Artificial Intelligence Constraint satisfaction problems

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Artificial IntelligenceConstraint satisfaction
problems

• Fall 2008
• professor Luigi Ceccaroni

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Problem characterization

• A constraint satisfaction problem (or CSP) is a
  special kind of problem that satisfies some
  additional structural properties beyond the basic
  requirements for problems in general.
• In a CSP, the states are defined by the values of
  a set of variables and the goal test specifies a
  set of constraints that the values have to obey.

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Problem characterization

• State components
• Variables
• Domains (possible values for the variables)
• (Binary) constraints between variables
• Goal to find a state (a complete assignment of
  values to variables), which satisfies the
  constraints
• Examples
• map coloring
• crossword puzzles
• n-queens
• resource assignment/distribution/location

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Representation

• State constraint graph
• variables (n) node tags
• domains node content
• constraints directed and tagged arcs between
  nodes
• Example map coloring

C1
blue, red
?
C2
blue
C3
C1
?
?
C3
C4
C2
C4
blue, red, green
blue, green
?
initial state
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Representation

• In the search tree, a variable is assigned at
  each level.
• Solutions have to be complete assignment,
  therefore they appear at depth n, the number of
  variable and maximum depth of the tree.
• Depth-first search algorithms are popular in
  CSPs.
• The simplest class of CSP (map coloring,
  n-queens) are characterized by
• discrete variables
• finite domains

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Finite domains

• If the maximum size of the domain of any variable
  is d, then the number of possible complete
  assignments is O(dn), exponential in the number
  of variables.
• CSPs with finite domain include Boolean CSPs,
  whose variables can only be true or false.
• In most practical applications, CSP algorithms
  can solve problems with domains orders of
  magnitude larger than the ones solvable by
  uninformed search algorithms.

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Infinite domains

• With infinite domains (e.g., integers and
  strings), to describe constraints enumerating all
  legal combinations of values is not possible.
• A constraint language has to be used, for
  example
• job-start1 5 job-start3
• There exist algorithms for linear constraints
  over integer variables.
• No algorithm exists for general non-linear
  constraints over integer variables.
• In cases, infinite-domain problems can be reduced
  to a finite domain, just restricting the values
  of all variables (e.g., setting limits to the
  dates in which jobs can start).

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Search-based algorithms

• Incremental formulation
• Initial state empty assignment , with all
  unassigned variables
• Successor function assignment of a value to any
  variable not yet assigned, provided it does not
  conflict with assigned variables
• Goal test check if the current assignment is
  complete
• Path cost a constant cost (e.g., 1) for every
  step
• Complete formulation
• Each state is a complete assignment, which
  satisfies or not the constraints.
• Local-search methods work well in this case.
• Constraint propagation
• Before the search
• During the search

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Constraints

• The simplest type is the unary constraint, which
  constraints the values of just one variable.
• A binary constraint relates two variables.
• Higher-order constraints involve three or more
  variables. Cryptarithmetic puzzles are an example

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Cryptarithmetic puzzles

• Variables F, T, U, W, R, O, X1, X2, X3
• Domains 0,1,2,3,4,5,6,7,8,9
• Constraints
• Alldiff (F,T,U,W,R,O)
• O O R 10 X1
• X1 W W U 10 X2
• X2 T T O 10 X3
• X3 F, T ? 0, F ? 0

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Depth-first search with backtracking

• Standard depth-first search on a CSP wastes time
  searching when constraints have already been
  violated.
• Because of the way that the operators have been
  defined, an operator can never redeem a
  constraint that has already been violated.
• A first improvement is
• To test constraints after each variable
  assignment
• If all possible values violate some constraint,
  then the algorithm backtracks to the last valid
  assignment
• Variables are classified as past, current,
  future.

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Backtracking search algorithm
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Backtracking search algorithm

• Set each variable as undefined. Empty stack. All
  variables are future variables.
• Select a future variable as current variable.
• If it exists, delete it from FUTURE and stack it
  (top current variable),
• if not, the assignment is a solution.
• Select an unused value for the current variable.
• If it exists, mark the value as used,
• if not, set current variable as undefined,
• mark all its values as unused,
• unstack the variable and add it to FUTURE,
• if stack is empty, there is no solution,
• if not, go to 3.
• Test constraints between past variables and the
  current one.
• If they are satisfied, go to 2,
• if not, go to 3.
• (It is possible to use heuristics to select
  variables (2.) and values (3.).

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Backtracking search algorithm
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Example 4-queens

• Place 4 queens, one per row, so that they do not
  attack each others
• Variables R1 R4 (queens)
• Domains 1 4 for each Ri (columns)
• Constraints Ri does not attack Rj

not attacking
Ri
Rj
1 .. 4
1 .. 4
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R11
R12
R21 NO R22 NO R23
R24
R21 NO R22 NO R23 NO R24
R31 NO R32 NO R33 NO R34 NO
R31 NO R32
R31
R41 NO R42 NO R43 NO R44 NO
R41 NO R42 NO R43
Backtracking R2
Backtracking R3, R2, R1
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Propagating information through constraints

• So far the algorithm considers the constraints on
  a variable only at the time that the variable is
  chosen (e.g., by Select-Unassigned-Variable).
• By looking at some of the constraints earlier in
  the search, or even before the search has
  started, the search space can be drastically
  reduced.

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Forward checking

• A way to make better use of constraints during
  search.
• 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 Ys domain any value that is
  inconsistent with the value chosen for X.

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Forward checking algorithm
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Forward checking example
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Forward checking example
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Forward checking example
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Forward checking example

• Idea
• Keep track of remaining legal values for
  unassigned variables
• Terminate search when any variable has no legal
  values

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Forward checking example

• Idea
• Keep track of remaining legal values for
  unassigned variables
• Terminate search when any variable has no legal
  values

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Forward checking example

• Idea
• Keep track of remaining legal values for
  unassigned variables
• Terminate search when any variable has no legal
  values

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Forward checking example

• Idea
• Keep track of remaining legal values for
  unassigned variables
• Terminate search when any variable has no legal
  values

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Forward checking 4-queens example

• R11? propagation R23,4 R32,4 R42,3 ?
  R11
• R23? propagation R3?
• R24? propagation R32 R43 ?
  R24
• R32? propagation R4 ?
• No other value for R3. Backtracking to R2
• No other value for R2. Backtracking to R1
• R12? propagation R24 R31,3 R41,3,4 ?
  R12
• R24? propagation R31 R41,3 ?
  R24
• R31? propagation R43
  ? R31
• R43? No propagations ? R43

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Constraint propagation

• Forward checking propagates information from
  assigned to unassigned variables, but doesn't
  provide early detection for all failures
• NT and SA cannot both be blue!
• Constraint propagation repeatedly enforces
  constraints locally

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Constraint propagation

• Forward checking does not detect the blue
  inconsistency, because it does not look far
  enough ahead.
• Constraint propagation is the general term for
  propagating the implications of a constraint on
  one variable onto other variables.
• The idea of arc consistency provides a fast
  method of constraint propagation that is
  substantially stronger than forward checking.

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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

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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

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

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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
• 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 preprocess or after each
  assignment

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Algorithm for arc consistency
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Algorithm for arc consistency AC-3

• It uses a queue to keep track of the arcs that
  need to be checked for inconsistency.
• Each arc (Xi, Xj) in turn is removed from the
  agenda and checked.
• If any values need to be deleted from the domain
  of Xi, then every arc (Xk, Xj) pointing to Xi
  must be reinserted on the queue for checking.

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Algorithm for arc consistency AC-3

• (C1,C2) eliminate AZUL
• (C2,C1) ok
• (C2,C3) ok
• (C3,C2) eliminate AZUL
• (C2,C4) ok
• (C4,C2) eliminate AZUL
• (C3,C4) eliminate VERDE
• add (C2,C3)
• (C4,C3) ok
• (C2,C3) ok

C1
AZUL, ROJO
?
C2
AZUL
?
?
C3
C4
AZUL, ROJO, VERDE
AZUL, VERDE
?
Initial list (C1,C2), (C2,C1), (C2,C3), (C3,C2),
(C2,C4), (C4,C2), (C3,C4), (C4,C3)