FP Laboratory 6

Operators

• Define following functions that performs corresponding logic operations: not', and', or', nand', xor', impl', equ'
• Define the 'standard' priority for all these functions, if they are used as operators.
• Create a function that prints the truth table of a given logical expression for two variables.

table :: (Bool -> Bool -> Bool) -> IO ()

table (\a b -> (and' a (or' a b)))                                                                              
True  True  True
True  False True
False True  False
False False False

Show solution

not' :: Bool -> Bool
not' True = False
not' False = True
infixl 5 `not'` 

and' :: Bool -> Bool -> Bool
and' True True = True
and' _ _ = False
infixl 4 `and'` 

or' :: Bool -> Bool -> Bool
or' False False = False
or' _ _ = True
infixl 3 `or'` 

nand' :: Bool -> Bool -> Bool
nand' x y = not' (and' x y)
infixl 4 `nand'` 

xor' :: Bool -> Bool -> Bool
xor' x y = x/=y
infixl 3 `xor'` 

impl' :: Bool -> Bool -> Bool
impl' True False = False
impl' _ _ = True
infixl 2 `impl'` 

equ' :: Bool -> Bool -> Bool
equ' x y = x == y
infixl 7 `equ'` 

table :: (Bool -> Bool -> Bool) -> IO ()
table expr = putStr (concat [nicePrint [x,y,(expr x y)] |x<-[True,False], y<-[True,False]])

nicePrint :: [Bool] -> String
nicePrint xs = concat [show x++"\t"| x<-xs] ++ "\n"

• Extend the previously defined function to accept any number of variables (the number of variables will be given as a first parameter).

tablen :: Int -> ([Bool] -> Bool) -> IO ()

 tablen 3 (\[a,b,c] -> a `and'` (b `or'` c) `equ'` a `and'` b `or'` a `and'` c)
True   True   True   => True
True   True   False  => True
True   False  True   => True
True   False  False  => False
False  True   True   => False
False  True   False  => False
False  False  True   => False
False  False  False  => False

Show solution

tablen :: Int -> ([Bool] -> Bool) -> IO ()
tablen n f = putStr(concat [nicePrint x ++ " => " ++ show(f x) ++ "\n" |x<-allValues n]) where 
  allValues 1 = [[True], [False]]
  allValues n = [x:y| x<-[True,False], y<-allValues (n-1)]

  nicePrint :: [Bool] -> String
  nicePrint xs = concat [show x++"\t"| x<-xs]

Complex function - Huffman Codes

• Create a function that will compute Huffman codes for a given list of characters and their frequencies.

huffman :: [(Char, Int)] -> [(Char, String)]

*Main>  huffman [('a',45),('b',13),('c',12),('d',16),('e',9),('f',5)]
[('a',"0"),('b',"101"),('c',"100"),('d',"111"),('e',"1101"),('f',"1100")]

Show solution

import Data.List (sortBy)

huffman :: [(Char, Int)] -> [(Char, String)] 
huffman input = 
  let 
    prep = [ (y, [(x,"")] ) | (x,y)<-input]            
  in sortBy (\ (x,_) (y,_) -> compare x y) (step prep) where
     step :: [(Int, [(Char, String)])] -> [(Char, String)]
     step [(_, result) ] = result
     step list = let ((a1, as2):(b1,bs2):rest) = sortBy (\ (x,_) (y,_) -> compare x y) list
                 in step ((a1+b1, [(x,'0':a2)|(x,a2)<-as2]++[(x,'1':b2)|(x,b2)<-bs2])  : rest)

Additional exercises

• Create a function that divides a list of elements into the list lists using a separator.

splitByElement :: Eq a => [a] -> a -> [[a]]

*Main> splitByElement "I love functional programming!" ' '
["I","love","functional","programming!"]
*Main> splitByElement [1,2,1,2,3,4,5,5,6,4,1,2,0,1,4] 1
[[2],[2,3,4,5,5,6,4],[2,0],[4]]
*Main> splitByElement [1,2,1,2,3,4,5,5,6,4,1,2,0,1,4] 5
[[1,2,1,2,3,4],[6,4,1,2,0,1,4]]