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------------------------------------------------------------------------ --- 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 <b>commutative</b> 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 ------------------------------------------------------------------------ |