===== Lab 11: Functors and IO =====
**Exercise 1:** This is a warm-up exercise. Write a function converting a string into a CamelCase format. It takes a string, splits particular words separated by whitespace characters, changes the first letter of each word to uppercase, and joins all the words into a single string. E.g. ''" no air"'' is converted into ''"NoAir"''. Moreover, make the function polymorphic so that it works over any functor instance over ''String'', i.e., our function should have the following type:
toCamelCaseF :: Functor f => f String -> f String
//Solution:// First, we need a function converting an alphabetic character into uppercase. In the library ''Data.Char'' there is a function ''toUpper'' doing that. We will implement this function ourselves. To represent the relation between lowercase and uppercase letters, we take a list of tuples ''[('a','A'), ('b','B'),...]''. This can be created by zipping ''['a'..'z']'' and ''['A'..'Z']''. For a character ''c'' if it is a lowercase letter, then we return the corresponding uppercase letter; otherwise we return just ''c''. To do that we can use the function
lookup :: Eq a => a -> [(a, b)] -> Maybe b
that takes an element of type ''a'' and a list of pairs and lookups the element among first components of those pairs. If it is there, it returns ''Just'' the second component and otherwise ''Nothing''. Using the case expression, we can distinguish both cases by pattern matching.
++++ Code|
toUpper :: Char -> Char
toUpper c = case lookup c $ zip ['a'..'z'] ['A'..'Z'] of
Nothing -> c
Just c' -> c'
++++
To split the input string into particular words, we can apply the function
words :: String -> [String]
Then we have to apply ''toUpper'' to the first letter of each word. Finally, concatenate the resulting words. Thus we have a function converting a string into a CamelCase string.
++++ Code|
toCamelCase :: String -> String
toCamelCase = concat . map toUpperHead . words where
toUpperHead "" = ""
toUpperHead (x:xs) = toUpper x:xs
++++
It remains to lift the above function by ''fmap'' so that we can apply ''toCamelCase'' over any functor instance.
++++ Code|
toCamelCaseF :: Functor f => f String -> f String
toCamelCaseF = fmap toCamelCase
++++
Examples:
> toCamelCaseF [" no air ", " get back"] -- over the list functor
["NoAir","GetBack"]
> toCamelCaseF (Just " no air ") -- over the Maybe functor
Just "NoAir"
> toCamelCaseF getLine -- over IO functor
no air -- user's input
"NoAir"
**Exercise 2:** A deterministic finite automaton (DFA) is a tuple $\langle Q,\Sigma,\delta,init,F\rangle$, where $Q$ is a set of states, $\Sigma$ is a finite alphabet, $\delta\colon Q\times\Sigma\to Q$ is a transition function, $init\in Q$ is an initial state and $F\subseteq Q$ is a set of final states. DFAs play a crucial role in applications of regular expressions as each regular expression can be converted into an equivalent DFA accepting the language defined by the regular expression. For instance, the regular expression ''[0-9]+\.[0-9][0-9]'' defines a language of numbers having the decimal point followed by two digits, e.g. $123.00$, $0.12$, $3476.25$. The equivalent automaton is depicted below. It has states ''Before, Digit, Dot, First, Second''. ''Before'' is the initial state and ''Second'' is the only final state. Automaton reads the input characters and changes its state according to $\delta$. After the whole input is read, it accepts the input string iff it is in a final state. At the beginning, it is in ''Before''. Once it reads a digit, the state changes to ''Digit'' and remains there until ''.'' is read. Then the next digit changes the state to ''First'' and finally the second digit after the decimal point changes the state to ''Second'' which is final. Anything else leads to the state ''Fail''.
{{ :courses:fup:tutorials:automaton.png?600 |}}
Our task is to define a parametric data type ''DFA a'' modelling a DFA and implement the function
evalDFA :: DFA a -> String -> Bool
taking an automaton, a string ''w'' and returning true if ''w'' is accepted by the automaton and false otherwise.
Further, define the above automaton and use it to implement a function
parseNum :: String -> Maybe Float
taking a string and returning ''Just'' the parsed floating number if the string is accepted by the automaton or ''Nothing''.
Finally, lift ''parseNum'' to any functor instance
parseNumF :: Functor f => f String -> f (Maybe Float)
//Solution:// To model an automaton, we need the transition function $\delta\colon Q\times\Sigma\to Q$, the initial and final states.
We make the type ''DFA a'' parametric over a type ''a'' representing states as we wish to work with automata whose states might be integers or strings or other data types. We could also make ''DFA a'' parametric over a type ''b'' representing the alphabet $\Sigma$ but for this example we set $\Sigma=$ ''Char''. Thus the transition function $\delta$ is of type ''a -> Char -> a''. The initial state is of type ''a'' and the set of final states can be represented as a predicate of type ''a -> Bool''.
data DFA a = Automaton (a->Char->a) a (a->Bool)
Now we can write the function simulating the automaton computation. It starts with the initial states and repeatedly applies the transition function to the current state and the current letter. This can be done by folding as I explained in the lecture introducing folding in Scheme. In the comment below, you can see how to implement the automaton computation directly without folding. Finally, the predicate defining the final states is applied.
++++ Code|
evalDFA :: DFA a -> String -> Bool
evalDFA (Automaton dlt s inF) w =
inF (foldl dlt s w)
-- inF (deltaStar s w)
-- where deltaStar q [] = q
-- deltaStar q (a:ws) = deltaStar (dlt q a) ws
++++
Now we represent the above automaton as an instance of ''DFA a''. We first define a type representing the states. Then we define the automaton over these states.
data State = Before | Digit | Dot | First | Second | Fail
isNum :: Char -> Bool
isNum c = c `elem` ['0'..'9']
final :: State -> Bool
final Second = True
final _ = False
delta :: State -> Char -> State
delta Before c | isNum c = Digit
| otherwise = Fail
delta Digit c | isNum c = Digit
| c == '.' = Dot
| otherwise = Fail
delta Dot c | isNum c = First
| otherwise = Fail
delta First c | isNum c = Second
| otherwise = Fail
delta Second _ = Fail
delta Fail _ = Fail
automaton :: DFA State
automaton = Automaton delta Before final
Next, the function ''parseNum'' takes a string, and uses the automaton to check if the string has the correct format. If yes, it is read by the ''read'' function and otherwise ''Nothing'' is returned.
++++ Code|
parseNum :: String -> Maybe Float
parseNum w = if evalDFA automaton w then Just (read w)
else Nothing
++++
Now, we can lift it via ''fmap''.
++++ Code|
parseNumF :: Functor f => f String -> f (Maybe Float)
parseNumF = fmap parseNum
++++
Examples:
> parseNumF ["234", "123.12", ".5", "0.50"] -- the list functor instance
[Nothing,Just 123.12,Nothing,Just 0.5]
> parseNumF getLine -- IO functor instance
1234.34 -- user's input
Just 1234.34
> parseNumF getLine -- IO functor instance
1.234 -- user's input
Nothing
**Exercise 3:** Using the function ''parseNumF'' from the previous exercise, write a function ''parseIO :: IO ()'' that displays a string "Enter number:\n" and then reads from the keyboard a string. If the string has the correct format (i.e., number with two digits after the decimal point), then it displays "Ok"; otherwise it asks for the user's input again.
//Solution:// First, we execute the action ''putStrLn'' displaying the string "Enter number:". Then we execute the action ''parseNumF getLine :: IO (Maybe Float)''. Depending of its result, we either display "Ok" or execute the whole action ''parseIO'' again. We can either use the monadic operators as follows:
++++ Code using the bind operator|
parseIO :: IO ()
parseIO = putStrLn "Enter number:"
>> parseNumF getLine
>>= \x -> case x of
Nothing -> parseIO
Just _ -> putStrLn "Ok"
++++
or we can use the do-syntax as follows:
++++ Code using the do-notation|
parseIO :: IO ()
parseIO = do putStrLn "Enter number:"
x <- parseNumF getLine
case x of
Nothing -> parseIO
Just _ -> putStrLn "Ok"
++++
**Task 1:** Consider the following data type representing Boolean propositional formulas built up from atoms by negations, conjunctions, and disjunctions.
data Expr a = Atom a
| Neg (Expr a)
| And (Expr a) (Expr a)
| Or (Expr a) (Expr a)
deriving (Eq, Show)
The type constructor ''Expr'' has a single parameter ''a'' representing a data type for atoms. So for instance ''Expr Bool'' is a Boolean expression that can be directly evaluated, e.g. the expression $(True\wedge \neg False)\vee False$ is represented as
expr :: Expr Bool
expr = Or (And (Atom True) (Neg (Atom False))) (Atom False)
On the other hand, ''Expr String'' might represent propositional formulas whose atoms are variables represented as strings, e.g. the formula
$(\neg x\vee x)\wedge y$ is represented as
fle :: Expr String
fle = And (Or (Neg (Atom "x")) (Atom "x")) (Atom "y")
Write a function ''eval :: Expr Bool -> Bool'' evaluating a given Boolean expression. Thus it should evaluate ''expr'' to ''True''. Further, implement a function ''getAtoms :: Expr a -> [a]'' returning the list of atoms for a given expression, e.g. ''getAtoms fle'' should return
''["x","x","y"]''.
//Hint:// Logical operations negation, conjunction and disjunction can be respectively computed by ''not, &&, ||''. The last two are infix operators.
++++ Solution|
eval :: Expr Bool -> Bool
eval (Atom c) = c
eval (Neg e) = not (eval e)
eval (And e1 e2) = eval e1 && eval e2
eval (Or e1 e2) = eval e1 || eval e2
getAtoms :: Expr a -> [a]
getAtoms (Atom c) = [c]
getAtoms (Neg e) = getAtoms e
getAtoms (And e1 e2) = getAtoms e1 ++ getAtoms e2
getAtoms (Or e1 e2) = getAtoms e1 ++ getAtoms e2
++++
**Task 2:** The type constructor ''Expr'' from the previous task can be made into an instance of ''Functor'' as follows:
instance Functor Expr where
fmap f (Atom c) = Atom (f c)
fmap f (Neg e) = Neg (fmap f e)
fmap f (And e1 e2) = And (fmap f e1) (fmap f e2)
fmap f (Or e1 e2) = Or (fmap f e1) (fmap f e2)
Thus if we have a map ''f :: a -> b'', it can be lifted by ''fmap'' to a map of type ''Expr a -> Expr b''. This might be handy if we need to rename variables or we want to assign concrete Boolean values to variables. Write a polymorphic function
subst :: Functor f => [String] -> f String -> f Bool
taking a list of strings (variables) and a data structure over strings returning the same data structure where the strings (variables) in the input list are replaced by ''True'' and the rest by ''False''. Use the lifting by ''fmap''.
++++ Solution|
subst :: Functor f => [String] -> f String -> f Bool
subst xs = fmap (`elem` xs)
++++
Next, apply the function ''subseqs :: [a] -> [[a]]'' from the previous lab returning a list of all sublists of a given list.
subseqs :: [a] -> [[a]]
subseqs [] = [[]]
subseqs (x:xs) = subseqs xs ++ [x:ys | ys <- subseqs xs]
The above function can generate all possible evaluations of a propositional formula if we apply it to the result of ''getAtoms''. Implement functions
isTaut, isSat :: Expr String -> Bool
testing whether a given formula is a tautology (resp. satisfiable). A propositional formula is satisfiable if there exists an evaluation of atoms
such that the Boolean expression resulting from the replacing atoms by the respective Boolean values is evaluated to ''True''. A propositional formula is called tautology if it is satisfied by all possible evaluations of its atoms.
//Hint:// To check that there exists an evaluation satisfying a formula or if all evaluations satisfy the formula, use the functions ''or'', ''and'' respectively. These functions are applicable to any list of Boolean values.
++++ Solution|
check :: ([Bool] -> Bool) -> Expr String -> Bool
check g e = g [ eval $ subst vs e | vs <- vss]
where vss = subseqs $ getAtoms e
isTaut, isSat :: Expr String -> Bool
isTaut = check and
isSat = check or
++++