FP Laboratory 9

User defined data types and type classes

Consider following representation of expressions

data Expr = Num Int
          | Add Expr Expr
          | Sub Expr Expr
          | Mul Expr Expr
          | Div Expr Expr
          | Var Char
	  deriving (Eq)

• Create function eval that evaluates expresions.

eval :: Expr -> Int

*Main> eval (Add (Num 1) (Num 2))
3
*Main> eval (Mul (Add (Num 1) (Num 2)) (Num 3))
9

eval :: Expr -> Int
eval (Num x) = x
eval (Add l r) =  (eval l) + (eval r)
eval (Sub l r) =  (eval l) - (eval r)
eval (Mul l r) =  (eval l) * (eval r)
eval (Div l r) =  (eval l) `div` (eval r)

• Create function showExpr that shows expression as a String.

showExpr :: Expr -> String

*Main> showExpr (Add (Num 1) (Num 2))
"1+2"
*Main> showExpr (Mul (Add (Num 1) (Num 2)) (Num 3))
"(1+2)*3"
*Main> showExpr (Mul (Add (Num 1) (Mul (Num 2) (Var 'x'))) (Mul (Num 3) (Var 'x')))
"(1+2*x)*3*x"
*Main> showExpr (Mul (Num 2) (Mul (Var 'x') (Var 'x')))                            
"2*x*x"

showExpr :: Expr -> String
showExpr expr = showExpr' expr NoOp

data Operation = Hi | HiDiv | Lo | LoSub | NoOp deriving (Eq)

showExpr' :: Expr -> Operation -> String
showExpr' (Num x) _ = show x
showExpr' (Var x) _ =  [x]
showExpr' (Add l r) op = let
  x = showExpr' l Lo ++"+"++showExpr' r Lo
  in if op == Hi || op == HiDiv || op==LoSub 
     then "(" ++ x ++")"
     else x
showExpr' (Sub l r) op = let
  x = showExpr' l Lo ++"-"++showExpr' r LoSub
  in if op == Hi || op == HiDiv || op==LoSub 
     then "(" ++ x ++")"
     else x     
showExpr' (Mul l r) op = let
  x = showExpr' l Hi ++"*"++showExpr' r Hi
  in if op == HiDiv
     then "(" ++ x ++")"
     else x
showExpr' (Div l r) op = let
  x = showExpr' l Hi ++"/"++showExpr' r HiDiv
  in if op == HiDiv
     then "(" ++ x ++")"
     else x

• Extend class Show to be usable with our expressions.

*Main> Add (Num 1) (Num 2)
"1+2"
*Main> Mul (Add (Num 1) (Num 2)) (Num 3)
"(1+2)*3"
*Main> Mul (Add (Num 1) (Mul (Num 2) (Var 'x'))) (Mul (Num 3) (Var 'x'))
"(1+2*x)*3*x"
*Main> Mul (Num 2) (Mul (Var 'x') (Var 'x'))           
"2*x*x"

instance (Show Expr) where
  show = showExpr

• Create function derivation representing symbolic derivation of a given expression.

deriv :: Expr -> Char -> Expr

*Main> deriv (Add (Num 1) (Num 2)) 'x'
0+0
*Main> deriv (Mul (Num 2) (Mul (Var 'x') (Var 'x'))) 'x'
0*x*x+2*(1*x+x*1)
*Main> deriv (Mul (Num 2) (Mul (Var 'x') (Var 'x'))) 'x'
0*x*x+2*(1*x+x*1)

deriv :: Expr-> Char -> Expr
deriv (Num _) _ = (Num 0)     
deriv (Var x) y | x==y = (Num 1)
                | otherwise = (Num 0)
deriv (Add l r) x = Add (deriv l x) (deriv r x)                
deriv (Sub l r) x = Sub (deriv l x) (deriv r x)
deriv (Mul l r) x = Add (Mul (deriv l x) r) (Mul l (deriv r x))
deriv (Div l r) x = 
   Div
    (Sub (Mul (deriv l x) r) (Mul l (deriv r x)))
    (Mul r r)

Additional exercises

• Define a function that counts the number of operators in an expression.

size :: Expr -> Int

*Main> size (Add (Num 1) (Num 2))
1
*Main> size (Mul (Add (Num 1) (Mul (Num 2) (Var 'x'))) (Mul (Num 3) (Var 'x')))
4

• It is possible to extend the type Expr so that it contains conditional expressions. Consider the following representation of boolean expressions:

data BExpr = Val Bool
        | And BExpr BExpr
        | Not BExpr 
        | Equal Expr Expr
        | Greater Expr Expr
    deriving (Eq)

• Define a function that evaluates boolean expressions.

bEval :: BExpr -> Bool

*Main> bEval (And (Equal (Num 4) (Num 5)) (Greater (Num 5) (Num 1))) 
False
*Main> bEval (And (Equal (Add (Num 1) (Num 4)) (Num 5)) (Greater (Num 5) (Num 1)))
True

• Define a function that shows a boolean expression as a string. Use symbols "/\","!","=",">" for logical conjunction, negation, equality, and comparison respectively.

showBExpr :: BExpr -> String

*Main> showBExpr((And (Val True) (Not (Greater (Num 5) (Num 1)))))
"(True/\\!(5>1))"
*Main> showBExpr(And (Equal (Add (Num 1) (Num 4)) (Num 5)) (Greater (Num 5) (Num 1)))
"(((1+4)=5)/\\(5>1))"

• Extend class Show to be usable with our boolean expressions.

*Main> (And (Val True) (Not (Greater (Num 5) (Num 1))))
(True/\!(5>1))
*Main> And (Equal (Add (Num 1) (Num 4)) (Num 5)) (Greater (Num 5) (Num 1))
(((1+4)=5)/\(5>1))

• Extend the Expr type with the If-Then-Else statement as follows:

data Expr = Num Int
          | Add Expr Expr
          | Sub Expr Expr
          | Mul Expr Expr
          | Div Expr Expr
          | If BExpr Expr Expr
 deriving (Eq)

• Modify function eval to evaluate If-Then-Else statements.

*Main> eval (If (Greater (Num 5)(Num 1)) (Num 5) (Num 6))     
5
*Main> eval (Mul (If (Greater (Add (Num 0) (Num 1))(Num 1)) (Num 5) (Num 6)) (Num 7))
42

• Modify function showExpr to show If-Then-Else statements as string.

*Main> showExpr (If (Greater (Num 5)(Num 1)) (Num 5) (Num 6)) 
"(If (5>1) then 5 else 6)"
*Main> showExpr (Mul (If (Greater (Add (Num 0) (Num 1))(Num 1)) (Num 5) (Num 6)) (Num 7))
"((If ((0+1)>1) then 5 else 6)*7)"