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Functional Programming Tommy “is awesome” Montgomery 2009-03-06

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What is functional programming? <ul><li>Uses functions </li></ul><ul><li>Lambda Calculus </li></ul><ul><li>Very academic </li></ul><ul><li>Kinda goofy </li></ul>

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Elements of functional languages <ul><li>Recursion </li></ul><ul><li>Functions (der) </li></ul><ul><li>No state </li></ul>

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Functional Languages <ul><li>Haskell </li></ul><ul><li>Erlang </li></ul><ul><li>F# </li></ul><ul><li>OCaml </li></ul><ul><li>Scheme </li></ul><ul><li>Smalltalk </li></ul><ul><li>J </li></ul><ul><li>K </li></ul><ul><li>Mathematica </li></ul><ul><li>XSLT </li></ul><ul><li>LISP (kinda) </li></ul>

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Fun Fact #1 <ul><li>Tommy was once a professional musician </li></ul>

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λ -c alculus <ul><li>Anonymous functions </li></ul><ul><ul><li>JavaScript </li></ul></ul><ul><ul><li>PHP 4.0.1 – PHP 5.2.x (kinda) </li></ul></ul><ul><ul><li>PHP 5.3 (more kinda) </li></ul></ul><ul><ul><li>C# 2.0 </li></ul></ul><ul><ul><li>Java sucks, as usual </li></ul></ul><ul><li>Unary </li></ul><ul><ul><li>Functions take one argument, and return one value </li></ul></ul>

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Example <ul><li>λ x . x + 2 </li></ul>

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Example <ul><li>λ x . x + 2 </li></ul>input

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Example <ul><li>λ x . x + 2 </li></ul>input return value

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Example <ul><li>λ x . x + 2 </li></ul>input return value f(x) = x + 2

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Higher order functions <ul><li>Functions that take functions as arguments and return functions </li></ul><ul><li>Where the real power of functional programming lies </li></ul>

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Higher order function example <ul><li>Mathematical derivative </li></ul>

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Higher order function example //f is a function function derivative ( f ) { return function ( x ) { //approximation of derivative return ( f ( x + 0.00001 ) – f ( x )) / 0.00001 ; } }

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Higher order function example //evaluate derivative of x 2 : var deriv_x_squared = derivative ( function ( x ) { return x * x ; } ); alert ( deriv_x_squared ( 3 )); //alerts 6ish

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Fun Fact #2 <ul><li>Tommy used to be a gangsta </li></ul>

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Bound variables <ul><li>The λ operator binds its variables to its scope </li></ul><ul><li>All other variables are “free” </li></ul>Freedom 

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Bound variables <ul><li>λ x . x + y </li></ul>

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Bound variables <ul><li>λ x . x + y </li></ul>bound variable bound variable free variable

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Bound variables <ul><li>λ x . x + y </li></ul>bound variable bound variable free variable f(x) = x + y

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Natural numbers <ul><li>Everything is a function, remember? </li></ul><ul><li>How do we define the natural numbers in functional programming (1, 2, 3, …)? </li></ul>

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<ul><li>very painfully </li></ul>

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Church numerals <ul><li>0 ≡ λ f . λ x . x </li></ul><ul><li>1 ≡ λ f . λ x . f x </li></ul><ul><li>2 ≡ λ f . λ x . f (f x) </li></ul><ul><li>3 ≡ λ f . λ x . f (f (f x)) </li></ul><ul><li>... </li></ul>

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Church numerals in JavaScript //identity function // λ x . x function identity ( x ) { return x ; } //Church numeral zero // λ f . λ x . x function zero ( f ) { return identity ; }

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Church numerals in JavaScript //successor function (succ(n) = n + 1) // λ n . λ f . f (n f x) function succ ( n ) { return function ( f ) { return function ( x ) { return f ( n ( f )( x )); } } }

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Church numerals in JavaScript //gets a function representing //the nth church number function getChurchNumber ( n ) { var ret = zero ; for ( var i = 0 ; i < n ; i ++) { ret = succ ( ret ); } return ret ; }

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Church numerals in JavaScript //gets the nth church number function getNaturalNumber ( n ) { var value = 0 ; for ( var i = 0 ; i < n ; i ++) { value += getChurchNumber ( i )( function ( x ) { return x + 1 ; } )( 0 ); } return value ; }

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Fun fact #3 <ul><li>Wombats are unfairly cute </li></ul>

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Addition <ul><li>// λ m . λ n . λ f . n f (m f x) </li></ul><ul><li>function add ( m ) { </li></ul><ul><li>return function ( n ) { </li></ul><ul><li>return function ( f ) { </li></ul><ul><li>return function ( x ) { </li></ul><ul><li>return n ( f )( m ( f )( x )); </li></ul><ul><li>} </li></ul><ul><li>} </li></ul><ul><li>} </li></ul><ul><li>} </li></ul>

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Addition using successor <ul><li>// λ m . λ n . m succ n </li></ul><ul><li>function addWithSucc ( m ) { </li></ul><ul><li>return function ( n ) { </li></ul><ul><li>return m ( succ )( n ); </li></ul><ul><li>} </li></ul><ul><li>} </li></ul>

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Holy crap <ul><li>http://uxul.wordpress.com/2009/03/02/generating-church-numbers-with-javascript/ </li></ul>

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Recursion <ul><li>Local variables are the devil! </li></ul><ul><li>Also, a paradox </li></ul><ul><li>In lambda calculus, you cannot define a function that includes itself </li></ul><ul><li>i.e. the definition of recursion </li></ul>

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The Y Combinator <ul><li>Cool name </li></ul><ul><li>Also known as “The Paradoxical Operator” </li></ul>

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Recursion To properly define recursion, a recursive function g must take as an argument a function f , which expands to g which takes an argument f .

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<ul><li>Got that? </li></ul>

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Recursion <ul><li>f = g(f) </li></ul><ul><li>f is a fixed point of g </li></ul><ul><li>Known as the Y-Combinator </li></ul>

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The Y-Combinator <ul><li>Y = λ g . (λ x . g (x x)) (λ x . g (x x)) </li></ul>

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The Y-Combinator in JavaScript <ul><li>function Y ( g ) { </li></ul><ul><li>return function ( x ) { </li></ul><ul><li>return g ( </li></ul><ul><li>function ( y ) { </li></ul><ul><li>return x ( x )( y ); </li></ul><ul><li>} </li></ul><ul><li>); </li></ul><ul><li>}( </li></ul><ul><li>function ( x ) { </li></ul><ul><li>return g ( </li></ul><ul><li>function ( y ) { </li></ul><ul><li>return x ( x )( y ); </li></ul><ul><li>} </li></ul><ul><li>); </li></ul><ul><li>} </li></ul><ul><li>); </li></ul><ul><li>} </li></ul><ul><li>http://matt.might.net/articles/implementation-of-recursive-fixed-point-y-combinator-in-javascript-for-memoization/ </li></ul>

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Functional Factorial <ul><li>Y factorial n </li></ul><ul><li>Y : definition of Y-Combinator </li></ul><ul><li>factorial : functional definition of factorial </li></ul><ul><li>n : integer </li></ul>

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Functional Factorial <ul><li>function factorial ( f ) { </li></ul><ul><li>return function ( n ) { </li></ul><ul><li>return ( n == 0 ) ? 1 : n * f ( n – 1 ); </li></ul><ul><li>} </li></ul><ul><li>} </li></ul><ul><li>//call like so: </li></ul><ul><li>alert ( Y ( factorial )( 5 )); //alerts 120 </li></ul>

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What is happening? <ul><li>Recursive calls are abstracted to an inner function, which is evaluated as a lambda function (which by very definition are not recursive, remember?) </li></ul><ul><li>Interesting, but not all that useful… </li></ul>

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Other topics… <ul><li>Currying </li></ul><ul><li>Mapping </li></ul><ul><li>Reduction </li></ul><ul><li>Substitution </li></ul><ul><li>Elaboration on statelessness </li></ul><ul><li>Imperative vs. Functional </li></ul><ul><li>Performance </li></ul>

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Questions…