Difference between revisions of "FP Laboratory 9"

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* It is possible to extend the type Expr si that it contains conditional expressions.

Revision as of 08:45, 18 August 2023

User defined data types and type classes

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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.
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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)
Try it!
  • Create function showExpr that shows expression as a String.
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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
Try it!
  • Extend class Show to be usable with our expressions.
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*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
Try it!
  • 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)
Try it!

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 si that it contains conditional expressions.