lecture 1: constraint networks overview
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Constraint NetworksOverview
Suggested reading
Russell and Norvig. Artificial Intelligence: Modern Approach. Chapter 5.
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Good source of advanced information
Rina Dechter,
Constraint Processing,Morgan Kaufmann
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Outline
CSP: Definition, and simple modeling examples Representing constraints Basic search strategy Improving search:
– Consistency algorithms– Look-ahead methods– Look-back methods
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Outline
CSP: Definition, and simple modeling examples Representing constraints Basic search strategy Improving search:
– Consistency algorithms– Look-ahead methods– Look-back methods
A Bred greenred blackgreen redgreen blackblack greenblack red
Constraint Satisfaction
Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (e.g., red, green, black) Constraints:
A≠B, A≠D, D≠E , etc .
C
A
B
D
E
F
G
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Constraint Network; Definition
A constraint network is: R=(X,D,C)X variables
D domain
C constraints
R expresses allowed tuples over scopes
A solution is an assignment to all variables that satisfies all constraints (join of all relations).
Tasks: consistency?, one or all solutions, counting, optimization
X={X1 , .. . ,X n}
D={D1 ,. . . , Dn}, Di={v1 , . .. vk }
C={C1 ,. .. C t },,, C i= S i , Ri
Example The 4-queen problem
Q
Q QQ
Place 4 Queens on a chess board of 4x4 such that no two queens reside in the same row, column or diagonal.
Standard CSP formulation of the problem:• Variables: each row is a variable.
X1
X 4
X 3
X 2
1 2 3 4
• Domains: Di={1,2,3,4}.
• Constraints: There are = 6 constraints involved:42( )
R12={1,31,4 2,4 3,1 4,1 4,2 }R13={1,2 1,4 2,12,3 3,2 3,4 4,1 4,3}R14={1,2 1,3 2,12,3 2,4 3,1 3,2 3,4 4,2 4,3 }R23={1,31,4 2,4 3,1 4,1 4,2 }R24={1,2 1,4 2,1 2,3 3,2 3,4 4,1 4,3 }R34={1,3 1,4 2,4 3,1 4,1 4,2 }
• Constraint Graph :X1
X 2 X 4
X 3
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Outline
CSP: Definition, and simple modeling examples Representing constraints Basic search strategy Improving search:
– Consistency algorithms– Look-ahead methods– Look-back methods
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Constraint’s representations
Relation: allowed tuples
Algebraic expression:
Propositional formula:
Constraint graph
X Y 2≤ 10 , X ≠Y
a∨b ¬c
X Y Z1 3 22 1 3
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Figure 1.8: Example of set operations intersection, union, and difference applied to relations.
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Constraint Graphs: Primal, Dual and Hypergraphs
A (primal) constraint graph: a node per variable arcs connect constrained variables. A dual constraint graph: a node per constraint’s scope, an
arc connect nodes sharing variables =hypergraph
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Outline
CSP: Definition, and simple modeling examples Representing constraints Basic search strategy Improving search:
– Consistency algorithms– Look-ahead methods– Look-back methods
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Backtracking search
Search space
Backtracking
The search space
A tree of all partial solutions A partial solution: (a1,…,aj) satisfying all
relevant constraints The size of the underlying search space depends
on: Variable ordering Level of consistency possessed by the problem
Search space and the effect of ordering
Backtracking
Complexity of extending a partial solution: Complexity of consistent O(e log
t), t bounds tuples, e constraints Complexity of selectvalue O(e k
log t)
A coloring problem example
Backtracking search for a solution
Backtracking Search for a Solution
Backtracking Search for All Solutions
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Outline
CSP: Definition, and simple modeling examples Representing constraints Basic search strategy Improving search:
– Consistency algorithms– Look-ahead methods– Look-back methods
• Before search: (reducing the search space)– Arc-consistency, path-consistency, i-consistency– Variable ordering (fixed)
• During search:– Look-ahead schemes:
• Value ordering/pruning (choose a least restricting value), • Variable ordering (Choose the most constraining variable)
– Look-back schemes:• Backjumping• Constraint recording• Dependency-directed backtracking
Improving Backtracking O(exp(n))
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Consistency methods
Constraint propagation – inferring new constraints
Can get such an explicit network that the search will find the solution without dead-ends.
Approximation of inference: Arc, path and i-consistency
Methods that transform the original network into a tighter and tighter representations
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Arc-consistency
32,1,
32,1, 32,1,
1 X, Y, Z, T 3X YY = ZT ZX T
X Y
T Z
32,1,
=
- infer constraints based on pairs of variables
Insures that every legal value in the domain of a single variable hasa legal match In the domain of any other selected variable
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1 X, Y, Z, T 3X YY = ZT ZX T
X Y
T Z
=
1 3
2 3
Arc-consistency
Arc-consistency algorithm
domain of x domain of y
Arc is arc-consistent if for any value of there exist a matching value of
Algorithm Revise makes an arc consistent
Begin
1. For each a in Di if there is no value b in Dj that matches a then delete a from the Dj.
End.
Revise is , k is the number of value in each domain.
)( ji ,XX iX iX
)( ji ,XX
)O(k 2
Algorithm AC-3
• Begin– 1. Q <--- put all arcs in the queue in both directions– 2. While Q is not empty do,– 3. Select and delete an arc from the queue Q
• 4. Revise• 5. If Revise cause a change then add to the queue all arcs
that touch Xi (namely (Xi,Xm) and (Xl,Xi)).– 6. end-while
• End• Complexity:
– Processing an arc requires O(k^2) steps– The number of times each arc can be processed is 2·k– Total complexity is
)( ji ,XX
)( ji ,XX
)O(ek 3
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Sudoku –Constraint Satisfaction
Each row, column and major block must be alldifferent
“Well posed” if it has unique solution: 27 constraints
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•Variables: empty slots
•Domains = {1,2,3,4,5,6,7,8,9}
•Constraints: 27 all-different
•Constraint •Propagation
•Inference
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Path-consistency
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I-consistency
The Effect of Consistency Level
• After arc-consistency z=5 and l=5 are removed
• After path-consistency– R’_zx– R’_zy– R’_zl– R’_xy– R’_xl– R’_yl
Tighter networks yield smaller search spaces
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Outline
CSP: Definition, and simple modeling examples Representing constraints Basic search strategy Improving search:
– Consistency algorithms– Look-ahead methods– Look-back methods
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Outline
CSP: Definition, and simple modeling examples Representing constraints Basic search strategy Improving search:
– Consistency algorithms– Look-ahead methods– Look-back methods
Look-back: Backjumping / Learning
• Backjumping: – In deadends, go back to the most recent culprit.
• Learning: – constraint-recording, no-good recording.– good-recording
Backjumping
• (X1=r,x2=b,x3=b,x4=b,x5=g,x6=r,x7={r,b})• (r,b,b,b,g,r) conflict set of x7• (r,-,b,b,g,-) c.s. of x7• (r,-,b,-,-,-,-) minimal conflict-set• Leaf deadend: (r,b,b,b,g,r)• Every conflict-set is a no-good
The cycle-cutset method
• An instantiation can be viewed as blocking cycles in the graph
• Given an instantiation to a set of variables that cut all cycles (a cycle-cutset) the rest of the problem can be solved in linear time by a tree algorithm.
• Complexity (n number of variables, k the domain size and C the cycle-cutset size):
)( 2knkO C
Tree Decomposition
GSAT – local search for SAT(Selman, Levesque and Mitchell, 1992)
1. For i=1 to MaxTries2. Select a random assignment A3. For j=1 to MaxFlips4. if A satisfies all constraint, return A5. else flip a variable to maximize the score 6. (number of satisfied constraints; if no variable 7. assignment increases the score, flip at random)8. end9. end
Greatly improves hill-climbing by adding restarts and sideway moves
WalkSAT (Selman, Kautz and Cohen, 1994)
With probability p random walk – flip a variable in some unsatisfied constraintWith probability 1-p perform a hill-climbing step
Adds random walk to GSAT:
Randomized hill-climbing often solves large and hard satisfiable problems
More Stochastic Search: Simulated Annealing, reweighting• Simulated annealing:
– A method for overcoming local minimas– Allows bad moves with some probability:
• With some probability related to a temperature parameter T the next move is picked randomly.
– Theoretically, with a slow enough cooling schedule, this algorithm will find the optimal solution. But so will searching randomly.
• Breakout method (Morris, 1990): adjust the weights of the violated constraints
