# Module CLP.FD

Library for finite domain constraint solving.

An FD problem is specified as an expression of type `FDConstr` using the constraints and expressions offered in this library. FD variables are created by the operation `domain`. An FD problem is solved by calling `solveFD` with labeling options, the FD variables whose values should be included in the output, and a constraint. Hence, the typical program structure to solve an FD problem is as follows:

```main :: [Int]
main =
let fdvars  = take n (domain u o)
fdmodel = {description of FD problem}
in solveFD {options} fdvars fdmodel```

where `n` are the number of variables and `[u..o]` is the range of their possible values.

Author: Michael Hanus

Version: December 2016

## Summary of exported operations:

 ```domain :: Int -> Int -> [FDExpr]```    Operations to construct basic constraints. ```fd :: Int -> FDExpr```    Represent an integer value as an FD expression. ```(=#) :: FDExpr -> FDExpr -> FDConstr```    Equality of FD expressions. ```(/=#) :: FDExpr -> FDExpr -> FDConstr```    Disequality of FD expressions. ```(<#) :: FDExpr -> FDExpr -> FDConstr```    "Less than" constraint on FD expressions. ```(<=#) :: FDExpr -> FDExpr -> FDConstr```    "Less than or equal" constraint on FD expressions. ```(>#) :: FDExpr -> FDExpr -> FDConstr```    "Greater than" constraint on FD expressions. ```(>=#) :: FDExpr -> FDExpr -> FDConstr```    "Greater than or equal" constraint on FD expressions. ```true :: FDConstr```    The always satisfied FD constraint. ```(/\) :: FDConstr -> FDConstr -> FDConstr```    Conjunction of FD constraints. ```andC :: [FDConstr] -> FDConstr```    Conjunction of a list of FD constraints. ```allC :: (a -> FDConstr) -> [a] -> FDConstr```    Maps a constraint abstraction to a list of FD constraints and joins them. ```allDifferent :: [FDExpr] -> FDConstr```    "All different" constraint on FD variables. ```sum :: [FDExpr] -> FDRel -> FDExpr -> FDConstr```    Relates the sum of FD variables with some integer of FD variable. ```scalarProduct :: [FDExpr] -> [FDExpr] -> FDRel -> FDExpr -> FDConstr```    `(scalarProduct cs vs relop v)` is satisfied if `(sum (cs*vs) relop v)` is satisfied. ```count :: FDExpr -> [FDExpr] -> FDRel -> FDExpr -> FDConstr```    `(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. ```solveFD :: [Option] -> [FDExpr] -> FDConstr -> [Int]```    Computes (non-deterministically) a solution for the FD variables (second argument) w.r.t. ```solveFDAll :: [Option] -> [FDExpr] -> FDConstr -> [[Int]]```    Computes all solutions for the FD variables (second argument) w.r.t. ```solveFDOne :: [Option] -> [FDExpr] -> FDConstr -> [Int]```    Computes a single solution for the FD variables (second argument) w.r.t.

## Exported datatypes:

FDRel

Possible relations between FD values.

Constructors:

• ```Equ :: FDRel``` : Equal
• ```Neq :: FDRel``` : Not equal
• ```Lt :: FDRel``` : Less than
• ```Leq :: FDRel``` : Less than or equal
• ```Gt :: FDRel``` : Greater than
• ```Geq :: FDRel``` : Greater than or equal

Option

This datatype defines options to control the instantiation of FD variables in the solver (`solveFD`).

Constructors:

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

FDExpr

Constructors:

FDConstr

Constructors:

## Exported operations:

 ```domain :: Int -> Int -> [FDExpr]```    Operations to construct basic constraints. Returns infinite list of FDVars with a given domain. Example call: `(domain min max)` Parameters: `min` : minimum value for all variables in xs `max` : maximum value for all variables in xs Further infos: solution complete, i.e., able to compute all solutions
 ```fd :: Int -> FDExpr```    Represent an integer value as an FD expression. Further infos: solution complete, i.e., able to compute all solutions
 ```(=#) :: FDExpr -> FDExpr -> FDConstr```    Equality of FD expressions. Further infos: defined as non-associative infix operator with precedence 4 solution complete, i.e., able to compute all solutions
 ```(/=#) :: FDExpr -> FDExpr -> FDConstr```    Disequality of FD expressions. Further infos: defined as non-associative infix operator with precedence 4 solution complete, i.e., able to compute all solutions
 ```(<#) :: FDExpr -> FDExpr -> FDConstr```    "Less than" constraint on FD expressions. Further infos: defined as non-associative infix operator with precedence 4 solution complete, i.e., able to compute all solutions
 ```(<=#) :: FDExpr -> FDExpr -> FDConstr```    "Less than or equal" constraint on FD expressions. Further infos: defined as non-associative infix operator with precedence 4 solution complete, i.e., able to compute all solutions
 ```(>#) :: FDExpr -> FDExpr -> FDConstr```    "Greater than" constraint on FD expressions. Further infos: defined as non-associative infix operator with precedence 4 solution complete, i.e., able to compute all solutions
 ```(>=#) :: FDExpr -> FDExpr -> FDConstr```    "Greater than or equal" constraint on FD expressions. Further infos: defined as non-associative infix operator with precedence 4 solution complete, i.e., able to compute all solutions
 ```true :: FDConstr```    The always satisfied FD constraint. Further infos: solution complete, i.e., able to compute all solutions
 ```(/\) :: FDConstr -> FDConstr -> FDConstr```    Conjunction of FD constraints. Further infos: defined as right-associative infix operator with precedence 3 solution complete, i.e., able to compute all solutions
 ```andC :: [FDConstr] -> FDConstr```    Conjunction of a list of FD constraints.
 ```allC :: (a -> FDConstr) -> [a] -> FDConstr```    Maps a constraint abstraction to a list of FD constraints and joins them.
 ```allDifferent :: [FDExpr] -> FDConstr```    "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. Further infos: solution complete, i.e., able to compute all solutions
 ```sum :: [FDExpr] -> FDRel -> FDExpr -> FDConstr```    Relates the sum of FD variables with some integer of FD variable. Further infos: solution complete, i.e., able to compute all solutions
 ```scalarProduct :: [FDExpr] -> [FDExpr] -> FDRel -> FDExpr -> FDConstr```    `(scalarProduct cs vs relop v)` is satisfied if `(sum (cs*vs) relop v)` is satisfied. The first argument must be a list of integers. The other arguments are as in `sum`. Further infos: solution complete, i.e., able to compute all solutions
 ```count :: FDExpr -> [FDExpr] -> FDRel -> FDExpr -> FDConstr```    `(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`. Further infos: solution complete, i.e., able to compute all solutions
 ```solveFD :: [Option] -> [FDExpr] -> FDConstr -> [Int]```    Computes (non-deterministically) a solution for the FD variables (second argument) w.r.t. constraint (third argument), where the values in the solution correspond to the list of FD variables. The first argument contains options to control the labeling/instantiation of FD variables.
 ```solveFDAll :: [Option] -> [FDExpr] -> FDConstr -> [[Int]]```    Computes all solutions for the FD variables (second argument) w.r.t. constraint (third argument), where the values in each solution correspond to the list of FD variables. The first argument contains options to control the labeling/instantiation of FD variables.
 ```solveFDOne :: [Option] -> [FDExpr] -> FDConstr -> [Int]```    Computes a single solution for the FD variables (second argument) w.r.t. constraint (third argument), where the values in the solution correspond to the list of FD variables. The first argument contains options to control the labeling/instantiation of FD variables.