To restrict, for example, arithmetic functions to require numeric That represent numbers and arithmetic, but there is no way You can use these simple building blocks to construct functions Sugar, all you are left with is functions that take arguments and return Operations is going to look kinda weird at first.Lambda calculus is a very simple language. The upcoming treatment of conditional expressions and boolean Conditional Expressions and Boolean Operations We can create conditional expressions and boolean operations purely Starting with just four basic constructs – names, functions,Īpplications, expressions – and five simple rules for performingįunction application, we've been able to construct pairs of argumentsĪnd things are about to get even more fun! We're now ready to see how Two arguments and which you then apply to a selection function.

This is super freaking cool! A pair is a function which has 'captured' Select_second to get the argument you want. The result is aįunction which you can apply to either select_first or Returns a function with takes a second argument.

So, to reiterate, make_pair works by taking a first argument. (( func identity ) apply ) select_first ) => (( select_first identity ) apply ) (( func identity ) apply ) Application of select_first and select_second (( select_first identity ) apply ) (( select_second identity ) apply ) Apply the result of make_pair to select_first ( λ func. Result of (( make_pair identity ) apply ) λ func. Let's start to flesh out this purely abstract notion of functionĪpplication with some examples, starting with the identity function: Without any regard for how actual hardware will carry out the Keep in mind that we're talking about a mathematical system here, When you apply a function to an argument expression, you replaceĪll instances of name within the function's body with the Go over some basic but crucial functions. There, let's first look at how function application works. By the end of this article you'll see howĬonditions and boolean operations can be represented in terms ofįunctions and functions that operate on functions. Theoretical underpinnings of functional programming.īut it gets way, way cooler. This is interesting in itself as it gives you a glimpse of the Where you have first class functions and higher order functions. You can see how this is directly related to functional programming, That's awesome! Second,įunction bodies and function application arguments can be anyĮxpression at all, meaning that a) functions can take functions asĪrguments and b) functions can return functions. 'expressions' which can be any of the above. Really boils down to four elements: names, functions, application, and There are two super-cool things about this specification. first the body of a function can itself be a function λ fn. y equivalent to above we 'll get into that more λ first. ::= ::= ( ) ::= ::= Examples Names x joey queen - amidala Functions Note that functions always have one and only one parameter λ x. ::= | | ::= any sequence of non - blank characters ::= λ. If you don't know what something means, don't worryĪbout it at this point - this is just an overview and we'll dig into The λ calculus is all about manipulating λ expressions. Enough history! Tell Me About λ Expressions! Semantics, the very definition of mathematical elegance, yet it'sĬapable of representing all computable functions. The λ calculus had a lean syntax and dead-simple Indeed, by examining the λ calculus, you can see where Lisp derives Theory of gravity or the theory of evolution) by Perhaps most famous for serving as the basis of Lisp, invented (orĭiscovered, if you prefer to think of Lisp as being on par with the The λ calculus is one of the foundations of computer science. The Entscheidungsproblem inspired another computational model which I hope it's exciting for you, too! A Bit of HistoryĪs every aspiring greybeard knows, the λ calculus was invented by It was a very different way of thinking for me, and exciting

We'll then look at the surprising,Ĭounterintuitive way that the λ calculus lets us represent conditionalĮxpressions and boolean operations - all with functions as the only Using examples from that book, this article will walk you through theīasics of λ calculus. I've been having a brain-bending good time readingĪn Introduction to Functional Programming Through Lambda Calculus.

Lambda calculus boolean