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Recap Consistency Arc Consistency

CSPs: Arc Consistency

CPSC 322 CSPs 3

Textbook 4.5

CSPs: Arc Consistency CPSC 322 CSPs 3, Slide 1
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Lecture Overview

1 Recap

2 Consistency

3 Arc Consistency

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Constraint Satisfaction Problems: Definition

Definition

A constraint satisfaction problem consists of:

a set of variables

a domain for each variablea set of constraints

DefinitionA model of a CSP is an assignment of values to variables thatsatisfies all of the constraints.


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CSPs as Search Problems

We map CSPs into search problems:

nodes: assignments of values to a subset of the variables

neighbours of a node: nodes in which values are assigned to

one additional variablestart node: the empty assignment (no variables assignedvalues)

goal node: a node which assigns a value to each variable, and

satisfies all of the constraintsNote: the path to a goal node is not important

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Lecture Overview

1 Recap

2 Consistency

3 Arc Consistency


R C i A C i
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Consistency Algorithms

Idea: prune the domains as much as possible before selectingvalues from them.

DefinitionA variable is domain consistent if no value of the domain of thenode is ruled impossible by any of the constraints.

Example: dom(B) = {1, 2, 3, 4} isnt domain consistent if wehave the constraint B = 3.


R C i t A C i t
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Constraint Networks

Domain consistency only talked about constraints involving a

single variablewhat can we say about constraints involving multiple variables?

Definition

A constraint network is defined by a graph, with

one node for every variable

one node for every constraint

and undirected edges running between variable nodes andconstraint nodes whenever a given variable is involved in a given

constraint.

When all of the constraints are binary, constraint nodes arenot necessary: we can drop constraint nodes and use edges toindicate that a constraint holds between a pair of variables.

why cant we do the same with general k-ary constraints?



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Example Constraint Network

A A

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Lecture Overview

1 Recap

2 Consistency

3 Arc Consistency


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Arc Consistency

DefinitionAn arc

X, r(X, Y)

is arc consistent if for each value of X in

dom(X) there is some value Y in dom(Y) such that r(X, Y) issatisfied.

In symbols, X dom(X), Y dom(Y) such that r(X, Y)is satisfied.

A network is arc consistent if all its arcs are arc consistent.

If an arc

X, Y

is not arc consistent, all values of X in

dom(X) for which there is no corresponding value in dom(Y)may be deleted from dom(X) to make the arc

X, Y

consistent.

This removal can never rule out any models (do you see why?)


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Arc Consistency Outcomes

Three possible outcomes (when all arcs are arc consistent):

One domain is empty no solution

Each domain has a single value unique solutionSome domains have more than one value may or may notbe a solution

in this case, arc consistency isnt enough to solve the problem:

we need to perform search


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Arc Consistency Algorithm

Consider the arcs in turn making each arc consistent.

An arc

X, r(X, Y)

needs to be revisited if the domain of Yis reduced.

Regardless of the order in which arcs are considered, we will

terminate with the same result: an arc consistent network.Worst-case complexity of this procedure:

let the max size of a variable domain be dlet the number of constraints be ecomplexity is O(ed3)

Some special cases are faster

e.g., if the constraint graph is a tree, arc consistency is O(ed)


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p y y

Arc Consistency Algorithm (binary constraints case)



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Adding edges back to TDA (binary constraints case)

When we change the domain of a variable X in the course ofmaking an arc X, r arc consistent, we add every arc Z, rwhere r involves X and:

r = r

Z= X

Thus we dont add back the same arc:

This makes senseits definitely arc consistent.



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r = r

Z= X

We dont add back other arcs that involve the same variableX

Weve just reduced the domain of X

If an arc X, r was arc consistent before, it will still be arcconsistent

in the for all well just check fewer values


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r = r

Z= X

We dont add back other arcs that involve the same constraintand a different variable:

Imagine that such an arcinvolving variable Yhad been arcconsistent before, but was no longer arc consistent after Xsdomain was reduced.

This means that some value in Ys domain could satisfy r onlywhen X took one of the dropped valuesBut we dropped these values precisely because there were novalues of Y that allowed r to be satisfied when X takes thesevaluescontradiction!


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Arc Consistency Example

A A

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