In this video, he scales the technique up to calculating a compiler! Predecessor woes Graham Hutton is the absolute master of this technique. This way of thinking – considering an incomplete function and by checking the result on each argument refining its definition – is called program calculation. As a result, our definition of isZero becomes: const isZero = n => n(x => false_)(true_) One such choice is the constantly-false function x => false. And, in both cases, we need the final result to be false. That is, whenever n is not zero, the result of isZero(n) is going to be the application of unknown f to either true (the value we have found for unknown x) or other recursive application of unknown f. On the other hand, we know that isZero(zero) should equal true_, so it must be the case that unknown x is true.īy the end of part 2, we understood how a Church numeral could be seen as a repeated application of a function. isZero(zero) = zero(unknown_f)(unknown_x) = unknown_x When n is the numeral zero, the definition collapses into unknown_x, because zero applies its first argument no times. const isZero = n => n(unknown_f)(unknown_x) Remember that Church numerals are functions themselves as we did in part 2, we are going to figure out which arguments unknown_f and unknown_x should be given to obtain the desired outcome. Are you zero?įocusing on our first task, we want to define a function isZero(n) such that it returns true (our encoded Boolean) whenever n is zero, and false otherwise. We are missing a way to figure out whether a number is zero, and to obtain the predecessor of a number. We already know how to represent some of the elements namely the conditional, the number 0, and the addition. If we were not talking about lambda calculus, the solution would be quite simple: const littleGaussHelper = n => n = 0 ? 0 : n + littleGaussHelper(n - 1) Not by developing a formula, but by programming the computer to do the sum. Stories aside, our goal is to help Little Gauss with his punishment. Have I mentioned Gauss is considered “The Prince of Mathematics” for his broad advancements in many areas of the discipline? How? by developing a simple formula to do so. As a punishment, his teacher asked him to sum all the numbers from 1 to 100, which he did in a blast. They say that when Carl Friedrich Gauss was a child – let’s call him “Little Gauss” from now on – he was a bit naughty. If you’ve ever listened to some of my beginner talks, you’ve definitely heard this story, because I love it. In this post, we look at the more familiar technique of recursion, leading us to talk about fixed points and, ultimately, to the swamps of non-termination. But, let’s be honest, defining addition and multiplication was kind of a magic trick. We’re moving fast through this lambda calculus series! In part one, we laid out the foundations for working with JavaScript as a lambda calculus, and, through part 2, we added Booleans and numbers to our repertory. There needs to be a revival or symbolic manipulation and lambda calculus for AI to truly progress beyond function fitting.This article was originally published at on January 14, 2021. The computational world got lost in other aspects like data processing, black-box model fitting (including ANN). Although this happened more than 50 years ago, this approach towards AI has not been given as much importance. The need for performing operations on symbols has led to the development of Lambda calculus and the language group of Lisp. In a way digital machines use 0’s and 1’s as symbols at the very base level and create abstractions around them already. If a machine can truly learn symbols, their associations to combine symbols to form complex symbols and form abstractions, it will be closer to humans in learning ability. But in this case the machine is not learning that symbol, rather the programmer is in a way hardcoding the symbol. To some extant this already happens when a variable is a given a name and it is referred by that name in subsequent code. If we want machines to emulate humans, machines should also understand symbols. This symbolic representation gives us the power of abstraction. Our arithmetic calculations are also symbol based. Even when we remember numbers (like for example phone number or value of Pi) we store it as a series of symbols rather than as float/in values. We do not remember data in quantitative fashion. Most of human learning happens through symbols.
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