3) Higher–Order Function, Closure

# Tag Archives: Functional programming

# Higher Order Function

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.

# You should learn Functional Programming in 2017

WhatsApp is written in Erlang – a Functional Programming Language. It supports 900 million users worldwide with only 50 programmers.

# Functional Programming for the Object Oriented – Øystein Kolsrud

Part 1: Compare 3 paradigms:

- Imperative
- Object- Oriented
- Functional Programming

Introduction to Haskell

Part 2: Example – The 8 Queens Problem

Note: A simpler Haskell coding here.

# Programming and Math

# BM Category Theory : Motivation and Philosophy

Object-Oriented has 2 weaknesses for Concurrency and Parallel programming :

- Hidden Mutating States;
- Data Sharing.

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:

ObjectsMorphism (Arrow)

2) Composition: Associative

3) Identity

**Notes: **

**Small Category with “Set” as object.****Large Category without Set as object.****Morphism is a Set : “Hom” Set.**

**
Example in Programming**:

- Object : Types Set
- Morphism : Function “Sin” converts degree to R:

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 **

**Surjective (greek: epic / epimorphism 满射),****Injective (greek : monic / monomorphism 单射)**

**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 -> ()

unit :: a -> ()

[to be continued 3.1 ….]

# Recommended Reading for Functional Programming

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