Lecture 6: Constraint satisfaction problems: Given: a set of n - PowerPoint PPT Presentation

CS440/ECE448: Intro to Artificial Intelligence � Tuesday ʼ s key concepts � Lecture 6: 
 Constraint satisfaction problems: � Given: a set of n variables X 1 ..X n , with More on constraint domains (sets of possible values) D 1 …D n , satisfaction problems � and a set of m constraints C 1 … C m � � Task: assign a value from D i for each X i Prof. Julia Hockenmaier � subjects to the constraints. � juliahmr@illinois.edu � � � http://cs.illinois.edu/fa11/cs440 � � � � Map coloring: 
 a solution for N=3 � CSP 1: 
 Map coloring 
 (Binary constraints) � CS440/ECE448: Intro AI � 4 �


 Constraint satisfaction problems States and solutions � are defined by… � - a set of variables X : � Each state is a complete or partial assignment of values to variables: 
 {WA, NT, QLD, NSW, VA, SA, TA} � - a set of domains D i 
 state35 = {WA=red, NT=blue, QLD= green, NSW= red, 
 (possible values for variable x i ): � � � � VA= green, SA= blue, TA= red}; 
 D WA = {red, blue, green} � state23 = {WA = red} � - a set of constraints C : � � Legal assignments don ʼ t violate any constraints. � { 〈 (WA,NT), WA ≠ NT 〉 , 〈 (WA,QLD), WA ≠ QLD 〉 ,…} 
 Solutions are complete legal assignments � scope relation � � CS440/ECE448: Intro AI � 5 � CS440/ECE448: Intro AI � 6 � Binary constraints: 
 Consistency � constraint graph � Node consistency: X is node-consistent iff 
 QLD � NT � each element in D X satisfies unary constraints on X � � Arc consistency: X is arc-consistent iff 
 WA � NSW � for each C(X, Y) and for each x ∈ D X there is a y ∈ D Y such that the assignment {X=x, Y=y} satisfies C(X,Y). � � Path consistency: {X,Y} are path consistent wrt. Z iff 
 SA � VA � for every x ∈ D X and y ∈ D X there is a z ∈ D Z such that the assignment {X=x,Y=y,Z=z} satisfies C(X,Z) and C(Y,Z) � � TA � � CS440/ECE448: Intro AI � 7 � CS440/ECE448: Intro AI � 8 � �

Tree-structured 
 AC-3 � constraint graphs � - // Is the CSP c arc-consistent? � Any two nodes connected by a single path � function AC3(CSP c) � - � input: CSP c = (X,D,C) �� With n vertices, there are n − 1 edges � � local: queue q ← all arcs C(X,Y) in c � - Can be solved in linear time ( O(nd 2 ) ) � � while q ≠ () do: � � � // Can C(X,Y) be satisfied? � � � (X,Y) = pop(q); � E � � // if domain(X) needs to be shrunk: � E B � � if revise(c,X,Y): � A B F � � � // Exit if CSP can’t be solved: � D � � � if domain(X) == () return false; � F C � � � // Are X’s neighbors still okay? � A C D � � � foreach Z in X.NEIGHBORS\{Y}: � � q ← push(q,(Z, X)); � � � � � return true; � CS440/ECE448: Intro AI � 9 � CS440/ECE448: Intro AI � 10 � The 8-queens puzzle � ♕ ! ! ! ! ♕ ! ! !" ! ! CSP 2: 
 ♕ ! ! ! ! ♕ ! ! ! ! ! ! ! 8 queens puzzle 
 ! ! ! ! ! ! ! ! ! ! ! ! ♕ ! ! (Combining search 
 ♕ ! ! ! ! and CSP inference) � The 8-queens puzzle has multiple solutions � Cannot be solved by constraint propagation alone � [ underdetermined CSP] � � � CS440/ECE448: Intro AI � 12 �

Search tree for 8-queens � Branching factor of 8-queens � ! ! ! ! ! ! !" ! ! The branching factor is at most 8: � ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! � ! ! ! ! ! ! ! ! ! ! - We can order the variables: 
 ! ! ! ! (CSPs are commutative ) � ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !" ! ! ! ! !" ! ! ! ! !" ! ! ! ! !" ! ! ! ! !" ! ! ! ! !" ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1. place queen in column A, � ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 2. place queen in column B, � ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 3. ... � � - The constraints restrict the domains of the ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !" ! ! !" ! ! ! ! !" ! ! ! ! !" ! ! ! ! !" ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! remaining variables � ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! � CS440/ECE448: Intro AI � 13 � CS440/ECE448: Intro AI � 14 � Ordering the variables � Caveat: implied constraints � D X = {R,G,B} C(X,Y): X ≠ Y � D X = D Y = D Z = {0,1,2,3,4,5,6,7,8,9} � D Y = {R} � C(X,Z): X ≠ Z � � D Z = {G,B} � C(X, Y): Y = X 2 � � X � Y � Z � C(Y, Z): Z = Y 2 � Which variable should we consider first? � � Minimum remaining values heuristic: � � The variable with the smallest domain � If we choose X as the root of the tree, we Degree heuristic: � need to run DAC down the tree and up The variable that has the most constraints � again. � CS440/ECE448: Intro AI � 15 � CS440/ECE448: Intro AI � 16 �


 Interleaving search 
 Ordering the values � and inference � ! ! ! ! ! ! ! ! ! ! ! ! ! ! !" ! ! !" ! ! !" ! ! ! ! !" ! ! Search: assign (guess) value x for variable X � ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Forward checking algorithm: � Check that X is arc-consistent with all Which values of X should we consider first? � remaining variables Y 
 � � Least-constraining value heuristic: � Maintaining-Arc-Consistency algorithm: � Pick the most likely value first 
 Run AC3 on all C(Y,X) constraints to check (= the one which rules out the fewest choices for other overall arc-consistency � variables) � CS440/ECE448: Intro AI � 17 � CS440/ECE448: Intro AI � 18 � Cryptarithmetic as CSP � TWO � + TWO � CSP 3: 
 = FOUR � Cryptarithmetic 
 Task: assign a digit to each letter � � (Global constraints) � Constraints : � Each letter has a unique digit 
 (= AllDiff constraint) � The result has to be a valid sum � CS440/ECE448: Intro AI �

Global (n-ary) constraints: 
 Hypergraphs � Constraint Hypergraph � In a hypergraph, hyperedges connect TWO � multiple vertices: � + TWO � = FOUR � F � T � U � W R � O � C 1000 � C 100 � C 10 � CS440/ECE448: Intro AI � 21 � CS440/ECE448: Intro AI � 22 � Global (n-ary) constraints: 
 Global (n-ary) constraints: 
 Constraint Hypergraph � Constraint Hypergraph � TWO � TWO � + TWO � + TWO � = FOUR � = FOUR � F � T � U � W R � O � F � T � U � W R � O � Variables: � Variables: � letters � Carry digits � C 1000 � C 100 � C 10 � C 1000 � C 100 � C 10 � CS440/ECE448: Intro AI � 23 � CS440/ECE448: Intro AI � 24 �

Global (n-ary) constraints: 
 Global (n-ary) constraints: 
 Constraint Hypergraph � Constraint Hypergraph � Alldiff TWO � TWO � constraint � + TWO � + TWO � = FOUR � = FOUR � F � T � U � W R � O � F � T � U � W R � O � 1 column � addition � constraint � C 1000 � C 100 � C 10 � C 1000 � C 100 � C 10 � CS440/ECE448: Intro AI � 25 � CS440/ECE448: Intro AI � 26 � Global (n-ary) constraints: 
 Global (n-ary) constraints: 
 Constraint Hypergraph � Constraint Hypergraph � TWO � TWO � + TWO � + TWO � = FOUR � = FOUR � F � T � U � W R � O � F � T � U � W R � O � 10 column � 100 column � addition � addition � constraint � constraint � C 1000 � C 100 � C 10 � C 1000 � C 100 � C 10 � CS440/ECE448: Intro AI � 27 � CS440/ECE448: Intro AI � 28 �

Global (n-ary) constraints: 
 Constraint propagation: 
 Constraint Hypergraph � Global constraints � Some n-ary constraints can be directly TWO � translated into a set of binary constraints: � + TWO � � = FOUR � Global constraint: � AllDiff(X,Y,Z) � New binary constraints: � 
 F � T � U � W R � O � � � � � � � � C(X,Y): � X ≠ Y 
 1000 column � � � � � C(X,Z): � X ≠ Z 
 addition � � � C(Y,Z): � Y ≠ Z 
 constraint � � NB: Special purpose algorithms are often faster. � C 1000 � C 100 � C 10 � CS440/ECE448: Intro AI � 29 � CS440/ECE448: Intro AI � 30 � Global constraints � With additional auxiliary variables, any n-ary constraint can be translated into a set of binary CSP 3: 
 constraints: � � Scheduling 
 Ternary constraint: � C(X,Y,Z): X+Y = Z � � d D = { 〈 a 1 ,a 2 〉 | a 1 ∈ d X , a 2 ∈ d Y } � Aux. variable A: � (Continuous domains) � New constraints: � � C(A,Z): a 1 + a 2 = Z 
 � � � C(A,X): a 1 = X 
 � � � C(A,Y): a 2 = Y � � NB: Special purpose algorithms are often faster � CS440/ECE448: Intro AI � 31 �