Y-Combinator in PHP

This talk is about Y-Combinator and implementation in PHP

Article associated to this talk

https://blog.darkwood.fr/article/y-combinator-en-php

Video associated to this talk

https://www.youtube.com/watch?v=QSS_ZcO8Q1g

Transcript

1. Y-Combinator
   @matyo91
   25-04-2023
   In PHP
   

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2. ⾠ I’m not an expert ⾠
   
   
   I will speak with my own words
   @matyo91
   @matyo91
   25-04-2023
   

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3. 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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4. How I discovered Y-Combinator ?
   @matyo91
   25-04-2023
   Why I need them ?
   

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5. 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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6. Why Y-Combinator ?
   @matyo91
   25-04-2023
   What does it solve ?
   

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7. Do recursion in a language that
   do not have any recursion
   mechanism at all
   “
   @matyo91
   25-04-2023
   

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8. Recursive
   
   
   Iterative
   
   
   @matyo91
   @matyo91
   25-04-2023
   

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9. Pipe
   
   
   @matyo91
   @matyo91
   25-04-2023
   

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10. 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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11. Lambda Calculus
    @matyo91
    25-04-2023
    

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12. Alonzo Church
    
    
    ( 1903 - 1995 )
    
    
    Inventor of Lambda Calculus
    @matyo91
    @matyo91
    25-04-2023
    

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13. Church Turing
    @matyo91
    @matyo91
    25-04-2023
    

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14. λ x . x + 1
    
    
    ( λ x . x + 1 ) ( 5 ) = 5 + 1 = 6
    @matyo91
    @matyo91
    25-04-2023
    X X + 1
    

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15. 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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16. 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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17. True ?
    
    
    False ?
    @matyo91
    @matyo91
    25-04-2023
    

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18. @matyo91
    @matyo91
    25-04-2023
    

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19. True = λ x . λ y . x
    
    
    False = λ x . λ y . y
    @matyo91
    @matyo91
    25-04-2023
    X
    X
    Y
    True
    X
    Y
    Y
    False
    

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20. True = λ x . λ y . x
    @matyo91
    @matyo91
    25-04-2023
    False = λ x . λ y . y
    

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21. Not ?
    @matyo91
    @matyo91
    25-04-2023
    

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22. @matyo91
    @matyo91
    25-04-2023
    

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23. Not = λ b . b False True
    @matyo91
    @matyo91
    25-04-2023
    b False True
    b Not
    

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24. Not = ( λ b . b False True ) ( True )
    
    
    = True False True
    
    
    = ( λ x . λ y . x ) ( False True )
    
    
    = False
    @matyo91
    @matyo91
    25-04-2023
    

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25. @matyo91
    @matyo91
    25-04-2023
    Not = λ b . b False True
    

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26. Y-Combinator
    @matyo91
    25-04-2023
    

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27. Y ( f ) = f ( Y ( f ) )
    
    
    = f ( f ( Y ( f ) ) )
    
    
    = f ( f ( f ( f ( ... ) ) ) )
    @matyo91
    @matyo91
    25-04-2023
    

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28. 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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29. Fac = Y ( ? )
    @matyo91
    @matyo91
    25-04-2023
    YCombinator in Lambda Calculus
    

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30. Fac = Y ( ? )
    
    
    = Y ( λ f . λ n . ? )
    
    
    = Y ( λ f . λ n . max( n , 1 ) * Fac ( n - 1 ) )
    @matyo91
    @matyo91
    25-04-2023
    YCombinator in Lambda Calculus
    

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31. Loop = ?
    @matyo91
    @matyo91
    25-04-2023
    YCombinator in Lambda Calculus
    

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32. Loop = Loop
    @matyo91
    @matyo91
    25-04-2023
    YCombinator in Lambda Calculus
    

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33. Loop = ( λ x . x . x ) ( λ x . x . x )
    @matyo91
    @matyo91
    25-04-2023
    YCombinator in Lambda Calculus
    

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34. Loop = ( λ x . x . x ) ( λ x . x . x )
    @matyo91
    @matyo91
    25-04-2023
    YCombinator in Lambda Calculus
    

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35. 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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36. 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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37. Loop = Y ( ? )
    @matyo91
    @matyo91
    25-04-2023
    YCombinator in Lambda Calculus
    

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38. 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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39. 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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40. https://github.com/darkwood-fr/flow
    @matyo91
    @matyo91
    25-04-2023
    YCombinator in PHP
    

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41. https://github.com/loophp/combinator
    @matyo91
    @matyo91
    25-04-2023
    

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42. https://github.com/igorw/lambda-php
    @matyo91
    @matyo91
    25-04-2023
    

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43. In Real Life
    @matyo91
    25-04-2023 @matyo91
    

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44. Hello!
    
    
    I Am Mathieu Ledru
    You can contact me at @matyo91
    25-04-2023 @matyo91
    Thanks!
    Any questions?
    

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