Outline What are CSPs? Standard search and CSPs Improvements. Constraint Satisfaction. Introduction. Introduction

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1 Constraint Satisfaction CS 86/686 May 6, 006 University of Waterloo Outline What are CSPs? Standard search and CSPs Improvements acktracking acktracking + heuristics Forward checking Introduction In the last couple of lectures we have been solving problems by searching in a space of states reating states as black boxes, ignoring any structure inside them Using problem-specific routines oday we study problems where the state structure is important States: all arrangements of 0,,, or 8 queens on the board Initial state: 0 queens on the board Successor function: Add a queen to the board Goal test: 8 queens on the board with no two of them attacking each other 6x6x 57 x0 states States: all arrangements k queens (0 k 8), one per column in the leftmost k columns, with no queen attacking another Initial state: 0 queens on the board Successor function: Add a queen to the leftmost empty column such that it is not attacked Goal test: 8 queens on the board 057 States 5 Introduction Earlier search methods studied often make choices in an arbitrary order In many problems the same state can be reached independent of the order in which the moves are chosen (commutative actions) Can we solve problems efficiently by being smart in the order in which we take actions? 6

2 -queens Constraint Propagation -queens Constraint Propagation Place a queen in a square Place a queen in a square emove conflicting squares from consideration 7 emove conflicting squares from consideration 8 -queens Constraint Propagation -queens Constraint Propagation Place a queen in a square Place a queen in a square emove conflicting squares from consideration 9 emove conflicting squares from consideration 0 CSP Definition CSP Definition A constraint satisfaction problem (CSP) is defined by {,D,C} where ={,,, n } is a set of variables D={D,,D n } is the set of domains, D i is the domain of possible values for variable i C={C,,C m } is the set of constraints Each constraint involves some subset of the variables and specifies the allowable combinations of values for that subset A state is an assignment of values to some or all of the variables { i =x i, j =x j, } An assignment is consistent if it does not violate any constraints A solution is a complete, consistent assignment ( hard constraints ) Some CSPs also require an objective function to be optimized ( soft constraints )

3 Example : 8-ueens Example 8 queens 6 variables ij, i= to 8, j= to 8 Domain of each variable is {0,} Constraints ij = ik =0 for all k j ij = kj =0 for all k i Similar constraint for diagonals i,j ij =8 inary constraints relate two variables 8 variables i, i= to 8 Domain of each variable is {,,,8} Constraints i =k j k for all j i Similar constraints for diagonals inary constraints relate two variables Example - Map Coloring Example from and N, Annotations from Stanford CS Example - Street Puzzle 7 variables {,,,,,,} Each variable has the same domain: {red, green, blue} No two adjacent variables have the same value:,,,,,,,, Constraint graph 5 5 C i = {ed, Green, White, Yellow, lue} D i = {ea, Coffee, Milk, Fruit-juice, Water} J i = {Painter, Sculptor, Diplomat, iolinist, Doctor} he Englishman lives in the ed house he Spaniard has a Dog he Japanese is a Painter he Italian drinks ea he Norwegian lives in the first house on the left he owner of the Green house drinks Coffee he Green house is on the right of the White house he Sculptor breeds Snails he Diplomat lives in the Yellow house he owner of the middle house drinks Milk he Norwegian lives next door to the lue house he iolinist drinks Fruit juice he Fox is in the house next to the Doctor s he Horse is next to the Diplomat s Who owns the Zebra? Who drinks Water? 6 Example from and N, Annotations from Stanford CS Street Puzzle 5 C i = {ed, Green, White, Yellow, lue} D i = {ea, Coffee, Milk, Fruit-juice, Water} J i = {Painter, Sculptor, Diplomat, iolinist, Doctor} he Englishman lives in the ed house (N i = English) (C i = ed) he Spaniard has a Dog he Japanese is a Painter (N i = Japanese) (J i = Painter) he Italian drinks ea he Norwegian lives in the first house on the left (N = Norwegian) he owner of the Green house drinks Coffee he Green house is on the right of the White house he Sculptor breeds Snails he Diplomat lives in the Yellow house (C i = White) (C i+ = Green) he owner of the middle house drinks Milk (C 5 White) he Norwegian lives next door to the lue house (C Green) he iolinist drinks Fruit juice he Fox is in the house next to the Doctor s he Horse is next to the Diplomat s left as an exercise 7 Example from and N, Annotations from Stanford CS Street Puzzle 5 C i = {ed, Green, White, Yellow, lue} D i = {ea, Coffee, Milk, Fruit-juice, Water} J i = {Painter, Sculptor, Diplomat, iolinist, Doctor} he Englishman lives in the ed house (N i = English) (C i = ed) he Spaniard has a Dog he Japanese is a Painter (N i = Japanese) (J i = Painter) he Italian drinks ea he Norwegian lives in the first house on the left (N = Norwegian) he owner of the Green house drinks Coffee he Green house is on the right of the White house he Sculptor breeds Snails he Diplomat lives in the Yellow house (C i = White) (C i+ = Green) he owner of the middle house drinks Milk (C 5 White) he Norwegian lives next door to the lue house (C Green) he iolinist drinks Fruit juice he Fox is in the house next to the Doctor s he Horse is next to the Diplomat s unary constraints 8

4 Example from and N, Annotations from Stanford CS Street Puzzle 5 C i = {ed, Green, White, Yellow, lue} D i = {ea, Coffee, Milk, Fruit-juice, Water} J i = {Painter, Sculptor, Diplomat, iolinist, Doctor} i,j [,5], i j, N i N j he Englishman lives in the ed house he Spaniard has a Dog i,j [,5], i j, C i C j he Japanese is a Painter... he Italian drinks ea he Norwegian lives in the first house on the left he owner of the Green house drinks Coffee he Green house is on the right of the White house he Sculptor breeds Snails he Diplomat lives in the Yellow house he owner of the middle house drinks Milk he Norwegian lives next door to the lue house he iolinist drinks Fruit juice he Fox is in the house next to the Doctor s he Horse is next to the Diplomat s 9 Example from and N, Annotations from Stanford CS 5 Street Puzzle C i = {ed, Green, White, Yellow, lue} D i = {ea, Coffee, Milk, Fruit-juice, Water} J i = {Painter, Sculptor, Diplomat, iolinist, Doctor} he Englishman lives in the ed house he Spaniard has a Dog he Japanese is a Painter he Italian drinks ea he Norwegian lives in the first house on the left N = Norwegian he owner of the Green house drinks Coffee he Green house is on the right of the White house he Sculptor breeds Snails he Diplomat lives in the Yellow house he owner of the middle house drinks Milk D = Milk he Norwegian lives next door to the lue house he iolinist drinks Fruit juice he Fox is in the house next to the Doctor s he Horse is next to the Diplomat s 0 Example from and N, Annotations from Stanford CS Street Puzzle Example 5 - Scheduling 5 C i = {ed, Green, White, Yellow, lue} D i = {ea, Coffee, Milk, Fruit-juice, Water} J i = {Painter, Sculptor, Diplomat, iolinist, Doctor} he Englishman lives in the ed house C ed he Spaniard has a Dog A Dog he Japanese is a Painter he Italian drinks ea he Norwegian lives in the first house on the left N = Norwegian he owner of the Green house drinks Coffee he Green house is on the right of the White house he Sculptor breeds Snails he Diplomat lives in the Yellow house he owner of the middle house drinks Milk D = Milk he Norwegian lives next door to the lue house he iolinist drinks Fruit juice J iolinist he Fox is in the house next to the Doctor s he Horse is next to the Diplomat s Four tasks,,, and are related by time constraints: must be done during must be achieved before starts must overlap with must start after is complete Are the constraints compatible? What are the possible time relations between two tasks? What if the tasks use resources in limited supply? Example 6 - -Sat n oolean variables,,, n K constraints of the form i v j v k where i is either true or false NP-complete ecall G and LK Properties of CSPs ypes of variables Discrete and finite Map colouring, 8-queens, boolean CSPs Discrete variables with infinite domains Scheduling jobs in a calendar equire a constraint language (Job + Job ) Continuous domains Scheduling on the Hubble telescope Linear programming

5 Properties of CSPs ypes of contraints Unary constraint relates a variable to a single value ueensland=lue, Green inary constraints relates two variables Can use a constraint graph to represent CSPs with only binary constraints Higher order constraints involve three of more variables Alldiff(,, n ) Can use a constraint hypergraph to represent the problem 5 CSPs and search N variables,, n alid assignment: { =x,, k =x k } for 0 k n such that values satisfy constraints on the variables States: valid assignments Initial state: empty assignment Successor: { =x,, K =x k } { =x,, k =x k, k+ =x k+ } Goal test: complete assignment If all domains all have size d, then there are O(d n ) complete assignments 6 CSPs and commutativity CSPs are commutative! he order of application of any given set of actions has no effect on the outcome When assigning values to variables we reach the same partial assignment, no matter the order All CSP search algorithms generate successors by considering possible assignments for only a single variable at each node in the search tree CSPs and commutativity variables,, Let the current assignment be A={ =x } Pick variable Let domain of be {a,b,c} he successors of A are { =x, =a} { =x, =b} { =x, =c} 7 8 acktracking Search acktracking 0 Depth first search which chooses values for one variable at a time acktracks when a variable has no legal values to assign 9 0 5

6 acktracking acktracking 0 0 =blue =red =green =blue =red =green =blue =red =green acktracking acktracking and efficiency =blue =blue =red =blue =red 0 =red =green =green =green acktracking search is an uninformed search method Not very efficient We can do better by thinking about the following questions Which variable should be assigned next? In which order should its values be tried? Can we detect inevitable failure early (and avoid the same failure in other paths)? Most constrained variable Choose the variable which has the fewest legal moves AKA minimum remaining values (M) heuristic Most constraining variable Most constraining variable: choose the variable with the most constraints on remaining variables ie-breaker among most constrained variables D ={green, blue} D ={green, blue} D ={blue} D ={blue, red} is involved in 5 constraints D others ={red, green, blue} D others ={red,green,blue} 5 6 6

7 Least-constraining value Given a variable, choose the least constraining value: the one that rules out the fewest values in the remaining variables Forward checking he third questions was Is there a way to detect failure early? Forward checking Keep track of remaining legal values for unassigned variables erminate search when any variable has no legal values 7 8 Forward Checking in Map Coloring Forward Checking in Map Coloring Forward checking removes the value ed of and of 9 0 Forward Checking in Map Coloring Forward Checking in Map Coloring G G G G G G G G G 7

8 Forward Checking in Map Coloring Example: ueens Empty set: the current assignment {( ), ( G), ( )} does not lead to a solution X X G G G G X X Example: ueens Example: ueens X X X X {,,,} X X X X {,,,} {,,, } 5 6 Example: ueens Example: ueens X X {,,,} X X {,,,} X X X X {,,,} {,,, } {,,, } {,,, } 7 8 8

9 Example: ueens Example: ueens X X {,,,} X X {,,,} X X X X {,,,} {,,,} {,,, } {,,,} 9 50 Example: ueens Example: ueens X X {,,,} X X {,,,} X X X X {,,, } {,,,} {,,, } {,,,} 5 5 Example: ueens Example: ueens X X {,,,} X X {,,,} X X X X {,,, } {,,,} {,,, } {,,,} 5 5 9

10 Example: ueens Summary X X {,,, } X {,,,} X {,,,} What you should know How to formalize problems as CSPs acktracking search Heuristics ariable ordering alue ordering Forward checking Next class Adversarial search ussell and Norvig, Chapter