[PDF] - Constraint Satisfaction Problems Problems Artificial Intelligence PDF Document

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Constraint Satisfaction Problems Problems

Artificial Intelligence AIMA 2nd edition, chapter 5 Section 1 – 3 Hadi Moradi

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Outline

Constraint Satisfaction Problems (CSP)

B kt ki h f CSP

Backtracking search for CSPs Local search for CSPs

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Constraint satisfaction problems (CSPs)

CSP:

State
goal test
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Example: Map-Coloring

Variables

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Domains
Constraints:

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Example: Map-Coloring

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Solutions are complete and consistent assignments,

Constraint graph

Binary CSP: each constraint relates two variables

Constraint graph: nodes are variables, arcs are constraints
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Varieties of CSPs

Discrete variables

finite domains: infinite domains:

7 Continuous variables

Varieties of constraints

Unary constraints involve a single variable, Binary constraints involve pairs of variables,

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Higher-order constraints involve 3 or more

variables,

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Real-world CSPs

Assignment problems Timetabling problems Transportation scheduling

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p g

Factory scheduling

Standard search formulation (incremental)

Let's start with the straightforward approach, then fix it

Initial state:

Initial state:

Successor function
Goal test:

B anching facto

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1.

Branching factor:

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Backtracking search

Variable assignments are commutative}, i.e.,

[ WA = red then NT = green ] same as [ NT = green then WA = red ]

Depth-first search for CSPs with single-variable assignments is called

backtracking search

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Can solve n-queens for n ≈ 25

Backtracking search

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Backtracking example

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Backtracking example

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Backtracking example

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Backtracking example

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Improving backtracking efficiency

General-purpose methods can give huge

gains in speed: gains in speed:

Which variable should be assigned next? In what order should its values be tried?

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Can we detect inevitable failure early?

Minimum Remaining Variable

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Degree Heuristic

Choose the Most constraining variable:

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Least constraining value

Given a variable, choose the least

constraining value: constraining value:

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Forward checking

Idea:

Keep track of remaining legal values for unassigned variables

p g g g

Terminate search when any variable has no legal values

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Forward checking

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Forward checking

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Forward checking

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Constraint propagation

Forward checking doesn't provide early detection for all

failures: failures:

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Arc consistency

Simplest form of propagation makes each arc consistent

X Y i i t t iff

X Y is consistent iff

for every value x of X there is some allowed y

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Arc consistency

Simplest form of propagation makes each arc consistent

X Y i i t t iff

X Y is consistent iff

for every value x of X there is some allowed y

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Arc consistency

If X loses a value, neighbors of X need to be rechecked

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Arc consistency

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Arc consistency algorithm AC-3

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Local search for CSPs

Hill-climbing, simulated annealing typically work with

"complete" states, i.e., all variables assigned

To apply to CSPs: 31

Example: 4-Queens

States: 4 queens in 4 columns (44 = 256 states)

Actions: move queen in column
Goal test: no attacks
Evaluation: h(n) = number of attacks
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Summary

CSPs are a special kind of problem:
states defined by values of a fixed set of variables
goal test defined by constraints on variable values
goal test defined by constraints on variable values
Backtracking = depth-first search with one variable assigned per node
Variable ordering and value selection heuristics help significantly
Forward checking prevents assignments that guarantee later failure
Constraint propagation (e.g., arc consistency) does additional work to

constrain values and detect inconsistencies

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constrain values and detect inconsistencies

Iterative min-conflicts is usually effective in practice