Module CLPFD

Library for finite domain constraint solving. p The general structure of a specification of an FD problem is as follows: 

domain_constraint & fd_constraint & labeling

where:

domain_constraint specifies the possible range of the FD variables (see constraint domain)

fd_constraint specifies the constraint to be satisfied by a valid solution (see constraints #+, #-, allDifferent, etc below)

labeling is a labeling function to search for a concrete solution.

Note: This library is based on the corresponding library of Sicstus-Prolog but does not implement the complete functionality of the Sicstus-Prolog library. However, using the PAKCS interface for external functions, it is relatively easy to provide the complete functionality.

Author: Michael Hanus

Version: June 2012

Summary of exported operations:

domain :: [Int] -> Int -> Int -> Bool  Deterministic 
Constraint to specify the domain of all finite domain variables.
(+#) :: Int -> Int -> Int  Deterministic 
Addition of FD variables.
(-#) :: Int -> Int -> Int  Deterministic 
Subtraction of FD variables.
(*#) :: Int -> Int -> Int  Deterministic 
Multiplication of FD variables.
(=#) :: Int -> Int -> Bool  Deterministic 
Equality of FD variables.
(/=#) :: Int -> Int -> Bool  Deterministic 
Disequality of FD variables.
(<#) :: Int -> Int -> Bool  Deterministic 
"Less than" constraint on FD variables.
(<=#) :: Int -> Int -> Bool  Deterministic 
"Less than or equal" constraint on FD variables.
(>#) :: Int -> Int -> Bool  Deterministic 
"Greater than" constraint on FD variables.
(>=#) :: Int -> Int -> Bool  Deterministic 
"Greater than or equal" constraint on FD variables.
(#=#) :: Int -> Int -> Constraint  Deterministic 
Reifyable equality constraint on FD variables.
(#/=#) :: Int -> Int -> Constraint  Deterministic 
Reifyable inequality constraint on FD variables.
(#<#) :: Int -> Int -> Constraint  Deterministic 
Reifyable "less than" constraint on FD variables.
(#<=#) :: Int -> Int -> Constraint  Deterministic 
Reifyable "less than or equal" constraint on FD variables.
(#>#) :: Int -> Int -> Constraint  Deterministic 
Reifyable "greater than" constraint on FD variables.
(#>=#) :: Int -> Int -> Constraint  Deterministic 
Reifyable "greater than or equal" constraint on FD variables.
neg :: Constraint -> Constraint  Deterministic 
The resulting constraint is satisfied if both argument constraints are satisfied.
(#/\#) :: Constraint -> Constraint -> Constraint  Deterministic 
The resulting constraint is satisfied if both argument constraints are satisfied.
(#\/#) :: Constraint -> Constraint -> Constraint  Deterministic 
The resulting constraint is satisfied if both argument constraints are satisfied.
(#=>#) :: Constraint -> Constraint -> Constraint  Deterministic 
The resulting constraint is satisfied if the first argument constraint do not hold or both argument constraints are satisfied.
(#<=>#) :: Constraint -> Constraint -> Constraint  Deterministic 
The resulting constraint is satisfied if both argument constraint are either satisfied and do not hold.
solve :: Constraint -> Bool  Deterministic 
Solves a reified constraint.
sum :: [Int] -> (Int -> Int -> Bool) -> Int -> Bool  Deterministic 
Relates the sum of FD variables with some integer of FD variable.
scalarProduct :: [Int] -> [Int] -> (Int -> Int -> Bool) -> Int -> Bool  Deterministic 
(scalarProduct cs vs relop v) is satisfied if ((cs*vs) relop v) is satisfied.
count :: Int -> [Int] -> (Int -> Int -> Bool) -> Int -> Bool  Deterministic 
(count v vs relop c) is satisfied if (n relop c), where n is the number of elements in the list of FD variables vs that are equal to v, is satisfied.
allDifferent :: [Int] -> Bool  Deterministic 
"All different" constraint on FD variables.
all_different :: [Int] -> Bool  Deterministic 
For backward compatibility.
indomain :: Int -> Bool  Deterministic 
Instantiate a single FD variable to its values in the specified domain.
labeling :: [LabelingOption] -> [Int] -> Bool  Deterministic 
Instantiate FD variables to their values in the specified domain.

Exported datatypes:

Constraint

A datatype to represent reifyable constraints.

Constructors:

LabelingOption

This datatype contains all options to control the instantiated of FD variables with the enumeration constraint labeling.

Constructors:

  • LeftMost :: LabelingOption : The leftmost variable is selected for instantiation (default)
  • FirstFail :: LabelingOption : The leftmost variable with the smallest domain is selected (also known as first-fail principle)
  • FirstFailConstrained :: LabelingOption : The leftmost variable with the smallest domain and the most constraints on it is selected.
  • Min :: LabelingOption : The leftmost variable with the smalled lower bound is selected.
  • Max :: LabelingOption : The leftmost variable with the greatest upper bound is selected.
  • Step :: LabelingOption : Make a binary choice between x=#b and x/=#b for the selected variable x where b is the lower or upper bound of x (default).
  • Enum :: LabelingOption : Make a multiple choice for the selected variable for all the values in its domain.
  • Bisect :: LabelingOption : Make a binary choice between x<=#m and x>#m for the selected variable x where m is the midpoint of the domain x (also known as domain splitting).
  • Up :: LabelingOption : The domain is explored for instantiation in ascending order (default).
  • Down :: LabelingOption : The domain is explored for instantiation in descending order.
  • All :: LabelingOption : Enumerate all solutions by backtracking (default).
  • Minimize :: Int -> LabelingOption : Find a solution that minimizes the domain variable v (using a branch-and-bound algorithm).
  • Maximize :: Int -> LabelingOption : Find a solution that maximizes the domain variable v (using a branch-and-bound algorithm).
  • Assumptions :: Int -> LabelingOption : The variable x is unified with the number of choices made by the selected enumeration strategy when a solution is found.
  • RandomVariable :: Int -> LabelingOption : Select a random variable for instantiation where x is a seed value for the random numbers (only supported by SWI-Prolog).
  • RandomValue :: Int -> LabelingOption : Label variables with random integer values where x is a seed value for the random numbers (only supported by SWI-Prolog).

Exported operations:

domain :: [Int] -> Int -> Int -> Bool  Deterministic 

Constraint to specify the domain of all finite domain variables.

Example call:
(domain xs min max)
Parameters:
  • xs : list of finite domain variables
  • min : minimum value for all variables in xs
  • max : maximum value for all variables in xs

(+#) :: Int -> Int -> Int  Deterministic 

Addition of FD variables.

Further infos:
  • defined as left-associative infix operator with precedence 6

(-#) :: Int -> Int -> Int  Deterministic 

Subtraction of FD variables.

Further infos:
  • defined as left-associative infix operator with precedence 6

(*#) :: Int -> Int -> Int  Deterministic 

Multiplication of FD variables.

Further infos:
  • defined as left-associative infix operator with precedence 7

(=#) :: Int -> Int -> Bool  Deterministic 

Equality of FD variables.

Further infos:
  • defined as non-associative infix operator with precedence 4

(/=#) :: Int -> Int -> Bool  Deterministic 

Disequality of FD variables.

Further infos:
  • defined as non-associative infix operator with precedence 4

(<#) :: Int -> Int -> Bool  Deterministic 

"Less than" constraint on FD variables.

Further infos:
  • defined as non-associative infix operator with precedence 4

(<=#) :: Int -> Int -> Bool  Deterministic 

"Less than or equal" constraint on FD variables.

Further infos:
  • defined as non-associative infix operator with precedence 4

(>#) :: Int -> Int -> Bool  Deterministic 

"Greater than" constraint on FD variables.

Further infos:
  • defined as non-associative infix operator with precedence 4

(>=#) :: Int -> Int -> Bool  Deterministic 

"Greater than or equal" constraint on FD variables.

Further infos:
  • defined as non-associative infix operator with precedence 4

(#=#) :: Int -> Int -> Constraint  Deterministic 

Reifyable equality constraint on FD variables.

Further infos:
  • defined as non-associative infix operator with precedence 4
  • solution complete, i.e., able to compute all solutions

(#/=#) :: Int -> Int -> Constraint  Deterministic 

Reifyable inequality constraint on FD variables.

Further infos:
  • defined as non-associative infix operator with precedence 4
  • solution complete, i.e., able to compute all solutions

(#<#) :: Int -> Int -> Constraint  Deterministic 

Reifyable "less than" constraint on FD variables.

Further infos:
  • defined as non-associative infix operator with precedence 4
  • solution complete, i.e., able to compute all solutions

(#<=#) :: Int -> Int -> Constraint  Deterministic 

Reifyable "less than or equal" constraint on FD variables.

Further infos:
  • defined as non-associative infix operator with precedence 4
  • solution complete, i.e., able to compute all solutions

(#>#) :: Int -> Int -> Constraint  Deterministic 

Reifyable "greater than" constraint on FD variables.

Further infos:
  • defined as non-associative infix operator with precedence 4
  • solution complete, i.e., able to compute all solutions

(#>=#) :: Int -> Int -> Constraint  Deterministic 

Reifyable "greater than or equal" constraint on FD variables.

Further infos:
  • defined as non-associative infix operator with precedence 4
  • solution complete, i.e., able to compute all solutions

neg :: Constraint -> Constraint  Deterministic 

The resulting constraint is satisfied if both argument constraints are satisfied.

Further infos:
  • solution complete, i.e., able to compute all solutions

(#/\#) :: Constraint -> Constraint -> Constraint  Deterministic 

The resulting constraint is satisfied if both argument constraints are satisfied.

Further infos:
  • defined as right-associative infix operator with precedence 3
  • solution complete, i.e., able to compute all solutions

(#\/#) :: Constraint -> Constraint -> Constraint  Deterministic 

The resulting constraint is satisfied if both argument constraints are satisfied.

Further infos:
  • defined as right-associative infix operator with precedence 2
  • solution complete, i.e., able to compute all solutions

(#=>#) :: Constraint -> Constraint -> Constraint  Deterministic 

The resulting constraint is satisfied if the first argument constraint do not hold or both argument constraints are satisfied.

Further infos:
  • defined as right-associative infix operator with precedence 1
  • solution complete, i.e., able to compute all solutions

(#<=>#) :: Constraint -> Constraint -> Constraint  Deterministic 

The resulting constraint is satisfied if both argument constraint are either satisfied and do not hold.

Further infos:
  • defined as right-associative infix operator with precedence 1
  • solution complete, i.e., able to compute all solutions

solve :: Constraint -> Bool  Deterministic 

Solves a reified constraint.

sum :: [Int] -> (Int -> Int -> Bool) -> Int -> Bool  Deterministic 

Relates the sum of FD variables with some integer of FD variable.

scalarProduct :: [Int] -> [Int] -> (Int -> Int -> Bool) -> Int -> Bool  Deterministic 

(scalarProduct cs vs relop v) is satisfied if ((cs*vs) relop v) is satisfied. The first argument must be a list of integers. The other arguments are as in `sum`.

count :: Int -> [Int] -> (Int -> Int -> Bool) -> Int -> Bool  Deterministic 

(count v vs relop c) is satisfied if (n relop c), where n is the number of elements in the list of FD variables vs that are equal to v, is satisfied. The first argument must be an integer. The other arguments are as in sum.

allDifferent :: [Int] -> Bool  Deterministic 

"All different" constraint on FD variables.

Example call:
(allDifferent xs)
Parameters:
  • xs : list of FD variables
Returns:
satisfied if the FD variables in the argument list xs have pairwise different values.

all_different :: [Int] -> Bool  Deterministic 

For backward compatibility. Use allDifferent.

indomain :: Int -> Bool  Deterministic 

Instantiate a single FD variable to its values in the specified domain.

labeling :: [LabelingOption] -> [Int] -> Bool  Deterministic 

Instantiate FD variables to their values in the specified domain.

Example call:
(labeling options xs)
Parameters:
  • options : list of option to control the instantiation of FD variables
  • xs : list of FD variables that are non-deterministically instantiated to their possible values.