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 bthat 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.
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.
It remains to lift the above function by fmap
so that we can apply toCamelCase
over any functor instance.
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
.
Our task is to define a parametric data type DFA a
modelling a DFA and implement the function
evalDFA :: DFA a -> String -> Booltaking 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 Floattaking 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.
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.
Now, we can lift it via fmap
.
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:
or we can use the do-syntax as follows:
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.
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 Booltaking 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
.
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 -> Booltesting 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.