Constraint Satisfaction Problems (CSPs) Russell and Norvig Chapter 6

Transcription

1 Costrait Satisfactio Problems (CSPs) Russell ad Norvig Chapter 6

2 CSP example: map colorig Give a map of Australia, color it usig three colors such that o eighborig territories have the same color. 2

3 CSP example: map colorig 3

4 Costrait satisfactio problems A CSP is composed of: A set of variables X 1,X 2,,X with domais (possible values) D 1,D 2,,D A set of costraits C 1,C 2,,C m Each costrait C i limits the values that a subset of variables ca take, e.g., V 1 V 2 4

5 Costrait satisfactio problems A CSP is composed of: A set of variables X 1,X 2,,X with domais (possible values) D 1,D 2,,D A set of costraits C 1,C 2,,C m Each costrait C i limits the values that a subset of variables ca take, e.g., V 1 V 2 I our example: Variables: WA, NT, Q, NSW, V, SA, T Domais: D i ={red,gree,blue} Costraits: adjacet regios must have differet colors. E.g. WA NT (if the laguage allows this) or (WA,NT) i {(red,gree),(red,blue),(gree,red),(gree,blue),(blue,red), (blue,gree)} 5

6 Costrait satisfactio problems A state is defied by a assigmet of values to some or all variables. Cosistet assigmet: assigmet that does ot violate the costraits. Complete assigmet: every variable is metioed. Goal: a complete, legal assigmet. {WA=red,NT=gree,Q=red,NSW=gree,V=red,SA=blue,T=gree} 6

7 Costrait satisfactio problems Simple example of a formal represetatio laguage CSP beefits Stadard represetatio laguage Geeric goal ad successor fuctios Useful geeral-purpose algorithms with more power tha stadard search algorithms, icludig geeric heuristics Applicatios: Time table problems (exam/teachig schedules) Assigmet problems (who teaches what) 7

8 Varieties of CSPs Discrete variables Fiite domais of size d O(d ) complete assigmets. The satisfiability problem: a Boolea CSP Ifiite domais (itegers, strigs, etc.) Cotiuous variables Liear costraits solvable i poly time by liear programmig methods (dealt with i the field of operatios research). Our focus: discrete variables ad fiite domais 8

9 Varieties of costraits Uary costraits ivolve a sigle variable. e.g. SA gree Biary costraits ivolve pairs of variables. e.g. SA WA Global costraits ivolve a arbitrary umber of variables. Preferece (soft costraits) e.g. red is better tha gree ofte represetable by a cost for each variable assigmet; ot cosidered here. 9

10 Costrait graph Biary CSP: each costrait relates two variables Costrait graph: odes are variables, edges are costraits 10

11 Example: cryptarithmetic puzzles The costraits are represeted by a hypergraph 11

12 CSP as a stadard search problem Icremetal formulatio Iitial State: the empty assigmet {}. Successor: Assig value to uassiged variable provided there is o coflict. Goal test: the curret assigmet is complete. Same formulatio for all CSPs!!! Solutio is foud at depth ( variables). What search method would you choose? 12

13 Backtrackig search Observatio: the order of assigmet does t matter ca cosider assigmet of a sigle variable at a time. Results i d leaves (d: umber of values per variable). Backtrackig search: DFS for CSPs with siglevariable assigmets (backtracks whe a variable has o value that ca be assiged) The basic uiformed algorithm for CSP 13

14 Sudoku solvig A B C D E F G H I

15 Sudoku solvig Costraits: Alldiff(A1,A2,A3,A4,A5,A6,A7,A8,A9) Alldiff(A1,B1,C1,D1,E1,F1,G1,H1,I1) Alldiff(A1,A2,A3,B1,B2,B3,C1,C2,C3) Ca be traslated ito costraits betwee pairs of variables. A B C D E F G H I

16 Sudoku solvig A B C D E F G H I Let s see if we ca figure the value of the ceter grid poit. Images from wikipedia ad 16

17 Solvig Sudoku I this essay I tackle the problem of solvig every Sudoku puzzle. It turs out to be uite easy (about oe page of code for the mai idea ad two pages for embellishmets) usig two ideas: costrait propagatio ad search. Peter Norvig 17

18 Costrait propagatio Eforce local cosistecy Propagate the implicatios of each costrait 18

19 Arc cosistecy X Y is arc-cosistet iff for every value x of X there is some allowed value y of Y Example: X ad Y ca take o the values 0 9 with the costrait: Y=X 2. Ca use arc cosistecy to reduce the domais of X ad Y: X Y reduce X s domai to {0,1,2,3} Y X reduce Y s domai to {0,1,4,9} 19

20 The Arc Cosistecy Algorithm fuctio AC-3(csp) returs false if a icosistecy is foud ad true otherwise iputs: csp, a biary csp with compoets {X, D, C} local variables: ueue, a ueue of arcs iitially the arcs i csp while ueue is ot empty do (X i, X j ) REMOVE-FIRST(ueue) if REVISE(csp, X i, X j ) the if size of D i =0 the retur false for each X k i X i.neighbors {X j } do add (X k, X i ) to ueue fuctio REVISE(csp, X i, X j ) returs true iff we revise the domai of X i revised false for each x i D i do if o value y i D j allows (x,y) to satisfy the costraits betwee X i ad X j the delete x from D i revised true retur revised 20

21 Arc cosistecy limitatios X Y is arc-cosistet iff for every value x of X there is some allowed y of Y Cosider mappig Australia with two colors. Each arc is cosistet, ad yet there is o solutio to the CSP. So it does t help 21

22 Path Cosistecy Looks at triples of variables The set {X i, X j } is path-cosistet with respect to X m if for every assigmet cosistet with the costraits of X i, X j, there is a assigmet to X m that satisfies the costraits o {X i, X m } ad {X m, X j } The PC-2 algorithm achieves path cosistecy 22

23 K-cosistecy Stroger forms of propagatio ca be defied usig the otio of k-cosistecy. A CSP is k-cosistet if for ay set of k-1 variables ad for ay cosistet assigmet to those variables, a cosistet value ca always be assiged to ay k-th variable. Not practical! 23

24 Backtrackig example 24

25 Backtrackig example 25

26 Backtrackig example 26

27 Backtrackig example 27

28 Improvig backtrackig efficiecy Geeral-purpose methods/heuristics ca give huge gais i speed: Which variable should be assiged ext? I what order should its values be tried? Ca we detect ievitable failure early? 28

29 Backtrackig search fuctio BACKTRACKING-SEARCH(csp) retur a solutio or failure retur RECURSIVE-BACKTRACKING({}, csp) fuctio RECURSIVE-BACKTRACKING(assigmet, csp) retur a solutio or failure if assigmet is complete the retur assigmet var SELECT-UNASSIGNED-VARIABLE(VARIABLES[csp],assigmet,csp) for each value i ORDER-DOMAIN-VALUES(var, assigmet, csp) do retur failure if value is cosistet with assigmet accordig to CONSTRAINTS[csp] the add {var=value} to assigmet result RECURSIVE-BACTRACKING(assigmet, csp) if result failure the retur result remove {var=value} from assigmet 29

30 Most costraied variable var SELECT-UNASSIGNED-VARIABLE(csp) Choose the variable with the fewest legal values (most costraied variable) a.k.a. miimum remaiig values (MRV) or fail first heuristic What is the ituitio behid this choice? 30

31 Most costraiig variable Select the variable that is ivolved i the largest umber of costraits o other uassiged variables. Also called the degree heuristic because that variable has the largest degree i the costrait graph. Ofte used as a tie breaker e.g. i cojuctio with MRV. 31

32 Least costraiig value heuristic Guides the choice of which value to assig ext. Give a variable, choose the least costraiig value: the oe that rules out the fewest values i the remaiig variables why? 32

33 Forward checkig Ca we detect ievitable failure early? Ad avoid it later? Forward checkig: keep track of remaiig legal values for uassiged variables. Termiate search directio whe a variable has o legal values. 33

34 Forward checkig Assig {WA=red} Effects o other variables coected by costraits with WA NT ca o loger be red SA ca o loger be red 34

35 Forward checkig Assig {Q=gree} Effects o other variables coected by costraits with WA NT ca o loger be gree NSW ca o loger be gree SA ca o loger be gree 35

36 Forward checkig If V is assiged blue Effects o other variables coected by costraits with WA SA is empty NSW ca o loger be blue FC has detected that partial assigmet is icosistet with the costraits ad backtrackig ca occur. 36

37 Example: 4-Quees Problem X1 {1,2,3,4} X2 {1,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} 37

38 Example: 4-Quees Problem X1 {1,2,3,4} X2 {1,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} 38

39 Example: 4-Quees Problem X1 {1,2,3,4} X2 {,,3,4} X3 {,2,,4} X4 {,2,3, } 39

40 Example: 4-Quees Problem X1 {1,2,3,4} X2 {,,3,4} X3 {,2,,4} X4 {,2,3, } 40

41 Example: 4-Quees Problem X1 {1,2,3,4} X2 {,,3,4} X3 {,,, } X4 {,2,3, } 41

42 Example: 4-Quees Problem X1 {,2,3,4} X2 {1,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} 42

43 Example: 4-Quees Problem X1 {,2,3,4} X2 {,,,4} X3 {1,,3, } X4 {1,,3,4} 43

44 Example: 4-Quees Problem X1 {,2,3,4} X2 {,,,4} X3 {1,,3, } X4 {1,,3,4} 44

45 Example: 4-Quees Problem X1 {,2,3,4} X2 {,,,4} X3 {1,,, } X4 {1,,3, } 45

46 Example: 4-Quees Problem X1 {,2,3,4} X2 {,,,4} X3 {1,,, } X4 {1,,3, } 46

47 Example: 4-Quees Problem X1 {,2,3,4} X2 {,,,4} X3 {1,,, } X4 {,,3, } 47

48 Example: 4-Quees Problem X1 {,2,3,4} X2 {,,,4} X3 {1,,, } X4 {,,3, } 48

49 Forward checkig Solvig CSPs with combiatio of heuristics plus forward checkig is more efficiet tha either approach aloe. FC does ot provide early detectio of all failures. Oce WA=red ad Q=gree: NT ad SA caot both be blue! MAC (maitaiig arc cosistecy): calls AC-3 after assigig a value (but oly deals with the eighbors of a ode that has bee assiged a value). 49

50 The zebra puzzle There are five houses. The Eglishma lives i the red house. The Spaiard ows the dog. Coffee is druk i the gree house. The Ukraiia driks tea. The gree house is immediately to the right of the ivory house. The Old Gold smoker ows sails. Kools are smoked i the yellow house. Milk is druk i the middle house. The Norwegia lives i the first house. The ma who smokes Chesterfields lives i the house ext to the ma with the fox. Kools are smoked i the house ext to the house where the horse is kept. The Lucky Strike smoker driks orage juice. The Japaese smokes Parliamets. The Norwegia lives ext to the blue house. Now, who driks water? Who ows the zebra? 50

51 The zebra puzzle 51

52 The Zebra Puzzle 52

53 Local search for CSP Local search methods use a complete state represetatio, i.e., all variables assiged. To apply to CSPs Allow states with usatisfied costraits reassig variable values Select a variable: radom coflicted variable Select a value: mi-coflicts heuristic Value that violates the fewest costraits Hill-climbig like algorithm with the objective fuctio beig the umber of violated costraits Works surprisigly well i problems like -Quees 53

54 Mi-Coflicts fuctio MIN-CONFLICTS(csp, max_steps) returs a solutio or failure iputs: csp, a costrait satisfactio problem max_steps, the umber of steps allowed before givig up curret a iitial complete assigmet for csp for i = 1 to max_steps do retur failure if curret is a solutio for csp the retur curret var a radomly chose coflicted variable from csp.variables value the value v for var that miimizes CONFLICTS(var, v, curret, csp) set var=value i curret 54

55 Problem structure How ca the problem structure help to fid a solutio uickly? Subproblem idetificatio is importat: Colorig Tasmaia ad mailad are idepedet subproblems Idetifiable as coected compoets of costrait graph. Improves performace 55

56 Problem structure Suppose each problem has c variables out of a total of. Worst case solutio cost is O(/c d c ) istead of O(d ) Suppose =80, c=20, d= = 4 billio years at 1 millio odes/sec. 4 * 2 20 =.4 secod at 1 millio odes/sec 56

57 Tree-structured CSPs Theorem: if the costrait graph has o loops the CSP ca be solved i O(d 2 ) time Compare with geeral CSP, where worst case is O(d ) 57

58 Tree-structured CSPs Ay tree-structured CSP ca be solved i time liear i the umber of variables. Choose a variable as root, order variables from root to leaves such that every ode s paret precedes it i the orderig. (label var from X 1 to X ) For j from dow to 2, apply REMOVE-INCONSISTENT- VALUES(Paret(X j ), X j ) For j from 1 to assig X j cosistetly with Paret(X j ) 58

59 Tree-structured CSPs Ay tree-structured CSP ca be solved i time liear i the umber of variables. Fuctio TREE-CSP-SOLVER(csp) returs a solutio or failure iputs: csp, a CSP with compoets X, D, C umber of variables i X assigmet a empty assigmet root ay variable i X X TOPOLOGICALSORT(X, root) for j = dow to 2 do for i = 1 to do MAKE-ARC-CONSISTENT(PARENT(X j ),X j ) if it caot be made cosistet the retur failure assigmet[x i ] ay cosistet value from D i if there is o cosistet value the retur failure retur assigmet 59

60 Nearly tree-structured CSPs Ca more geeral costrait graphs be reduced to trees? Two approaches: Remove certai odes Collapse certai odes 60

61 Nearly tree-structured CSPs Idea: assig values to some variables so that the remaiig variables form a tree. Assig {SA=x} cycle cutset Remove ay values from the other variables that are icosistet. The selected value for SA could be the wrog: have to try all of them 61

62 Nearly tree-structured CSPs This approach is effective if cycle cutset is small. Fidig the smallest cycle cutset is NP-hard Approximatio algorithms exist This approach is called cutset coditioig. 62

63 Summary CSPs are a special kid of problem: states defied by values of a fixed set of variables, goal test defied by costraits o variable values Backtrackig=depth-first search with oe variable assiged per ode Variable orderig ad value selectio heuristics help sigificatly Forward checkig prevets assigmets that lead to failure. Costrait propagatio does additioal work to costrai values ad detect icosistecies. Structure of CSP affects its complexity. Tree structured CSPs ca be solved i liear time. 63

64 Iterim class summary We have bee studyig ways for agets to solve problems. Search Uiformed search Easy solutio for simple problems Basis for more sophisticated solutios Iformed search Iformatio = problem solvig power Adversarial search αβ-search for play agaist optimal oppoet Early cut-off whe ecessary Costrait satisfactio problems What s ext? Logical iferece Probabilistic iferece Machie learig Other fu stuff motio plaig, structural biology 64