```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 ``` ```------------------------------------------------------------------------------ --- Library for finite domain constraint solving. ---

--- 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 --- @category general ------------------------------------------------------------------------------ module CLPFD(domain, (+#), (-#), (*#), (=#), (/=#), (<#), (<=#), (>#), (>=#), Constraint, (#=#), (#/=#), (#<#), (#<=#), (#>#), (#>=#), neg, (#/\#), (#\/#), (#=>#), (#<=>#), solve, sum, scalarProduct, allDifferent, all_different, count, indomain, labeling, LabelingOption(..)) where -- The operator declarations are similar to the standard arithmetic -- and relational operators. infixl 7 *# infixl 6 +#, -# infix 4 =#, /=#, <#, <=#, >#, >=# infix 4 #=#, #/=#, #<#, #<=#, #>#, #>=# infixr 3 #/\# infixr 2 #\/# infixr 1 #=>#, #<=># --- Constraint to specify the domain of all finite domain variables. --- @param xs - list of finite domain variables --- @param min - minimum value for all variables in xs --- @param max - maximum value for all variables in xs domain :: [Int] -> Int -> Int -> Bool domain vs l u = ((prim_domain \$!! (ensureSpine vs)) \$# l) \$# u prim_domain :: [Int] -> Int -> Int -> Bool prim_domain external --- Addition of FD variables. (+#) :: Int -> Int -> Int x +# y = (prim_FD_plus \$! y) \$! x prim_FD_plus :: Int -> Int -> Int prim_FD_plus external --- Subtraction of FD variables. (-#) :: Int -> Int -> Int x -# y = (prim_FD_minus \$! y) \$! x prim_FD_minus :: Int -> Int -> Int prim_FD_minus external --- Multiplication of FD variables. (*#) :: Int -> Int -> Int x *# y = (prim_FD_times \$! y) \$! x prim_FD_times :: Int -> Int -> Int prim_FD_times external --- Equality of FD variables. (=#) :: Int -> Int -> Bool x =# y = (prim_FD_equal \$! y) \$! x prim_FD_equal :: Int -> Int -> Bool prim_FD_equal external --- Disequality of FD variables. (/=#) :: Int -> Int -> Bool x /=# y = (prim_FD_notequal \$! y) \$! x prim_FD_notequal :: Int -> Int -> Bool prim_FD_notequal external --- "Less than" constraint on FD variables. (<#) :: Int -> Int -> Bool x <# y = (prim_FD_le \$! y) \$! x prim_FD_le :: Int -> Int -> Bool prim_FD_le external --- "Less than or equal" constraint on FD variables. (<=#) :: Int -> Int -> Bool x <=# y = (prim_FD_leq \$! y) \$! x prim_FD_leq :: Int -> Int -> Bool prim_FD_leq external --- "Greater than" constraint on FD variables. (>#) :: Int -> Int -> Bool x ># y = (prim_FD_ge \$! y) \$! x prim_FD_ge :: Int -> Int -> Bool prim_FD_ge external --- "Greater than or equal" constraint on FD variables. (>=#) :: Int -> Int -> Bool x >=# y = (prim_FD_geq \$! y) \$! x prim_FD_geq :: Int -> Int -> Bool prim_FD_geq external --------------------------------------------------------------------------- -- Reifyable constraints. --- A datatype to represent reifyable constraints. data Constraint = FDEqual Int Int | FDNotEqual Int Int | FDGreater Int Int | FDGreaterOrEqual Int Int | FDLess Int Int | FDLessOrEqual Int Int | FDNeg Constraint | FDAnd Constraint Constraint | FDOr Constraint Constraint | FDImply Constraint Constraint | FDEquiv Constraint Constraint --- Reifyable equality constraint on FD variables. (#=#) :: Int -> Int -> Constraint x #=# y = FDEqual x y --- Reifyable inequality constraint on FD variables. (#/=#) :: Int -> Int -> Constraint x #/=# y = FDNotEqual x y --- Reifyable "less than" constraint on FD variables. (#<#) :: Int -> Int -> Constraint x #<# y = FDLess x y --- Reifyable "less than or equal" constraint on FD variables. (#<=#) :: Int -> Int -> Constraint x #<=# y = FDLessOrEqual x y --- Reifyable "greater than" constraint on FD variables. (#>#) :: Int -> Int -> Constraint x #># y = FDGreater x y --- Reifyable "greater than or equal" constraint on FD variables. (#>=#) :: Int -> Int -> Constraint x #>=# y = FDGreaterOrEqual x y --- The resulting constraint is satisfied if both argument constraints --- are satisfied. neg :: Constraint -> Constraint neg x = FDNeg x --- The resulting constraint is satisfied if both argument constraints --- are satisfied. (#/\#) :: Constraint -> Constraint -> Constraint x #/\# y = FDAnd x y --- The resulting constraint is satisfied if both argument constraints --- are satisfied. (#\/#) :: Constraint -> Constraint -> Constraint x #\/# y = FDOr x y --- The resulting constraint is satisfied if the first argument constraint --- do not hold or both argument constraints are satisfied. (#=>#) :: Constraint -> Constraint -> Constraint x #=># y = FDImply x y --- The resulting constraint is satisfied if both argument constraint --- are either satisfied and do not hold. (#<=>#) :: Constraint -> Constraint -> Constraint x #<=># y = FDEquiv x y --- Solves a reified constraint. solve :: Constraint -> Bool solve c = prim_solve_reify \$!! c prim_solve_reify :: Constraint -> Bool prim_solve_reify external --------------------------------------------------------------------------- -- Complex constraints. --- Relates the sum of FD variables with some integer of FD variable. sum :: [Int] -> (Int -> Int -> Bool) -> Int -> Bool sum vs rel v = seq (normalForm (ensureSpine vs)) (seq (ensureNotFree rel) (seq v (prim_sum vs rel v))) prim_sum :: [Int] -> (Int -> Int -> Bool) -> Int -> Bool prim_sum external --- (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. scalarProduct :: [Int] -> [Int] -> (Int -> Int -> Bool) -> Int -> Bool scalarProduct cs vs rel v = seq (groundNormalForm cs) (seq (normalForm (ensureSpine vs)) (seq (ensureNotFree rel) (seq v (prim_scalarProduct cs vs rel v)))) prim_scalarProduct :: [Int] -> [Int] -> (Int -> Int -> Bool) -> Int -> Bool prim_scalarProduct external --- (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. count :: Int -> [Int] -> (Int -> Int -> Bool) -> Int -> Bool count v vs rel c = seq (ensureNotFree v) (seq (normalForm (ensureSpine vs)) (seq (ensureNotFree rel) (seq c (prim_count v vs rel c)))) prim_count :: Int -> [Int] -> (Int -> Int -> Bool) -> Int -> Bool prim_count external --- "All different" constraint on FD variables. --- @param xs - list of FD variables --- @return satisfied if the FD variables in the argument list xs --- have pairwise different values. allDifferent :: [Int] -> Bool allDifferent vs = seq (normalForm (ensureSpine vs)) (prim_allDifferent vs) --- For backward compatibility. Use allDifferent. all_different :: [Int] -> Bool all_different = allDifferent prim_allDifferent :: [Int] -> Bool prim_allDifferent external --- Instantiate a single FD variable to its values in the specified domain. indomain :: Int -> Bool indomain x = seq x (prim_indomain x) prim_indomain :: Int -> Bool prim_indomain external --------------------------------------------------------------------------- -- Labeling. --- Instantiate FD variables to their values in the specified domain. --- @param options - list of option to control the instantiation of FD variables --- @param xs - list of FD variables that are non-deterministically --- instantiated to their possible values. labeling :: [LabelingOption] -> [Int] -> Bool labeling options vs = seq (normalForm (map ensureNotFree (ensureSpine options))) (seq (normalForm (ensureSpine vs)) (prim_labeling options vs)) prim_labeling :: [LabelingOption] -> [Int] -> Bool prim_labeling external --- This datatype contains all options to control the instantiated of FD variables --- with the enumeration constraint labeling. --- @cons LeftMost - The leftmost variable is selected for instantiation (default) --- @cons FirstFail - The leftmost variable with the smallest domain is selected --- (also known as first-fail principle) --- @cons FirstFailConstrained - The leftmost variable with the smallest domain --- and the most constraints on it is selected. --- @cons Min - The leftmost variable with the smalled lower bound is selected. --- @cons Max - The leftmost variable with the greatest upper bound is selected. --- @cons Step - 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). --- @cons Enum - Make a multiple choice for the selected variable for all the values --- in its domain. --- @cons Bisect - 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). --- @cons Up - The domain is explored for instantiation in ascending order (default). --- @cons Down - The domain is explored for instantiation in descending order. --- @cons All - Enumerate all solutions by backtracking (default). --- @cons Minimize v - Find a solution that minimizes the domain variable v --- (using a branch-and-bound algorithm). --- @cons Maximize v - Find a solution that maximizes the domain variable v --- (using a branch-and-bound algorithm). --- @cons Assumptions x - The variable x is unified with the number of choices --- made by the selected enumeration strategy when a solution --- is found. --- @cons RandomVariable x - Select a random variable for instantiation --- where `x` is a seed value for the random numbers --- (only supported by SWI-Prolog). --- @cons RandomValue x - Label variables with random integer values --- where `x` is a seed value for the random numbers --- (only supported by SWI-Prolog). data LabelingOption = LeftMost | FirstFail | FirstFailConstrained | Min | Max | Step | Enum | Bisect | Up | Down | All | Minimize Int | Maximize Int | Assumptions Int | RandomVariable Int | RandomValue Int -- end of library CLPFD ```