Added first session notes

haskell/sessions/13_03/notes.md

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+
+## Progress report - confidence scale
+In the first meeting, we went through the structure 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
+```
+
+<p align="center">
+    <img src="folds.png" alt="folds" width="700"/>
+</p>
+
+`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.