alpha is greater than r equal t beta. This prcedure is called by the player prcedure (alpha-beta-player depth evaluatin-functin). Initially, alpha is and beta is cnsider these t represent "negative infinity" and "psitive infinity". Prf. Patrick H. Winstn Page 8 f 9

9 2.6. A better evaluatin functin This prblem is ging t be a bit different. There's n single right way t d it - it will take a bit f creativity and thught abut the game. Yur gal is t write a new prcedure fr static evaluatin that utperfrms the ne we gave yu. It will be tested against simple-evaluate n 10 different games, with search depths frm 3 t 7. Yu get a pint fr each game yu win. S a well-frmed slutin t this prblem shuld easily get 5 pints, and yu can get a maximum f 10. Yu might nt get all 10 pints - it might nt even be feasible t get all 10 pints - but we expect that the mre thught yu put int it, the higher a scre yu can get. These pints cme entirely frm the public tester. If there are hidden tests invlving this prblem, they wn't hinge n whether yu win; they may, fr example, make sure yur functin desn't prduce errrs r evaluate mre psitins within the static evaluatr. (If yu did, that wuldn't be very static.) Write a new prcedure fr static evaluatin, (better-evaluate ps player depth), which wins against simple-evaluate when searching t the same depth. 3. Survey Please answer these questins at the bttm f yur ps3.scm file: Hw many hurs did this prblem set take? Which parts f this prblem set, if any, did yu find interesting? Which parts f this prblem set, if any, did yu find bring r tedius? (We'd ask which parts yu find cnfusing, but if yu're cnfused yu shuld really ask a TA.) When yu're dne, run the tester and submit yur.scm files t yur psets/ps3 directry n Athena. If the tester dies with an errr when it's run n yur cde, r if it stps and waits fr user input, yur submissin will nt cunt. 4. Prblems? If the tester dies immediately, cmplaining that instantiate is an unbund variable, yu frgt t lad the Graphical versin f the MzScheme language, as described in sectin 1.1. Prf. Patrick H. Winstn Page 9 f 9