Lambda Calculas

        

Scenario:

• λx.x [Function Notation,Argument.Body(Return)] like function (x) {return x}
• Function Application - λx.x (y). So substitute all x with y
• λx(bound variable).x y(free variable). 
• Substitution/Reduction
• Substitution is left-applicative as you apply from left (when value is applied to a function, it is substituted in the function itself)
• E.g. - λx.x5 => x[x: = 5] => 5
•  λx. λy.y x => λx. λy.y => λy.y
• Multiple Variables
• λx.λy.t also called currying, so its like Function x has body of λy.t [Function calling Function]
• Parenthesis:
• λx.(λy.y x) => λx.(λy.yx) => λx.(y[y: = x]) => λx.x
• λx.λy.y x => λx.λy.yx => λx.(λy.y) => λy.y
• Combinators [Combination of Functions]
• Boolean
• λx.λy.x [true]
• λx.λy.y [false] 
• Conditional
• λx.λy.λz.x y z
• Numbers
• λf.λx.x=0
• λf.λx.f(x)=1 => λx.x (like identity)
• λf.λx.f(f(x))=2
• λf.λx.f(f(f(x)))=3
• Omega Combinator [looping]
• Ω => λx.xx(λx.xx) => λx.xxλx.xx
• Y combinator [for each, only once]:
• Y = λf.(λx.f(xx))f.(λx.f(xx))

//Identity Function/I Combinator

function myFunc(x){return x}

console.log((x=>x)(10));

//Constant Function

let y = 5

console.log((x=>y)(10));

//Substitution

//Constant [Returns 5]
let y = 5

console.log((x=>y)(10));

//Identity [Returns 10]
let y = 5

console.log((x=>x)(10));

//Multiple

console.log(((x,y) => x+y)(5,10));

//5
console.log((x => y => x + y)(2)(3));

//Error
console.log((x => x) (y => y)(2)(3));

//2
console.log((x => x) (y => y)(2));

//Combinators 

//Boolean

let True = (x => y => x);

let False = (x => y => y);

console.log(True(true)(false));

console.log(False(true)(false));

let If = (x => y => z => x(y)(z));

console.log(If(True)("TRUE")("Yes"));

console.log(If(False)("NOT")("nah"));

//Numbers:

let calculate = f => f(x => x + 1) (0)

let zero = f => x => x;
let one = f => x => f(x);
let two = f => x => f(f(x));

console.log(calculate(zero));
console.log(calculate(one));
console.log(calculate(two));

//Omega [for]
let Omega = x => x(x);
console.log(Omega(Omega));

//Y Combinator [For each]
//Fibonnaci
let Y = f => (x => x(x))(x => f(y => x(x)(y)));
let fib = f => n => n <=1 ? n : f(n-1) + f(n-2);
let fibfunc = Y(fib);
console.log(fibfunc(10));