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|
{-# LANGUAGE OverlappingInstances, UndecidableInstances,
ExistentialQuantification, FlexibleInstances #-}
-- module Formula where
module Main where
import Prelude hiding (logBase)
import Data.Maybe
-------------------------------------------------------------------------------
-- Formula
-- The data type for formulas (algegraic expressions).
--
-- It should be an extensible type, so that users of
-- the library can add new kinds of formulas.
-- For example, in this prototype I explore:
-- integer constants (FInt)
-- unknown variables (FVar)
-- sums (FSum)
-- products (FPro)
-- powers (FPow)
-- logarithms (FLog)
-- The user of the library may want to extend it with
-- trigonometric formulas or derivative formulas, for
-- example.
--
-- The idea is to let each kind of formula be a new data
-- type. Similar operations with them are implemented
-- using overloading. So there is a class (FORMULA) to collect
-- them and each kind of formula should be an instance of it.
class (Eq f, Show f) => FORMULA f where
ty :: f -> FType
intVal :: f -> Integer
varName :: f -> String
argList :: f -> [Formula]
same :: (FORMULA f1) => f -> f1 -> Bool
intVal = error ""
varName = error ""
argList = error ""
same _ _ = False
-- By now extensibility is accomplished by existentialy
-- quantified type variables.
data Formula = forall f . ( FORMULA f
, AddT f
) =>
Formula f
instance Show Formula where
show (Formula f) = show f
instance Eq Formula where
(Formula x) == (Formula y) = same x y
instance FORMULA Formula where
ty (Formula f) = ty f
intVal (Formula f) = intVal f
varName (Formula f) = varName f
argList (Formula f) = argList f
same (Formula f) = same f
-------------------------------------------------------------------------------
-- How to uniquely identify the type of formula?
-- Each type of formula is associated to a key (FType)
-- that identifies it.
--
-- Here I use an enumated data type. When extending
-- the library, the user will have to modify this
-- data type adding a new constant constructor.
data FType = INT
| VAR
| SUM
| PRO
| POW
| LOG
deriving (Eq,Ord,Enum,Show)
-------------------------------------------------------------------------------
-- Integer formula
data FInt = FInt Integer
deriving (Eq,Show)
mkInt = Formula . FInt
instance FORMULA FInt where
ty _ = INT
intVal (FInt x) = x
same (FInt x) y = isInt y && x == intVal y
-- Variable formula
data FVar = FVar String
deriving (Eq,Show)
mkVar = Formula . FVar
instance FORMULA FVar where
ty _ = VAR
varName (FVar x) = x
same (FVar x) y = isVar y && x == varName y
-- Sum formula
data FSum = FSum [Formula]
deriving (Eq,Show)
mkSum = Formula . FSum
instance FORMULA FSum where
ty _ = SUM
argList (FSum xs) = xs
same (FSum xs) y = isSum y && xs == argList y
-- Product formula
data FPro = FPro [Formula]
deriving (Eq,Show)
mkPro = Formula . FPro
instance FORMULA FPro where
ty _ = PRO
argList (FPro xs) = xs
same (FPro xs) y = isPro y && xs == argList y
-- Exponentiation formula
data FPow = FPow Formula Formula
deriving (Eq,Show)
mkPow x y = Formula (FPow x y)
instance FORMULA FPow where
ty _ = POW
argList (FPow b e) = [b,e]
same (FPow b e) y = isPow y && [b,e] == argList y
-- Logarithm formula
data FLog = FLog Formula Formula
deriving (Eq,Show)
mkLog x b = Formula (FLog x b)
instance FORMULA FLog where
ty _ = LOG
argList (FLog x b) = [x,b]
same (FLog x b) y = isLog y && [x,b] == argList y
-------------------------------------------------------------------------------
-- Some predicates
isInt x = ty x == INT
isVar x = ty x == VAR
isSum x = ty x == SUM
isPro x = ty x == PRO
isPow x = ty x == POW
isZero x = isInt x && intVal x == 0
-------------------------------------------------------------------------------
-- Adding two formulas
-- This is a really very simple algorithm for adding
-- two formulas.
add :: Formula -> Formula -> Formula
add x y
| isJust u = fromJust u
| isJust v = fromJust v
| otherwise = mkSum [x,y]
where
u = addT x y
v = addT y x
class AddT a where
addT :: a -> Formula -> Maybe Formula
addT _ _ = Nothing
instance (FORMULA a) => AddT a where {}
instance AddT Formula where
addT (Formula f) = addT f
instance AddT FInt where
addT (FInt 0) y = Just y
addT (FInt x) y
| isInt y = Just (mkInt (x + intVal y))
| otherwise = Nothing
instance AddT FSum where
addT (FSum xs) y
| isSum y = Just (mkSum (merge xs (argList y)))
| otherwise = Just (mkSum (merge xs [y]))
where
merge = (++)
instance AddT FLog where
addT (FLog x b) y
| isLog y && b == logBase y = Just (mkLog (mkPro [x,logExp y]) b)
| otherwise = Nothing
where
merge = (++)
isLog x = ty x == LOG
logBase x
| isLog x = head (tail (argList x))
logExp x
| isLog x = head (argList x)
-------------------------------------------------------------------------------
-- Test addition of formulas
main = print [ add (mkInt 78) (mkInt 110)
, add (mkInt 0) (mkVar "x")
, add (mkVar "x") (mkInt 0)
, add (mkVar "x") (mkVar "y")
, add (mkSum [mkInt 13,mkVar "x"]) (mkVar "y")
, add (mkLog (mkVar "x") (mkInt 10))
(mkLog (mkVar "y") (mkInt 10))
, add (mkLog (mkVar "x") (mkInt 10))
(mkLog (mkVar "y") (mkVar "e"))
]
|
