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of my exam on a very surface level. +I reached the conclusion that, although I have a vague idea of what the topics are, I don't have a deep understanding of them. All of my knowledge is very superficial. + +Hence, this progress report will be a preamble of each session, where I will list the topics and the corresponding confidence I have in them. + +The scale of my confidence in the topics: +1. 😎 - Very confident +2. 😊 - Confident +3. 😐 - Can solve, no intuition +4. 😕 - No idea + +| Topic | Description | Confidence | +|-------|-------------|------------| +| Signatures | Checking whether the expression types are correct | 😊 | +| Programming | Writing code to solve an algorithmic problem | 😐 | +| Higher order functions | Demonstrate understanding of higher order functions, usually by chaining them | 😐 | +| List comprehensions | Demonstrate understanding of list comprehensions | 😐 | +| Recursion/infinite lists | Infinitely generating lists, usually by recursion and/or list comprehensions | 😕 | +| ADTs | Algebraic Data Types, usually defining a data type and writing functions that operate on it | 😕 | +| Proof on lists | Proving properties of functions that operate on lists | 😕 | +| Proof on trees/ADTs | Proving properties of functions that operate on trees or ADTs | 😕 | + +## Basics + +### Signatures + +Example given - the `.` operator a.k.a. $f \circ g = f(g(x))$ + +```haskell +(.) :: (b -> c) -> (a -> b) -> a -> c +f . g = \x -> f (g x) +``` +Why? Because the type of `f` is `b -> c`, the type of `g` is `a -> b`, and the type of `x` is `a`. Hence, the type of the output is `c`. + + +### Stack and heap +**What's the stack?** +> The stack is a data structure that stores information about the active subroutines of a computer program. It is used to store the return address of the function, the local variables of the function, and the parameters passed to the function. +> This is like CPU registers, but for functions. + +**What's the heap?** +> The heap is a region of memory that is not managed automatically for you. This is a large pool of memory from which you can request blocks of memory. +> This is like the RAM of the computer. + +**What's the difference between the stack and the heap?** +> The stack is used for static memory allocation and the heap is used for dynamic memory allocation, both stored in the computer's RAM. +i.e. in C: +```c +int main() { + int x = 5; // Stack + int* y = malloc(sizeof(int)); // Heap +} +``` + +**What's the difference between the stack and the heap in Haskell?** +> In Haskell, the stack is used for function calls and the heap is used for storing data. +> This is because Haskell is a lazy language, so it doesn't evaluate the function calls immediately. Instead, it stores them in the stack and evaluates them when needed. +> The heap is used for storing data because Haskell is a functional language, so it doesn't have mutable variables. Hence, it stores the data in the heap. +> This is why Haskell is a pure language. + + +### Recursion + +Recursion in Haskell is very similar to recursion in other languages. The only difference is that Haskell is a lazy language, so it doesn't evaluate the recursion immediately. Instead, it stores the recursion in the stack and evaluates it when needed. + +Tail recursion is a special case of recursion where the recursive call is the last thing in the function. + +Example of non-tail recursion: +```haskell +fib :: Int -> Int +fib 0 = 0 +fib 1 = 1 +fib n = fib (n-1) + fib (n-2) +``` + +To make the `fib` function tail recursive, we can use an accumulator[^1] +```haskell +fib :: Int -> Int +fib n = go n 0 1 + where + go 0 a b = a + go n a b = go (n-1) b (a+b) +``` + +### `map, foldr, foldl, filter, zip` + +These are higher order functions in Haskell. They are used to operate on lists. + +`map` applies a function to each element of a list. +```haskell +map :: (a -> b) -> [a] -> [b] +map f [] = [] +map f (x:xs) = f x : map f xs +``` + +

+ folds +

+ +`foldr` is a right fold. It is used to reduce a list to a single value. +```haskell +foldr :: (a -> b -> b) -> b -> [a] -> b +foldr f z [] = z +foldr f z (x:xs) = f x (foldr f z xs) +``` + +`foldl` is a left fold. It is used to reduce a list to a single value. +```haskell +foldl :: (b -> a -> b) -> b -> [a] -> b +foldl f z [] = z +foldl f z (x:xs) = foldl f (f z x) xs +``` + +`filter` is used to filter a list based on a predicate. +```haskell +filter :: (a -> Bool) -> [a] -> [a] +filter p [] = [] +filter p (x:xs) = if p x then x : filter p xs else filter p xs +``` + +`zip` is used to combine two lists into a list of tuples. +```haskell +zip :: [a] -> [b] -> [(a, b)] +zip [] _ = [] +zip _ [] = [] +zip (x:xs) (y:ys) = (x, y) : zip xs ys +``` + + +[^1]: This is a common pattern in functional programming. The idea is to pass an accumulator to the function and update it in each recursive call. This is to "circumvent" immutability by introducing a way to store the "state" of the function.