1 / 48

490 likes | 655 Views

Constraint Satisfaction Problems. Intro Example: 8-Queens. Generate-and-test: 8 8 combinations. Intro Example: 8-Queens. Constraint Satisfaction Problem. Set of variables {X 1 , X 2 , …, X n } Each variable X i has a domain D i of possible values Usually D i is discrete and finite

Download Presentation
## Constraint Satisfaction Problems

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**Intro Example: 8-Queens**Generate-and-test:88 combinations**Constraint Satisfaction Problem**• Set of variables {X1, X2, …, Xn} • Each variable Xi has a domain Di of possible values • Usually Di is discrete and finite • Set of constraints {C1, C2, …, Cp} • Each constraint Ck involves a subset of variables and specifies the allowable combinations of values of these variables • Assign a value to every variable such that all constraints are satisfied**Example: 8-Queens Problem**• 8 variables Xi, i = 1 to 8 • Domain for each variable {1,2,…,8} • Constraints are of the forms: • Xi = k Xj k for all j = 1 to 8, ji • Xi = ki, Xj= kj |i-j| | ki - kj| • for all j = 1 to 8, ji**NT**Q WA SA NT NSW Q V WA SA T NSW V T Example: Map Coloring • 7 variables {WA,NT,SA,Q,NSW,V,T} • Each variable has the same domain {red, green, blue} • No two adjacent variables have the same value: • WANT, WASA, NTSA, NTQ, SAQ, SANSW, SAV,QNSW, NSWV**Binary constraints**NT Q WA NSW SA V T Constraint Graph Two variables are adjacent or neighbors if they are connected by an edge or an arc**{}**NT Q WA WA=red WA=green WA=blue SA NSW V WA=red NT=green WA=red NT=blue T WA=red NT=green Q=red WA=red NT=green Q=blue Map Coloring**Backtracking Algorithm**CSP-BACKTRACKING(PartialAssignment a) • If a is complete then return a • X select an unassigned variable • D select an ordering for the domain of X • For each value v in D do • If v is consistent with a then • Add (X= v) to a • result CSP-BACKTRACKING(a) • If resultfailure then return result • Return failure CSP-BACKTRACKING({})**Questions**• Which variable X should be assigned a value next? • In which order should its domain D be sorted? • In which order should constraints be verified?**NT**WA NT Q WA SA SA NSW V T Choice of Variable • Map coloring**Choice of Variable**• 8-queen**Choice of Variable**Most-constrained-variable heuristic: Select a variable with the fewest remaining values = Fail First Principle**NT**Q WA SA SA NSW V T Choice of Variable Most-constraining-variable heuristic: Select the variable that is involved in the largest number of constraints on other unassigned variables = Fail First Principle again**NT**WA NT Q WA SA NSW V {} T Choice of Value**NT**WA NT Q WA SA NSW V {blue} T Choice of Value Least-constraining-value heuristic: Prefer the value that leaves the largest subset of legal values for other unassigned variables**Choice of Constraint to Test**Most-constraining-Constraint: Prefer testing constraints that are more difficult to satisfy = Fail First Principle**Constraint Propagation …**… is the process of determining how the possible values of one variable affect the possible values of other variables**Forward Checking**After a variable X is assigned a value v, look at each unassigned variable Y that is connected to X by a constraint and deletes from Y’s domain any value that is inconsistent with v**NT**Q WA NSW SA T V Map Coloring**NT**Q WA NSW SA T V Map Coloring**NT**Q WA NSW SA T V Map Coloring**NT**Q WA NSW SA T V Impossible assignments that forward checking do not detect Map Coloring**X1**{1,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} Example: 4-Queens Problem**X1**{1,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} Example: 4-Queens Problem**X1**{1,2,3,4} X2 { , ,3,4} 1 2 3 4 1 2 3 4 X3 { ,2,,4} X4 { ,2,3, } Example: 4-Queens Problem**X1**{1,2,3,4} X2 { ,,3,4} 1 2 3 4 1 2 3 4 X3 {,2,,4} X4 {,2,3,} Example: 4-Queens Problem**X1**{1,2,3,4} X2 { ,,3,4} 1 2 3 4 1 2 3 4 X3 { ,,,} X4 { ,2,3, } Example: 4-Queens Problem**X1**{ ,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} Example: 4-Queens Problem BT**X1**{,2,3,4} X2 {,,,4} X3 {1, ,3, } X4 {1, ,3,4} Example: 4-Queens Problem 1 2 3 4 1 BT 2 3 4**X1**{,2,3,4} X2 {,,,4} X3 {1, ,3, } X4 {1, ,3,4} Example: 4-Queens Problem 1 2 3 4 1 BT 2 3 4**X1**{,2,3,4} X2 {,,,4} X3 {1, , , } X4 {1, ,3, } Example: 4-Queens Problem 1 2 3 4 1 BT 2 3 4**X1**{,2,3,4} X2 {,,,4} X3 {1, , , } X4 {1, ,3, } Example: 4-Queens Problem 1 2 3 4 1 BT 2 3 4**X1**{,2,3,4} X2 {,,,4} X3 {1, , , } X4 { , ,3, } Example: 4-Queens Problem 1 2 3 4 1 BT 2 3 4**X1**{,2,3,4} X2 {,,,4} X3 {1, , , } X4 { , ,3, } Example: 4-Queens Problem 1 2 3 4 1 BT 2 3 4**Trihedral Objects**• Objects in which exactly three plane surfaces come together at each vertex. • Goal: label a 2-D object to produce a 3-D object**Labels of Edges**• Convex edge: • two surfaces intersecting at an angle greater than 180° • Concave edge • two surfaces intersecting at an angle less than 180° • + convex edge, both surfaces visible • − concave edge, both surfaces visible • convex edge, only one surface is visible and it is on the right side of **-**- + + - + + - - + + - - - - + + + - + Junction Label Sets (Waltz, 1975; Mackworth, 1977)**+**+ + + + - - + + + + + Edge Labeling**Edge Labeling as a CSP**• A variable is associated with each junction • The domain of a variable is the label set of the corresponding junction • Each constraint imposes that the values given to two adjacent junctions give the same label to the joining edge**-**+ + - - + + - + Edge Labeling**+**- - + + - - - - + + Edge Labeling**+**+ + + - + + - - + + Edge Labeling + +**-**- + + + + + - - Edge Labeling + +**Removal of Arc Inconsistencies**REMOVE-ARC-INCONSISTENCIES(J,K) • removed false • X label set of J • Y label set of K • For every label y in Y do • If there exists no label x in X such that the constraint (x,y) is satisfied then • Remove y from Y • If Y is empty then contradiction true • removed true • Label set of K Y • Return removed**CP Algorithm for Edge Labeling**• Associate with every junction its label set • Q stack of all junctions • while Q is not empty do • J UNSTACK(Q) • For every junction K adjacent to J do • If REMOVE-ARC-INCONSISTENCIES(J,K) then • If K’s domain is non-empty then STACK(K,Q) • Else return false (Waltz, 1975; Mackworth, 1977)

More Related