```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 ``` ```------------------------------------------------------------------------ --- This module contains an implementation of set functions. --- The general idea of set functions --- is described in: --- --- > S. Antoy, M. Hanus: Set Functions for Functional Logic Programming --- > Proc. 11th International Conference on Principles and Practice --- > of Declarative Programming (PPDP'09), pp. 73-82, ACM Press, 2009 --- --- Intuition: If `f` is an n-ary function, then `(setn f)` is a set-valued --- function that collects all non-determinism caused by f (but not --- the non-determinism caused by evaluating arguments!) in a set. --- Thus, `(setn f a1 ... an)` returns the set of all --- values of `(f b1 ... bn)` where `b1`,...,`bn` are values --- of the arguments `a1`,...,`an` (i.e., the arguments are --- evaluated "outside" this capsule so that the non-determinism --- caused by evaluating these arguments is not captured in this capsule --- but yields several results for `(setn...)`. --- Similarly, logical variables occuring in `a1`,...,`an` are not bound --- inside this capsule (but causes a suspension until they are bound). --- The set of values returned by a set function is represented --- by an abstract type 'Values' on which several operations are --- defined in this module. Actually, it is a multiset of values, --- i.e., duplicates are not removed. --- --- Restrictions: --- 1. The set is a multiset, i.e., it might contain multiple values. --- 2. The multiset of values is completely evaluated when demanded. --- Thus, if it is infinite, its evaluation will not terminate --- even if only some elements (e.g., for a containment test) --- are demanded. However, for the emptiness test, at most one --- value will be computed --- 3. The arguments of a set function are strictly evaluated before --- the set functions itself will be evaluated. --- --- Since this implementation is restricted and prototypical, --- the interface is not stable and might change. --- --- @author Michael Hanus --- @version July 2017 --- @category general ------------------------------------------------------------------------ module SetFunctions (set0,set1,set2,set3,set4,set5,set6,set7 ,Values,isEmpty,notEmpty,valueOf ,choose,chooseValue,select,selectValue ,mapValues,foldValues,filterValues,minValue,maxValue ,values2list,printValues,sortValues,sortValuesBy ) where import Findall import Sort(mergeSortBy) import List(delete) --- Combinator to transform a 0-ary function into a corresponding set function. set0 :: b -> Values b set0 f = Values (someValue f) (findall (=:=f)) --- Combinator to transform a unary function into a corresponding set function. set1 :: (a1 -> b) -> a1 -> Values b set1 f x | x=:=x = Values (someValue (f x)) (findall (=:=(f x))) --- Combinator to transform a binary function into a corresponding set function. set2 :: (a1 -> a2 -> b) -> a1 -> a2 -> Values b set2 f x1 x2 | x1=:=x1 & x2=:=x2 = Values (someValue (f x1 x2)) (findall (=:=(f x1 x2))) --- Combinator to transform a function of arity 3 --- into a corresponding set function. set3 :: (a1 -> a2 -> a3 -> b) -> a1 -> a2 -> a3 -> Values b set3 f x1 x2 x3 | x1=:=x1 & x2=:=x2 & x3=:=x3 = Values (someValue (f x1 x2 x3)) (findall (=:=(f x1 x2 x3))) --- Combinator to transform a function of arity 4 --- into a corresponding set function. set4 :: (a1 -> a2 -> a3 -> a4 -> b) -> a1 -> a2 -> a3 -> a4 -> Values b set4 f x1 x2 x3 x4 | x1=:=x1 & x2=:=x2 & x3=:=x3 & x4=:=x4 = Values (someValue (f x1 x2 x3 x4)) (findall (=:=(f x1 x2 x3 x4))) --- Combinator to transform a function of arity 5 --- into a corresponding set function. set5 :: (a1 -> a2 -> a3 -> a4 -> a5 -> b) -> a1 -> a2 -> a3 -> a4 -> a5 -> Values b set5 f x1 x2 x3 x4 x5 | x1=:=x1 & x2=:=x2 & x3=:=x3 & x4=:=x4 & x5=:=x5 = Values (someValue (f x1 x2 x3 x4 x5)) (findall (=:=(f x1 x2 x3 x4 x5))) --- Combinator to transform a function of arity 6 --- into a corresponding set function. set6 :: (a1 -> a2 -> a3 -> a4 -> a5 -> a6 -> b) -> a1 -> a2 -> a3 -> a4 -> a5 -> a6 -> Values b set6 f x1 x2 x3 x4 x5 x6 | x1=:=x1 & x2=:=x2 & x3=:=x3 & x4=:=x4 & x5=:=x5 & x6=:=x6 = Values (someValue (f x1 x2 x3 x4 x5 x6)) (findall (=:=(f x1 x2 x3 x4 x5 x6))) --- Combinator to transform a function of arity 7 --- into a corresponding set function. set7 :: (a1 -> a2 -> a3 -> a4 -> a5 -> a6 -> a7 -> b) -> a1 -> a2 -> a3 -> a4 -> a5 -> a6 -> a7 -> Values b set7 f x1 x2 x3 x4 x5 x6 x7 | x1=:=x1 & x2=:=x2 & x3=:=x3 & x4=:=x4 & x5=:=x5 & x6=:=x6 & x7=:=x7 = Values (someValue (f x1 x2 x3 x4 x5 x6 x7)) (findall (=:=(f x1 x2 x3 x4 x5 x6 x7))) ------------------------------------------------------------------------ --- Abstract type representing multisets of values. data Values a = Values (Maybe a) [a] -- The first component is `Nothing` iff the multiset is empty, -- otherwise it is the first value (this enables -- also an emptiness check or selection of a single value -- even for infinite sets of values). -- The second component contains the set of all values (it will be -- completely evaluated if its value is demanded). --- Computes some value of a given expression. --- This implementation is specific to PAKCS in order to --- to implement `notEmpty` and `selectValue` efficiently and --- also for possibly infinite result sets. someValue :: a -> Maybe a someValue e = let xs = findall (=:= (findfirst (=:=e))) in if null xs then Nothing else Just (head xs) --- Is a multiset of values empty? isEmpty :: Values _ -> Bool isEmpty (Values firstval _) = firstval == Nothing --- Is a multiset of values not empty? notEmpty :: Values _ -> Bool notEmpty vs = not (isEmpty vs) --- Is some value an element of a multiset of values? valueOf :: a -> Values a -> Bool valueOf e (Values _ s) = e `elem` s --- Chooses (non-deterministically) some value in a multiset of values --- and returns the chosen value and the remaining multiset of values. --- Thus, if we consider the operation `chooseValue` by --- --- chooseValue x = fst (choose x) --- --- then `(set1 chooseValue)` is the identity on value sets, i.e., --- `(set1 chooseValue s)` contains the same elements as the --- value set `s`. choose :: Values a -> (a,Values a) choose (Values _ vs) = (x, Values (if null xs then Nothing else Just (head xs)) xs) where x = foldr1 (?) vs xs = delete x vs --- Chooses (non-deterministically) some value in a multiset of values --- and returns the chosen value. --- Thus, `(set1 chooseValue)` is the identity on value sets, i.e., --- `(set1 chooseValue s)` contains the same elements as the --- value set `s`. chooseValue :: Values a -> a chooseValue s = fst (choose s) --- Selects (indeterministically) some value in a multiset of values --- and returns the selected value and the remaining multiset of values. --- Thus, `select` has always at most one value. --- It fails if the value set is empty. --- --- **NOTE:** --- The usage of this operation is only safe (i.e., does not destroy --- completeness) if all values in the argument set are identical. select :: Values a -> (a,Values a) select (Values _ (x:xs)) = (x, Values (if null xs then Nothing else Just (head xs)) xs) --- Selects (indeterministically) some value in a multiset of values --- and returns the selected value. --- Thus, `selectValue` has always at most one value. --- It fails if the value set is empty. --- --- **NOTE:** --- The usage of this operation is only safe (i.e., does not destroy --- completeness) if all values in the argument set are identical. --- It returns a single value even for infinite value sets --- (in contrast to `select` or `choose`). selectValue :: Values a -> a selectValue (Values (Just val) _) = val --- Maps a function to all elements of a multiset of values. mapValues :: (a -> b) -> Values a -> Values b mapValues f (Values mbval s) = Values (maybe Nothing (Just . f) mbval) (map f s) --- Accumulates all elements of a multiset of values by applying a binary --- operation. This is similarly to fold on lists, but the binary operation --- must be commutative so that the result is independent of the order --- of applying this operation to all elements in the multiset. foldValues :: (a -> a -> a) -> a -> Values a -> a foldValues f z (Values _ s) = foldr f z s --- Keeps all elements of a multiset of values that satisfy a predicate. filterValues :: (a -> Bool) -> Values a -> Values a filterValues p (Values _ s) = Values val xs where xs = filter p s val = if null xs then Nothing else Just (head xs) --- Returns the minimal element of a non-empty multiset of values --- with respect to a given total ordering on the elements. minValue :: (a -> a -> Bool) -> Values a -> a minValue leq (Values _ s) = minOf s where minOf [x] = x minOf (x:y:ys) = let m1 = minOf (y:ys) in if leq x m1 then x else m1 --- Returns the maximal element of a non-empty multiset of value --- with respect to a given total ordering on the elements. maxValue :: (a -> a -> Bool) -> Values a -> a maxValue leq (Values _ s) = maxOf s where maxOf [x] = x maxOf (x:y:ys) = let m1 = maxOf (y:ys) in if leq x m1 then m1 else x --- Puts all elements of a multiset of values in a list. --- Since the order of the elements in the list might depend on --- the time of the computation, this operation is an I/O action. values2list :: Values a -> IO [a] values2list (Values _ s) = return s --- Prints all elements of a multiset of values. printValues :: Values _ -> IO () printValues s = values2list s >>= mapIO_ print --- Transforms a multiset of values into a list sorted by --- the standard term ordering. As a consequence, the multiset of values --- is completely evaluated. sortValues :: Values a -> [a] sortValues = sortValuesBy (<=) --- Transforms a multiset of values into a list sorted by a given ordering --- on the values. As a consequence, the multiset of values --- is completely evaluated. --- In order to ensure that the result of this operation is independent of the --- evaluation order, the given ordering must be a total order. sortValuesBy :: (a -> a -> Bool) -> Values a -> [a] sortValuesBy leq (Values _ s) = mergeSortBy leq s ------------------------------------------------------------------------ ```