http://codinghelmet.com/articles/what-makes-functional-and-object-oriented-programming-equal
Comparison of 4 Functional Programming (FP : F#, Lisp/Clojure Haskell, Scala,… ) concepts in Object-Oriented (OO: C#, C++, Java,… ) :
[Not covered here] : There are other FP techniques lacking in OO: Functor (FoldMap), Monad, etc.
1. Function:
2. Closure (variable binding)
3. Currying
4. Function Composition
…
…Conclusion:
Advanced Programming needs Advanced Math: eg.
Video Game Animation: Verlet Integration
AI: Stats, Probability, Calculus, Linear Algebra
Search Engine : PageRank: Linear Algebra
Abstraction in Program “Polymorphism” : Monoid, Category, Functor, Monad
Program “Proof” : Propositions as Types, HoTT
https://awalterschulze.github.io/blog/post/neglecting-math-at-university/
Abstraction: Monoid, Category
Category
Functional Programming has the following key styles:
1) Lambda function:
2) Map, Filter, Reduce
Since April 2019 until I re-visit this Youtube video on 12 August 2019, I can now totally understand his speech after a pause of 4 months by viewing other related Youtube (below prerequisite) videos on Category Theory, Type Theory, Homotopy Type Theory.
That is the technique of self-study:
- First go through the whole video,
- Don’t understand? view other related simpler videos.
- Repeat 1.
Prerequisite knowledge:
Two Key Takeaway Points:
…
…
…
https://dev.to/juaneto/knowing-monads-through-the-category-theory-1mea
While Mathematicians like to talk non-sensical abstract idea, Informaticians want to know how to apply the idea concretely:
Mathematical Parlance:
Monad = Monoid +Endofunctor
Monoid = Identity + Associative
Endo-functor = functor between 2 same categories
IT Parlance:
Monad is a ‘function’ to wrap the ‘side effects’ (exception errors, I/O,… ) so that function composition in ‘pipeline‘ chained operation sequence is still possible in pure FP (Functional Programming, which forbids side-effects).
Some common Monads: ‘Maybe’, ‘List’, ‘Reader’…
This allows monads to simplify a wide range of problems, like handling potential undefined values (with the Maybe monad), or keeping values within a flexible, well-formed list (using the List monad). With a monad, a programmer can turn a complicated sequence of functions into a succinct pipeline that abstracts away additional data management, control flow, or side-effects.[2][3]
Exploring Monads in Scala Collections
Introduction:
Promise of Monads (A)
Promise of Monads (B)
(Part 1/3) – λC 2017
What is Category ?
Objects
Morphism (Arrows )
Rule1: Associative
Rule 2: Identity
A <– C –> B
Product of Categories : A x B
Unique
Sum of Categories: A + B
Unique
(Either a b)
Co-Product
Reverse all arrows.
Unique
Functor F: C-> D
Mapping of all objects (A, B) in categories C,D
Mapping of arrows f
f : A -> B
Ff : FA -> FB (preservation)
F Id = Id
F (f。g) = Ff。Fg
Example:
Constant C -> F
FC = k
Ff = Id
Arrow Functor F: C -> D
For any object A in C,
F A = D -> A
(Functional Type is also Type)
Functors compose !
Category of categories:
Objects: categories
Arrows : Functors
Haskell Category (Hask) is always Endo-Functor, ie Category Hask to itself.
Mapping of arrows.
Mapping of Objects = predefined
(Part 2/3) – λC 2017
(part 3/3) – λC 2017
3) Higher–Order Function, Closure
As Tikhon Jelvis explained in his response, functions map sets to sets, and functions themselves form sets. This is the essence of the untyped lambda calculus. Unfortunately, untyped lambda calculus suffers from the Kleene–Rosser paradox (later simplified to Curry’s paradox).
This paradox can be removed by introducing types, as in the typed lambda calculus. Simple types are equivalent to sets, but in order to pass a function as an argument to another function (or return one), we have to give this function a type. To really understand what a function type is, you need to look into category theory.
The categorical model for the typed lambda calculus is a category in which objects are types and morphism are functions. So if you want to have higher order functions, you have to be able to represent morphisms as objects — in other words, create a type for functions. This is possible only if the category is cartesian closed. In such a category you can define product types and exponential types. The latter correspond to function types.
So that’s a mathematical explanation for higher order.
WhatsApp is written in Erlang – a Functional Programming Language. It supports 900 million users worldwide with only 50 programmers.
Part 1: Compare 3 paradigms:
Introduction to Haskell
Part 2: Example – The 8 Queens Problem
Note: A simpler Haskell coding here.
Category Theory (CT) is like Design Pattern, only difference is CT is a better mathematical pattern which you can prove, also it has no “SIDE-EFFECT” and with strong Typing.
The examples use Haskell to explain the basic category theory : product, sum, isomorphism, fusion, cancellation, functor…
Object-Oriented has 2 weaknesses for Concurrency and Parallel programming :
Category Theory (CT): a higher abstraction of all different Math structures : Set , Logic, Computing math, Algebra… =>
Our brain works by: 1) Abstraction 2) Composition 3) Identity (to identify)
What is a Category ?
1) Abstraction:
- Objects
- Morphism (Arrow)
2) Composition: Associative
3) Identity
Notes:
Example in Programming:
Note: We just look at the Category “Types Set” from external Macroview, “forget ” what it contains, we only know the “composition” (Arrows) between the Category “Type Set”, also “forget” what these Arrows (sin,cosin, tgt, etc) actually are, we only study these arrows’ behavior (Associativity).
2.1 : Function of Set, Morphism of Category
Set: A function is
Category: [Surjective]
g 。f = h 。f
=> g = h (Right Cancellation )
2.2 Monomorphism
f 。g = f 。h
=> g = h (Left cancellation)
NOT Necessary !! Reason ( click here):
In Haskell, 2 foundation Types: Void, Unit
Void = False
Unit ( ) = True
Functions : absurd, unit
absurd :: Void -> a (a = anything)
unit :: a -> ()
[to be continued 3.1 ….]
Functional Programming, especially Haskell, requires the Math foundation in Abstract Algebra, and Category.
http://reinh.com/notes/posts/2014-07-25-recommended-reading-material.html
