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Chapter 5. Constraint Satisfaction Problems (CSPs). Date:2004/3/17 Presenter: Shih, Ya-Ting. Outline. CSPs Backtracking search for CSPs Local search for CSP Problem structure. 1 CSPs. a type of Assignment Problems for problem solving

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constraint satisfaction problems csps

Chapter 5

Constraint Satisfaction Problems (CSPs)

Date:2004/3/17Presenter: Shih, Ya-Ting

outline
Outline
  • CSPs
  • Backtracking search for CSPs
  • Local search for CSP
  • Problem structure
1 csps
1 CSPs
  • a type of Assignment Problems for problem solving
  • states & goal test : a standard, structured, & very simple representation
  • CSP is defined by –a set of variables, X1,X2,…,Xn,with values from domain D1,D2,…,Dn,and a set of constraints, C1,C2,…,Cn,specifying allowance combinations of values for subsets of variables
1 csps cont
1 CSPs (cont.)
  • State is defined by an assignment of values to some or all of the variables,{ Xii= vi , Xjj= vj , … }
  • Consistent ( or legal) assignment is an assignment that does not violate any constraints.
  • Complete assignment is one in which every variable is mentioned.
  • Solution is a complete assignment that satisfies all the constraints.
1 3 varieties of csps
1.3 Varieties of CSPs
  • Discrete variables- finite domains:eg. Mapping coloring、8-Queens puzzle - infinite domains: (integers , strings , etc.)need a constraint language eg. Job scheduling ,StartJob1+5<StartJob3
  • Continuous variables eg. Hubble Space Telescope observation Linear programming problems
1 4 varieties of constraints
1.4 Varieties of Constraints
  • Unary constraint:involves a single variableeg. SA ≠ green
  • Binary constraint:involves pairs of variableseg. SA ≠ WA
  • Higher-order constraint:involves 3 or more variableseg. Cryptarithemetic puzzles (F5.2(a))(p.148)
  • Preference constrainteg. red is better than greenoften representable by a cost for each variable assignment constrained optimization problems
2 backtracking search for csps
2 Backtracking search for CSPs
  • In all CSPs, variable assignment are commutative. (if…)eg. [ WA = red then NT = green ] same as [ NT = green then WA = red ]
  • Only need to consider assignment to a single value at each node.
  • Backtracking search-- a form of DSF search for CSP with single–variable assignments
2 2 some key questions of backtracking search
2.2 Some Key Questions of Backtracking Search
  • Variable and value orderingwhich variable should be assigned next, and in what order should its values be tried?
  • Propagating information through constraintswhat are the implications of the current variable assignments for the other unassigned variables?
  • Intelligent backtrackingwhen a path fails– that is, a state is reached in which a variable has no legal value – can the search avoid repeating this failure in subsequent paths?
2 2 1 most constrain ed variable
2.2.1 Most constrained variable
  • Minimum remaining variable (MRV) heuristicor Most constrained variable heuristicor Fail-First heuristic-- choose the variable with the fewer legal values
2 2 2 most constrain ing variable
2.2.2 Most constraining variable
  • Degree heuristic-- tie-breaker among most constrained variables-- choose the variable with the most constrains on remaining variables
2 2 3 least constrain ing value
2.2.3 Least constraining value
  • Least-constraining-value heuristic-- try to leave the maximum flexibility for subsequent variable assignment-- prefer the value that rules out the fewest choices for the neighboring variables in the constraint graph
2 3 constraint propagation
2.3 Constraint Propagation
  • Propagation the implications of a constraint on one variable onto other variables -- Forward checking-- Arc consistency (more stronger)
2 3 1 forward checking
2.3.1 Forward Checking
  • Idea:Keep track of remaining legal values for unassigned variables. Terminate search when any variable has no legal values.
  • Whenever a variable X is assigned, the forward checking process looks 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 the value chosen for X.
2 3 1 forward checking cont
2.3.1 Forward Checking (cont.)

WA

NT

Q

NSW

V

SA

T

Initial domains

After WA=red

After Q=green

After V=blue

Figure 5.6

Partial assignment { WA=red, Q=green V=blue } is inconsistent with the constraint of the problem

2 3 2 arc consistency
2.3.2 Arc Consistency
  • Simplest form of propagation makes each arc consistent.
  • X→Y is consistent iff for every value x of X, there is some allowed y
  • Applying arc consistency has result in early detection of an inconsistency that is not detected by pure forward checking.
2 4 intelligent backtracking
2.4 Intelligent backtracking
  • called chronological backtrackingcause the most recent decision point is revisited-- to go all the way back to one of the set of variables that caused the failure-- conflict set
  • The backjumping method backtracks to the most recent variable in the conflict set
  • Backjumping occurs when every value in a domain is in conflict with the current assignment .
2 4 intelligent backtracking cont
2.4 Intelligent backtracking (cont.)
  • fixed variable ordering Q, NSW, V, T, SA, WA, NT
  • partial assignment {Q=red, NSW=green, V=blue, T=red} (conflict set for SA) every value violates a constraint

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3 local search for csp
3 Local Search for CSP
  • min-conflicts heuristic choose value that violates the fewest constraints
  • 8-Queens problem
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