This article is part of the Lambda on Lisp series.

Tags: lambda-calculus, lisp

Exploring the lambda calculus using Common Lisp

A Lisper's commentary on the “Fundamentals of Lambda Calculus & Functional Programming in Javascript”.


In this series we’ll explore the lambda calculus and implement a lambda calculus syntax on Common Lisp. Our emphasis will be on the lambda calculus, with just enough Lisp to give us an environment to play in.

I said on, rather than in, Common Lisp because I would like to start with a thin-as-possible abstraction layer that transforms lambda calculus syntax into Lisp functions and function calls. See Paul Graham’s excellent book, “On Lisp”, for more details.

The title is intended to stress the importance of bottom-up programming in Lisp. Instead of just writing your program in Lisp, you can write your own language on Lisp, and write your program in that.

Paul Graham, “On Lisp: Advanced Techniques for Common Lisp”, 1993

As with anything else, this approach has some upsides and some downsides.

One advantage is that our lambda calculus will have full access to Lisp functions and values, which will be quite useful while we implement a language that, in the beginning, won’t even have the concept of natural numbers.

Another advantage is that Lisp will have full access to our lambda calculus. We’ll see examples of this later.

One disadvantage stems from my choice of Common Lisp rather than, say, Scheme or Clojure. Common Lisp is the lisp I know and love, but it is a Lisp-2. Don’t worry if you don’t know what that means right now — later on we’ll see how calling a function returned by a function is just a little bit tricky. This would have been easier in a Lisp-1 like Scheme, but it’s not going to be too much of a problem.

One cute side-benefit of doing it this way is that, because our lambda calculus functions will be transformed into Lisp functions, they can be byte-compiled — so your idle playing around with a language that no-one actually uses will at least be at breakneck speed.

What?! But, why?

“What do you mean no-one actually uses it?” someone asks, quite indignant. Someone else wonders, “Then why should I bother learning it?”

OK. So it’s true — as far as I know, no-one uses actual lambda calculus in production code — in the same way that no-one builds actual Turing machines to solve real-world problems. And, yes, you can learn functional programming without knowing a lambda calculus; just as you can learn imperative programming without every hearing of a Turing machine; but please consider:

  1. Lambda calculus is beautiful.

  2. The Turing machine and the lambda calculus are two of the purest forms of expressing computation that we have. They give us ways of thinking about what programming languages do and, possibly more importantly, what programming languages are.

  3. Lambda calculus and Turing machines are two very different ways of thinking about computation. At the risk of over-simplifying, Turing machines are the epitome of imperative programming, with their state, and with the machine’s head stepping along the tape one block at a time (albeit in either direction) — while the lambda calculus, consisting only of functions, offers the base model of functional programming. Exploring these differences can teach us a lot.

  4. Lambda calculus and Turing machines are exactly the same. OK, not exactly the same, but equivalent. The Church-Turing hypothesis teaches us that they are, in a sense, isomorphic. They are two sides of the same coin, two faces of the same Batman villain. Exploring these similarities can teach us a lot.

  5. The idea that everything that can be computed, can be computed using something as simple as the lambda calculus is mind-blowing. We’ll take a look at syntax soon, but for now I’ll just mention that the entire syntax can be expressed in only three rules.

  6. Lambda calculus is beautiful.

Why does this article’s teaser mention JavaScript?

This series was inspired by a Fullstack Academy talk presented by Gabriel Lebec.

Before you run off and watch the video, be aware that the flow of this series will follow that of the talk; and that each major section will contain a timestamped link to the next part of the video so that you can work through Gabriel’s talk and this series in parallel, while avoiding spoilers. But — if you’d prefer to watch the whole video first, here ya go: Lambda Calculus - Fundamentals of Lambda Calculus & Functional Programming in JavaScript.

Part 1 of Gabriel’s talk gave me a clearer understanding of the lambda calculus than anything I had previously read or watched. I enjoyed it so much that partway through, I decided I needed to stop watching and play around with what he had already presented before I continued.

I figured that implementing a lambda calculus in Lisp would further my understanding of both the calculus *and* Lisp --- a language I love but am still learning. Please, pull up a chair and join me for the ride!