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@@ -27,3 +27,8 @@ abstractions from Category Theory using command line options parsing as an
 example. No prior knowledge in either Category Theory or command line options
 parsing is required, being able to read Rust code and look at pretty pictures
 should be enough 🙂
+
+Slides (source, presented with [presenterm](https://github.com/mfontanini/presenterm)):
+- [slides.md](/content/2024-07-09/slides.md)
+- [pic1.png](/content/2024-07-09/pic1.png)
+- [pic2.png](/content/2024-07-09/pic2.png)
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+---
+title: Parsing command line options with Category Theory
+author: Michael Baykov
+---
+
+
+# Rust types and functions
+
+Two of the main building blocks in Rust are functions and values. Values come in different
+types, functions let you to convert between them. Not all functions are the same though, some
+depend on the inputs only and only produce some resulting value, some - do something else.
+
+Functions of first type are called pure and they are great - they are easy to work with, easy
+to test, easy to reason about.
+
+```rust
+let a = 3;
+
+fn inc(v: usize) -> usize [
+    v + 1
+}
+
+let b = inc(a)
+```
+
+
+<!-- end_slide -->
+
+# Pure functions, side effects
+
+Functions compose - that is given two functions: from A to B and from B to C you can make a
+function from A to C. We can think of a function as always taking a value in and always
+producing a value. If it takes none - it's `()`, if it takes multiple values - it's a tuple.
+Same with results - use tuples or `()`.
+
+![](pic1.jpg)
+
+<!-- end_slide -->
+# Parsers
+
+Then there's parsers. A typical shape of a parser looks like this
+
+
+```rust
+Fn(I) -> Result<(I, T), Error>
+```
+
+```rust
+Fn(&mut I) -> Result<T, Error>
+```
+
+First doesn't compose well, second - contains side effects. let's look into ways to manage
+those effects.
+<!-- end_slide -->
+
+# Modeling side effects
+
+We start with the easiest possible side effects - a computation that can fail. In Rust you'd
+use `Option` or `Result` for that.
+
+
+```rust
+fn bad(i: Option<u8>) -> Option<u8> {
+    match i {
+        None => Some(42),
+        Some(42) => Some(1),
+        _ => None
+    }
+}
+```
+
+```rust
+fn good(i: Option<u8>) -> Option<u8> {
+    match i {
+        None => None,
+        Some(i) => Some(i + 3),
+    }
+}
+```
+
+They are both valid functions, but for "bad" case pure computation and effects are mixed
+together, while in "good" one - effects are separate
+
+<!-- end_slide -->
+# Functor
+
+Second function represents something called a `Functor`
+
+
+Like pure functions Functors are also great. If you have something that obeys Functor laws -
+you can plug any pure computation inside and it will just work, easy to test too. Rust comes
+with a bunch of Functors: `Option::map`, `Result::map`. `Iterator<Item = A>::map` is also a
+functor
+
+Functor must preserve identity morphisms and compositions of morphisms:
+
+`a.map(id) == a`, `a.map(f).map(g) == a.map(g . f)`. But that's just a complicated way of
+saying - "don't throw away information and maintain the computation shape". In the previous
+example "bad" throws away and invents stuff, "good" does great.
+
+In general I'm going to use a notation `F<A>` for types and `map f` for arrows, where `F` is a
+type or a trait concrete for this Functor, and `A` is some variable Functor knows nothing about
+so it can only apply morphisms like `Fn(A) -> B` and nothing else
+
+![](pic2.png)
+
+
+<!-- end_slide -->
+# Applicative
+
+Before I said we'll assume functions take a single argument and for multiple - we make a tuple.
+
+Consider two values `A` and `B` and a function `f: Fn((A, B)) -> C` and think how corresponding
+Functor might look like.  A tuple is `(F<A>, F<B>)` and there's no corresponding pure
+functions. Not good. We need to introduce one more bit of abstraction.
+
+__Suppose `F` is a functor, then for two computations__
+__`F<A>` and `F<B>` we can make `Fn((F<A>, F<B>)) -> F<(A, B)>`__
+
+For `Option` this is `Option::zip`, for `Result`.... It can also be `zip`, as long as errors
+agree, but for some reason it's missing. There's also `Iterator::zip` that does it for two
+iterators. Same idea
+
+[functor like category, `(A, B) -> C` on the left, `(F<A>, F<B>) -> F<(A, B)> -> F<C>` on the right]
+
+<!-- end_slide -->
+# Alternative
+
+Getting somewhere. We know that the parser might fail so we must introduce some notion of
+failure. For `Option` this is `None`, for `Result` this is `Err`.
+
+We can add two functions to our abstract interface
+
+```
+pure: Fn(T) -> F<T>
+fail: Fn(E) -> F<T>
+```
+
+And now that we know that some computation can fail - we want something to be able to try
+several of them
+
+```rust
+alt: Fn(F<T>, F<T>) -> F<T>
+```
+
+In Rust this is `Option::or` and `Result::or` (and their variants)
+
+<!-- end_slide -->
+# Making a simple parser
+
+Now that we have all the basic methods we can look into making something with the parser
+
+```rust
+map: Fn(F<A>, Fn(A) -> B) -> F<B>
+zip: Fn(F<A>, F<B>) -> F<(A, B)>
+pure: Fn(A) -> F<A>
+fail: Fn(E) -> F<A>
+alt: Fn(F<A>, F<A>) -> F<A>
+```
+
+It doesn't matter which version of the parser we begin with -
+
+```rust
+Fn(I) -> Result<(I, T), Error> = ...
+Fn(&mut I) -> Result<T, Error> = ...
+```
+
+```rust
+struct Parser<T>(Box<Fn(&mut Args)> -> Result<T, Error>);
+```
+
+```rust
+trait Parser<A> {
+    fn map(self, impl Fn(A) -> B) -> impl Parser<B> {..}
+    fn zip(self, other: impl Parser<B>) -> impl Parser<(A, B)> {..}
+   ...
+}
+```
+
+
+<!-- end_slide -->
+
+# Methods we can implement
+```rust
+trait Parser<A> {
+    // Required to compose
+
+    fn map(self, f: impl Fn(F<A>, Fn(A) -> B) -> impl Parser<B> {..}
+    fn zip(self, other: impl Parser<B>) -> impl Parser<(A, B)> {..}
+    fn pure(val: A) -> impl Parser<A> {..}
+    fn fail(message: &'static str) -> impl Parser<A> {..}
+    fn alt(self, other: impl Parser<A>) -> impl Parser<A> {..}
+
+    // Convenience
+    fn optional(self) -> impl Parser<Option<A>> {..}
+    fn many(self) -> impl Parser<Vec<A>> {..}
+    fn some(self, error: &'static str) -> impl Parser<Vec<A>> {..}
+    fn collect<C: FromIterator<A>>(self) -> impl Parser<C> {..}
+    fn parse(self, impl Fn(A) -> Result<B, E>) -> impl Parser<B> {..}
+
+    // Required to run
+    fn run(self) -> A {..}
+}
+```
+<!-- end_slide -->
+
+# dynamic completionm
+
+- Easy to implement
+
+# Conclusions
+
+- High level overview of your app/API
+- Composing with CT reduces number of combinations from N^2 to N
+- Users don't have to know about it :)
+
+
+# See also
+- https://rustmagazine.org/issue-2/applicative-parsing/
+- https://en.wikipedia.org/wiki/Category_theory
+- https://crates.io/crates/bpaf
+- https://github.com/pacak
+- @manpacket@functional.cafe