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Y-Combinator
@matyo91
25-04-2023
In PHP
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⾠ I’m not an expert ⾠
I will speak with my own words
@matyo91
@matyo91
25-04-2023
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Objectives : Go back up to theoretical concepts that solve an issue
How I discovered Y-Combinator and why I do need them ?
Why Y-Combinator ?
Introduction to Lambda Calculus
Exemples of basic Lambda Calculus and equivalent in PHP
Y-Combinator in PHP (for me and others github repositories)
Plan
@matyo91
25-04-2023 @matyo91
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How I discovered Y-Combinator ?
@matyo91
25-04-2023
Why I need them ?
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In PHP https://github.com/darkwood-fr/flow
In PHP https://github.com/phpflo/phpflo
25-04-2023
Flow Based Railway Flow Based
MulByTwo MinusThree
AddOne
@matyo91
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Why Y-Combinator ?
@matyo91
25-04-2023
What does it solve ?
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Do recursion in a language that
do not have any recursion
mechanism at all
“
@matyo91
25-04-2023
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Recursive
Iterative
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25-04-2023
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Pipe
@matyo91
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25-04-2023
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A Y-combinator is a "functional" (a function that operates on other
functions) that enables recursion, when you can't refer to the
function from within itself.
In computer-science theory, it generalises recursion, abstracting its
implementation, and thereby separating it from the actual work of
the function in question.
This is applicable in languages that support Lambda-Calculus
functions
Y-Combinator
@matyo91
25-04-2023 @matyo91
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Lambda Calculus
@matyo91
25-04-2023
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Alonzo Church
( 1903 - 1995 )
Inventor of Lambda Calculus
@matyo91
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25-04-2023
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Church Turing
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λ x . x + 1
( λ x . x + 1 ) ( 5 ) = 5 + 1 = 6
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25-04-2023
X X + 1
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λ x . λ y . x+y
( λ x . λ y . x+y ) ( 5 , 6 ) = 5 + 6 = 11
@matyo91
@matyo91
25-04-2023
X
X + Y
Y
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There are variables.
We got a way to build functions.
We can apply functions to each others by providing elements of the
language itself.
All derivatives of lambda calculus language is something defined in
terms of this definition.
Lambda Calculus
@matyo91
25-04-2023 @matyo91
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True ?
False ?
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25-04-2023
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25-04-2023
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True = λ x . λ y . x
False = λ x . λ y . y
@matyo91
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25-04-2023
X
X
Y
True
X
Y
Y
False
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True = λ x . λ y . x
@matyo91
@matyo91
25-04-2023
False = λ x . λ y . y
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Not ?
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25-04-2023
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Not = λ b . b False True
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25-04-2023
b False True
b Not
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Not = ( λ b . b False True ) ( True )
= True False True
= ( λ x . λ y . x ) ( False True )
= False
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25-04-2023
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@matyo91
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25-04-2023
Not = λ b . b False True
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Y-Combinator
@matyo91
25-04-2023
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Y ( f ) = f ( Y ( f ) )
= f ( f ( Y ( f ) ) )
= f ( f ( f ( f ( ... ) ) ) )
@matyo91
@matyo91
25-04-2023
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Fac (n) = if n == 1 then 1
else n * Fac (n - 1)
Fac (3) = 3 * Fac (2)
= 3 * 2 * Fac (1)
= 3 * 2 * 1
= 6
@matyo91
@matyo91
25-04-2023
YCombinator in Lambda Calculus
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Fac = Y ( ? )
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25-04-2023
YCombinator in Lambda Calculus
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Fac = Y ( ? )
= Y ( λ f . λ n . ? )
= Y ( λ f . λ n . max( n , 1 ) * Fac ( n - 1 ) )
@matyo91
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25-04-2023
YCombinator in Lambda Calculus
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Loop = ?
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25-04-2023
YCombinator in Lambda Calculus
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Loop = Loop
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25-04-2023
YCombinator in Lambda Calculus
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Loop = ( λ x . x . x ) ( λ x . x . x )
@matyo91
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25-04-2023
YCombinator in Lambda Calculus
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Loop = ( λ x . x . x ) ( λ x . x . x )
@matyo91
@matyo91
25-04-2023
YCombinator in Lambda Calculus
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Loop = ( λ x . x . x ) ( λ x . x . x )
Loop = ( λ x . ( λ x . x . x ) . ( λ x . x . x ) )
@matyo91
@matyo91
25-04-2023
YCombinator in Lambda Calculus
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Loop = ( λ x . x . x ) ( λ x . x . x )
Loop = ( λ x . ( λ x . x . x ) . ( λ x . x . x ) )
Loop = ( λ x . x . x ) ( λ x . x . x )
@matyo91
@matyo91
25-04-2023
YCombinator in Lambda Calculus
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Loop = Y ( ? )
@matyo91
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25-04-2023
YCombinator in Lambda Calculus
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Loop = Y ( λ x . x . x )
= ( λ x . x . x ) ( Y ( λ x . x . x ) )
= ( λ x . x . x ) ( λ x . x . x ) ( Y ( λ x . x . x ) )
@matyo91
@matyo91
25-04-2023
YCombinator in Lambda Calculus
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Y ( f ) = λ f ( λ x . f ( x x ) ) ( λ x . f ( x x ) )
@matyo91
@matyo91
25-04-2023
YCombinator in Lambda Calculus
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https://github.com/darkwood-fr/flow
@matyo91
@matyo91
25-04-2023
YCombinator in PHP
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https://github.com/loophp/combinator
@matyo91
@matyo91
25-04-2023
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https://github.com/igorw/lambda-php
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25-04-2023
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In Real Life
@matyo91
25-04-2023 @matyo91
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Hello!
I Am Mathieu Ledru
You can contact me at @matyo91
25-04-2023 @matyo91
Thanks!
Any questions?