The dream of

λ Calculus

Vincent

RC F2'2019

What I'm doing here

Learn about programming languages design (mainly through "Types and programming languages" from Pierce) ...
... in the perspective of building my own later

Background Story

Invented by Alzonzo Church (1936)
Church Turing Thesis: functions computable in are function computable in a Turing machine
Fun fact: Church was the doctoral advisor of Turing

Syntax

How the hell can computability be captured with such a tiny syntax ?

x (define a variable)
λx. t (function of param x and body t)
f t (function application: f applied to t)

No builtin constant or operators, no numbers, no loops...everything is a function

Semantics

(λx.x) (λz. (λx.x z))
λz. (λx.x z)
λz.z

How to evaluate a term ? idea: substitute right hand side of a term for bound variable in the body of the function

Multiple arg function

Idea: use higher order functions

λx.λy.x y

This can equivalently be seen as a function which returns a function (ie given x, it returns the function λy. x y, and a function of two arguments. => currying

Boolean encoding

true = λt. λf. t
false = λt.λf. f
reminder: every value is a function
if = λb.λx.λy. b x y (where b is true or false)
not =λb.b false true

And many more...

numbers
arithmetic operation
recursion with infamous ycombinator:
λf. (λx. f (λy. x x y)) (λx. f (λy. x x y))

Conclusion

Not practical
but interesting to see that computability can be built from such a tiny core
gave rise to functional programming