Constraint Logic Programming CLP

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Constraint Logic Programming (CLP)

• Luis Tari
• March 10, 2005

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Why CLP?

• Generate-and-test approach is a common
  methodology for logic programming.
• Generate possible solutions
• Test and eliminate non-solutions
• Disadvantages of generate-and-test approach
• Passive use of constraints to test potential
  values
• Inefficient for combinatorial search problems
• CLP languages use the global search paradigm.
• Actively pruning the search space
• Recursively dividing a problem into subproblems
  until its subproblems are simple enough to be
  solved

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Why CLP and AnsProlog?

• AnsProlog is inefficient in dealing with
  numerical values, due to
• Implementation of the current solvers
• generate-and-test paradigm
• Goal of CLP is to pick numerical values from
  pre-defined domains for certain variables so that
  the given constraints on the variables are all
  satisfied.
• Idea use CLP to define and reason with numerical
  constraints and assignments

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List Notation in Prolog

• Example of list notation
• eggs, tea, milk, steak, spinach, toothpaste
• Head of a list
• First element of a list
• eggs
• Tail of a list
• The remaining list apart from the head
• tea, milk, steak, spinach, toothpaste
• HT
• H eggs
• T tea, milk, steak, spinach, toothpaste

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An example of List

• Use the predicate append to define appending two
  lists into one.
• append(XXL,YL,XZL) -
• append(XL,YL,ZL).
• append(,YL,YL).
• ?- append(1,2,X).
• X 1,2

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CLP with Sicstus Prolog

• To invoke the CLP library
• ?- - use_module(library(clpfd)).
• A constraint is called as if it is a Prolog
  predicate.
• ?- X in 1..5, Y in 2..8, XY T.
• X in 1..5,
• Y in 2..8,
• T in 3..13
• The above constraint says that given X,Y, assign
  values for T so that the constraint is
  satisfiable.
• ?- X in 1..5, T in 3..13, XY T.
• X in 1..5,
• T in 3..13,
• Y in -2..12

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Typical Steps of a CLP Program

• Step 1. Declare the domains of the variables
• Step 2. Post the problem constraints
• Step 3. Look for a feasible solution via
  backtrack search, or look for an optimal solution
  via branch-and-bound search

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Constraint Satisfaction Problem

• Lets consider the Send More Money puzzle.
• The variables are the letters S, E, N, D, M, O, R
  and Y.
• Each letter represents a digit between 0 and 9.
• Assign a value to each digit, such that SEND
  MORE equals MONEY.

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Send More Money Exampleusing CLP

• ?- mm(S,E,N,D,M,O,R,Y, ).
• D 7, E 5, M 1, N 6, O 0, R 8, S 9,
  Y 2

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Reified Constraints

• To reflect its truth value into a 0/1 variable B,
  so that
• The constraint is posted if B is set to 1
• The negation of the constraint is posted if B is
  set to 0
• B is set to 1 if the constraint becomes entailed
• B is set to 0 if the constraint becomes
  disentailed.
• A reified constraint is written as
• ?- Constraint ltgt B.

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An example using reified constraint

• Define exactly(X,L,N) so that it is true if X
  occurs exactly N times in the list L.
• If the constraint element X is the same as
  element Y becomes entailed, then assign B to be
  1 else 0.

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Other Constraints

• Arithmetic constraints
• ?- X in 1..2, Y in 3..5, XltY ltgt B.
• B 1,
• X in 1..2,
• Y in 3..5
• If the constraint value in X is lt value in Y
  becomes entailed, then assign B to be 1 else 0.
• Propositional constraints
• X 4 \/ Y 6
• Disjunction of the two equality constraints
• Many more.

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Defining Search Variants

• labeling(Options, Variables)
• Where Variables is a list of domain variables or
  integers and Options is a list of search options.
• ?- constraints(Variables),
• labeling(, Variables).
• same as leftmost,step,up,all
• leftmost the leftmost variable is selected
• step makes a binary choice between XB and
  X\B, where B is the lower or upper bound of X.
• up the domain is explored in ascending order
• all all solutions are enumerated by backtracking

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An Example - Cumulative Scheduling

• There are 7 tasks where each task has a fixed
  duration and a fixed amount of used resource.
• Goal find a schedule that minimizes the
  completion time for the schedule while not
  exceeding the capacity 13 of the resource.

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Cumulative Scheduling (2)

• - use_module(library(clpfd)).
• - use_module(library(lists), append/3).
• schedule(Ss, End) -
• length(Ss, 7),
• Ds16,6,13,7,5,18,4, Rs2,9,3,7,10,1,11,
• domain(Ss, 1, 30), domain(End, 1, 50),
• after(Ss, Ds, End), cumulative(Ss, Ds, Rs, 13),
• append(Ss, End, Vars),
• labeling(minimize(End), Vars).
• after(, , _).
• after(SSs, DDs, E) -
• E gt SD, after(Ss, Ds, E).

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Cumulative Scheduling (3)

• Ds 16,6,13,7, 5,18,4
• Rs 2 ,9, 3,7,10, 1,11
• Capacity 13
• ?- schedule(Ss, End).
• Ss 1,17,10,10,5,5,1,
• End 23

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References

• Sicstus Prolog Manual Version 3.12, Chapter 34 on
  Constraint Logic Programming over Finite Domains
• P. Van Hentenryck, H. Simonis, M. Dincbas,
  Constraint satisfaction using constraint logic
  programming, Journal of Artif. Intell., 58(1-3),
  pages 113--159, 1992.