lecture
Lectures Today's 5 & 6: Constraint Satisfaction
CMPSCI 383: Artificial Intelligence Instructor: Shlomo Zilberstein
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Formulating constraint satisfaction problems. Solving CSPs using backtracking search. Solving CSPs using local search. The relationship between problem structure and complexity.
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Shlomo Zilberstein University of Massachusetts
Constraint satisfaction problems
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Constraint satisfaction problems
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In a general search problem a state is a "black box"; any data structure can be used In a constraint satisfaction problem (CSP) a state is a set of variables X1...Xn. Each variable Xi has a domain Di of possible values. The goal test is a set of constraints C1...Cm over the values of the variables. A solution to a CSP: a complete assignment to all variables that satisfies all the constraints.
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Representation of constraints as predicates. Visualizing a CSP as a constraint graph. Allows useful general-purpose algorithms with more power than standard search algorithms
Shlomo Zilberstein University of Massachusetts
Shlomo Zilberstein University of Massachusetts
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Example 1: Map coloring
Map coloring
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Variables WA, NT, Q, NSW, V, SA, T Domains Di = {red,green,blue} Constraints: adjacent regions must have different colors e.g., WA ! NT, or (WA,NT) in {(red,green),(red,blue),(green,red), (green,blue),(blue,red),(blue,green)}
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Solutions are complete and consistent assignments, e.g., WA = red, NT = green,Q = red,NSW = green,V = red, SA = blue,T = green
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Constraint graph
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Example 2: Sudoku
Binary CSP: each constraint relates two variables Constraint graph: nodes are variables, arcs are constraints
NT" WA" SA" V"
NSW"
Q"
T"
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! What are the variables? domains? constraints?" ! How to can you generate easy/hard problem?"
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Shlomo Zilberstein University of Massachusetts
Example 3: N queens
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8 variables Xi, i = 1 to 8 Domain for each variable {1,2,...,8} Constraints are:
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Xi ! Xj for all j = 1 to 8, j!i |Xi - Xj| ! |i - j| for all j = 1 to 8, j!I
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Note that all constraints involve 2 variables Generate-and-test with no redundancies requires "only" NN combinations...
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! What are the variables? domains? constraints?"
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Example 4: Cryptarithmetic
T1 T2 T3 T1 T2 T2 T4 must must must must be done during T3 be achieved before T1 starts overlap with T3 start after T1 is complete
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Higer-order constraints:!
T4
T W O! + T W O! FOUR
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U!
W!
R!
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! What are the variables? domains? constraints?"
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O + O = R + 10X1! X3 X1 + W + W = U + 10X2! X2 + T + T = O + 10X3! X3 = F! alldiff(F,T,U,W,R,O)!
X2
X1
Shlomo Zilberstein University of Massachusetts
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Real-world CSPs
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Finite vs. infinite domains
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Assignment problems
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e.g., who teaches what class e.g., which class is offered when and where?
Finite domains: N queens, matching, cryptarithmetic, job assignment
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Timetabling problems
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Finite-domain " Boolean " 3SAT Cannot enumerate all possibilities Need a constraint language: Start-Job1 + 5 " Start-Job3 Choice of language affects complexity
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Infinite domains: job scheduling
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Transportation scheduling Factory scheduling Notice that many real-world problems involve real-valued variables
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Shlomo Zilberstein University of Massachusetts
Shlomo Zilberstein University of Massachusetts
Satisfaction vs. optimization
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Solving CSPs using search
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Representing preferences versus absolute constraints. Constraint optimization is generally more complicated. Can be solved using local search techniques. Hard to find optimal solutions.
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Initial state: the empty assignment Successor function: a value can be assigned to any variable as long as no constraint is violated. Goal test: the current assignment is complete. Path cost: irrelevant (a constant cost for every unsatisfied constraint in optimization).
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Shlomo Zilberstein University of Massachusetts
Shlomo Zilberstein University of Massachusetts
Commutativity
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Not just a successor function and goal test But also a means to propagate the constraints imposed by one queen on the others and an early failure test Thus, need explicit representation of constraints and constraint manipulation algorithms
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Nave application of search to CSPs:
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Branching factor is nd at the top level, then (n-1)d, and so on for n levels. The tree has n!dn leaves, even though there are only dn possible complete assignments!
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Nave formulation ignores commutativity of all CSPs. Solution: consider a single variable at each depth of the tree.
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Shlomo Zilberstein University of Massachusetts
Shlomo Zilberstein University of Massachusetts
Backtracking search
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Backtracking search
Depth-first search for CSPs with singlevariable assignments is called backtracking search Backtracking search is the basic uninformed algorithm for CSPs Can solve n-queens for n # 25
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Shlomo Zilberstein University of Massachusetts
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Shlomo Zilberstein University of Massachusetts
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Backtracking example
Heuristics that can help
Key questions:
1.! Which variable should be assigned next and in what order should the values be tried? 2.! What are the implications of the current variable assignments for the other unassigned variables? 3.! How to detect inevitable failure early? 4.! When a path fails, can the search avoid repeating this failure in subsequent paths?
Shlomo Zilberstein University of Massachusetts
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Shlomo Zilberstein University of Massachusetts
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Variable and value ordering
Variable ordering !! The most-constrained-variable heuristic (has the fewest "legal" values) !! The most-constraining-variable heuristic (involved in largest number of constraints) Value ordering !! The least-constraining-value heuristic (rules out the fewest choices for neighboring vars)
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Most constrained variable
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Most constrained variable: choose the variable with the fewest legal values
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a.k.a. minimum remaining values (MRV) heuristic
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Shlomo Zilberstein University of Massachusetts
Most constraining variable
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Least value
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Tie-breaker constraining among most constrained variables Most constraining variable:
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Given a variable, choose the least constraining value:
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choose the variable with the most constraints on remaining variables
the one that rules out the fewest values in the remaining variables
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Combining these heuristics makes 1000 queens feasible
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Shlomo Zilberstein University of Massachusetts
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Shlomo Zilberstein University of Massachusetts
... is the process of determining how the possible values of one variable affect the possible values of other variables
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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 Reduces the branching factor and help identify failures early.
Shlomo Zilberstein University of Massachusetts
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Shlomo Zilberstein University of Massachusetts
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Forward checking example
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Constraint propagation
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Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values
Forward checking propagates information from assigned to unassigned variables, but doesn't provide early detection for all failures:
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NT and SA cannot both be blue! Constraint propagation repeatedly enforces constraints
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Shlomo Zilberstein University of Massachusetts
Arc consistency
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Simplest form of propagation makes each arc consistent X "Y is consistent iff
for every value x of X there is some allowed y
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If X loses a value, neighbors of X need to be rechecked Arc consistency detects failure earlier than forward checking Can be run as a preprocessor or after each assignment
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!! Time complexity: O(n2d3)
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Shlomo Zilberstein University of Massachusetts
What is the result of applying arc consistency to this puzzle?"
Answer: (assuming inequality constraints)" 1,2 1,2
4 3 3 4
4 3
1,2
1,2 1,2,4
1,2,3 1,2
1,2
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1,2
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1,2
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1,2,3,4 1,2
Shlomo Zilberstein University of Massachusetts 33
Shlomo Zilberstein University of Massachusetts
Shlomo Zilberstein University of Massachusetts
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Shlomo Zilberstein University of Massachusetts
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Labels of edges
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Convex edge:
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two surfaces intersecting at an angle greater than 180
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Concave edge
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two surfaces intersecting at an angle less than 180
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+ 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 #
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Shlomo Zilberstein University of Massachusetts
Shlomo Zilberstein University of Massachusetts
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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
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(Waltz, 1975; Mackworth , 1977)
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Shlomo Zilberstein University of Massachusetts
Shlomo Zilberstein University of Massachusetts
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Shlomo Zilberstein University of Massachusetts
Complexity of arc consistency
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A binary CSP has at most O(n2) arcs Each arc (X$Y) can only be inserted on the agenda d times because at most d values of Y can be deleted. Checking consistency of an arc can be done in O(d2) time. Worst case time complexity is: O(n2d3). Does not reveal every possible inconsistency!
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Search:
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can find good solutions, but must examine nonsolutions along the way...