This blog post is a written up extension of work I did while reading SICP. You can look at the original file here. It is also an extension of the Simple Church Numerals post. I would also like to acknowledge “A Tutorial Introduction to the Lambda Calculus” and “Recursion in the Lambda Calculus” as extremely helpful resources.
I started out exploring lambda calculus with a simple question: how much functionality of a “standard” programming
language can I implement using only functions and a simple substition machine? Introductory computer science courses are
often filled with explanations of language keywords:
for loops and so on. But we’ve already seen in a
previous blog post as well as SICP that some of this
functionality is not really new; that it can be thought of as simple syntactic sugar for a series of lambda function
Thus my goal was to implement as much of the standard functionality of Racket I learned using only lambda calculus. Although most of the work was done via mathematical notation (on pen and paper), I will be showcasing the Racket implementation of that work.
The Rules of Lambda Calculus
Lambda calculus at it’s core is a very simple programming language. It consists of evaluating expressions, which are defined as follows:
Definition: An expression is either a name, a function, or an application.
Definition: A name, or variable is any identifier. For simplicity, we will use lowercase letters, such as a, or x.
Definition: A function consists of the following form: \(\lambda\) <\(name\)>.<\(expression\)>. The name before the period is the parameter of the function, and the expression after the period is the body. For example, \(\lambda x.x\) is the identity function.
Definition: An application consists of the following form: \(expression expression\). If the first expression is a function, we can substitute the parameter for the second expression in the function body. For example, \( (\lambda x.x)y \) would apply y to the identity function, resulting in y. This substitution is also known as a \(\beta\)-reduction.
In Racket we can express names as any symbol, functions as
(lambda (<name>) <expression>), and an application as
(<expression> <expression>). As you can see, lambda calculus itself is not much more than a simple substitution
It’s important to note that the actual symbol used for a name does not matter. That is, \(\lambda x.x\) and \(\lambda y.y\) both represent the same function (this is called \(\alpha\)-equivalence). This is important however, if you are substituting in an expression containing symbols already being used in the body. For example, consider the following substitution:
But this is incorrect since the y substituted in is different from the y inside the function body. If we had used a different variable name, such as t we would have gotten a different result, like so:
which is actually correct. It is important to avoid name collisions by renaming variables when necessary while doing substitution to get the correct answer.
I’ve glossed over some of these details rather quickly. If you would like a more formal definition of \(\alpha\)-equivalence, \(\beta\)-reduction, and the importance between bound and free variables, I suggest reading one of the resources cited at the beginning of the article.
Another important note is that there are no named functions. All functions are anonymous in lambda calculus: to call a
function, we write out the body and call it’s parameters. For the sake of convenience we will allow our functions to
have aliases by using Racket’s
Finally, one small limitation you might see is that our functions currently only take one variable. It is relatively simple to see that a multi-parameter function can be represented by nested uni-parameter functions. For example, if we wanted some function that took 3 parameters x, y, and z, then we could represent that as a function with parameter x that returned a function that takes in a parameter y, which then returns a function that takes in some parameter z, which then runs the function body with the 3 parameters.
In other words, in lambda calculus:
Or, in Racket:
(lambda (x y z) <body>) <=> (lambda (x) (lambda (y) (lambda (z) <body>)))
This process of transforming a multi-parameter function is also called currying. I won’t formally go through it but hopefully it is relatively clear that multi-parameter functions can easily be represented in our system. I will use curried and non-curried functions throughout this post based on what is more convenient and clear.
Building the Language
Now that we understand the basics of lambda calculus, let’s start seeing what it can do. Our goal is to utilize only lambdas and the simple substitution model of Racket to build as many language features as we can.
1. Church Numerals
We’ve already seen a system capable of representing numbers. For a full explanation, please see my previous blog post, as I will be glossing over most of the explanation here.
Let us first define zero, as well as a successor function:
(define zero (lambda (f) (lambda (x) x))) (define suc (lambda (n) (lambda (f) (lambda (x) (f ((n f) x))))))
We can then represent the set of positive integers. For example:
(define one (suc zero)) (define two (suc one)) (define three (suc two)) (define four (suc three)) (define five (suc four))
Noting that a Church Numeral n is a function that applies it’s first argument n times on it’s second argument, we can also form the functions addition and multiplication:
(define add (lambda (a b) ((a suc) b))) (define mult (lambda (a b) (lambda (f) (a (b f)))))
Unfortunately, since Racket just prints
#<procedure> instead of an actual expression, we’ll need more
testing methods to actually see the value of our Church numeral:
; We'll test the Church numeral n by applying the increment function to 0 n times. (define (inc n) (+ 1 x)) (define (test-numeral n) ((n inc) 0)) (test-numeral (add four three)) ; => 7 (test-numeral (mult four three)) ; => 12
Every language needs some form of if statement. Let’s start with booleans:
; We'll represent booleans as two parameter functions. ; True returns the first argument, false returns the second. (define TRUE (lambda (x) (lambda (y) x))) (define FALSE (lambda (x) (lambda (y) y)))
Because of our convinient definition, an if statement is just the application of a boolean to the two branches:
; If pred is TRUE, then we will return the truth expression. ; If pred is FALSE, then we will return the otherwise expression. (define IF (lambda (pred truth otherwise) ((pred truth) otherwise)))
It would be nice to have some basic boolean operations, like
; AND is a function that takes two booleans x and y. ; If x is TRUE, it will return y (thus returning TRUE if y is also TRUE, and ; FALSE if y is FALSE). Otherwise, x is FALSE, so we should return the function ; FALSE. (define AND (lambda (x y) ((x y) FALSE))) ; Similarly, these (define OR (lambda (x y) ((x TRUE) y))) (define NOT (lambda (x) ((x FALSE) TRUE)))
Again, we’ll need more testing methods to make sure our new boolean logic works:
(define test-bool (lambda (b) (IF b "TRUE" "FALSE"))) (IF (OR FALSE (AND TRUE (NOT FALSE))) "recurse" "center") ; => "recurse"
Let’s try connecting the two pieces we’ve built and write a function that tests if a numeral is zero:
(define isZero? (lambda (n) (((n FALSE) NOT) FALSE)))
Or in lambda calculus (where F is false, and T is true):
Let us see how this function works. Say we substitute in 0:
Remember that 0 is a function that applies it’s first argument 0 times to it’s second argument (thus just returning the second argument):
Thus returning true. Now let’s say we substitute any other number, say 2:
2 then applies F to \neg twice:
It is important to understand the order of function application, which I have emphasised with brackets above. Now since false is a function that always returns it’s second argument and ignores the first, false applied to one argument will always result in the identity function. More clearly, given \(F = \lambda xy.y\), then:
Thus, F applied to a single argument will always result in the identity function. Hence,
It is easy to generalize this and see that F applied any number of times to a single argument always leads to the identity function. Thus isZero always returns false for non-zero numbers.
Lists are a pretty fundamental structure in Lisp, so we should definitely try to implement them. First, we’ll implement the constructor for a pair:
(define pair ; like cons (lambda (a b) (lambda (z) ((z a) b))))
Note that a pair is a function that takes some function and applies it to the two elements in the pair. If we want to get the first element of the pair, we just need to pass in a function that takes two arguments and returns the first one, and similarly for the second element. Conviniently, this is the exact behaviour of our booleans true and false:
(define first ; like car (lambda (p) (p TRUE))) ; select the first arg (define rest ; like cdr (lambda (p) (p FALSE))) ; select the second arg
Now with our pair structure, we can write a function that increments a pair of consecutive numbers:
; increments a pair of number (n, n-1) to (n+1, n) (define inc-pair (lambda (p) ; p is our pair (pair (suc (first p)) (first p))))
We can then use this function to keep track of the previous number when incrementing through numbers, thus allowing us to write the predecessor function:
; predecessor function ; note one interesting property is that (pre zero) is zero. (define pre (lambda (n) (rest ((n inc-pair) ; apply inc-pair n times (pair zero zero))))) ; to (0, 0)
With the predecessor function, we can finally implement subtraction by applying
pre multiple times, and thus numerical
comparison. Note that we still cannot represent negative numbers. Since
(pre zero) is
zero, any subtraction that would
result in a negative simply evaluates to zero:
; greater than or equal (define GTE (lambda (x y) (isZero? ; is x - y zero? ((x pre) y)))) ; similarly... (define LTE (lambda (x y) (isZero? ((y pre) x)))) (define EQ (lambda (x y) (AND (GT x y) (LT x y)))) (define GT (lambda (x y) (AND (GTE x y) (NOT (EQ x y))))) (define LT (lambda (x y) (AND (LTE x y) (NOT (EQ x y))))) (define NEQ (lambda (x y) (NOT (EQ x y))))
One important thing we can’t do yet is recursion. Remember that all functions are anonymous; we cannot give them names. I have only allowed definitions in here for the sake of convinence but a function is not allowed to refer to itself in it’s own body.
Recursion however is still possible! Let us consider the following function:
(define Y (lambda (y) ((lambda (x) (y (x x))) (lambda (x) (y (x x))))))
Let’s just say I apply some argument R to Y. What does the substitution machine do?
(Y R) => ((lambda (x) (R (x x)) (lambda (x) (R (x x))
Substituting the second
(lambda (x) (R x x)) as
x in the first half, we get:
=> (R ((lambda (x) (R (x x)) (lambda (x) (R (x x)))) => (R (Y R)) ; thus, since we've shown (Y R) evaluates to (R (Y R)), it follows that ... => (R (R (Y R))) => (R (R (R (Y R)))) ; and e.t.c.
Let’s try this function out. We will write a function that sums all the positive numbers up to n as follows:
(define sum-to-n (lambda (r) ; r is the 'name' of our recursive function (lambda (n) ; n is the parameter we want our function to take in (IF (isZero? n) zero (add n (r (pre n)))))))
Then consider what happens when we evaluate:
((Y sum-to-n) three) ; since (Y R) becomes (R (Y R)), then => ((sum-to-n (Y sum-to-n)) three) ; substituting our two arguments (Y sum-to-n) and three into sum-to-n... => (IF (isZero? three) zero (add three ((Y sum-to-n) (pre three)))) => (add three ((Y sum-to-n) (pre three))) => (add three ((Y sum-to-n) two)) ; and the whole cycle begins again until we end up with ... => (add three (add two (add one ((Y sum-to-n) zero)))) => (add three (add two (add one (IF (isZero? zero) zero (add zero ((Y sum-to-n) (pre zero))))))) ; take the base case => (add three (add two (add one zero))) ... => six ; which is correct!
Thanks to our helper function Y, we have recursion! In fact, Y is so famous it has its own name: it’s called the Y-combinator, and was created by Haskell Curry.
Unfortunately, if any of you actually were to put this in a Racket interpreter and run it, you’ll find that it doesn’t actually work! In fact, the function will just evaluate forever, never terminating. This is because in our informal description, we actually glossed over a few crucial details.
The first is that we never discussed what order to evaluate things in: should we try to simplify function
arguments first, or substitute arguments into function bodies first? This difference is sometimes called
normal-ordered evaluation versus applicative-ordered evaluation, or strict vs lazy evaluation.
In the above examples, I’ve chosen the optimal ordering in order to give the proper answer, but Racket
defaults to strict ordering, always attempting to substitute
(Y R). This results in
(Y R) => (R (Y R)) => (R (R (Y R))) => (R (R (R (Y R)))) ... forever instead.
The second detail is that our implementation of
IF evaluates both branches; in which case it will
infinitely recurse on the second branch, even if the base case evaluates to true. Most programming languages
will only evaluate code in the branch that gets executed.
To get around this, we are going to tweak our definition of the Y-combinator to work with strict-order evaluation. Consider the following new definition of the Y-combinator:
(define Y (lambda (t) ((lambda (y) (t (lambda (x) ((y y) x)))) (lambda (y) (t (lambda (x) ((y y) x)))))))
This works similarly, as so:
(Y R) => ((lambda (y) (R (lambda (x) ((y y) x)))) (lambda (y) (R (lambda (x) ((y y) x))))) => (R (lambda (x) (((lambda (y) (R (lambda (x) ((y y) x)))) (lambda (y) (R (lambda (x) ((y y) x))))) x))) => (R (lambda (x) ((R (lambda (x) (((lambda (y) (R (lambda (x) ((y y) x)))) (lambda (y) (R (lambda (x) ((y y) x))))) x))) x))) => (R (lambda (x) ((R (lambda (x) ((R (lambda (x) (((lambda (y) (R (lambda (x) ((y y x))))) (lambda (y) (R (lambda (x) ((y y x)))))) x))) x))) x))) ... => (R (lambda (x) ((R (lambda (x) ((R (lambda (x) ... ((R (lambda (x) (((lambda (y) (R (lambda (x) ((y y) x)))) (lambda (y) (R (lambda (x) ((y y) x))))) x))) ... x))) x))) x)))
However, this version works with strict order evaluation. To solve our other problem, we’ll have to use Racket’s built in
For convenience, we can rewrite
isZero to return native boolean values:
(define Z (lambda (n) (((isZero? n) #t) #f)))
And thus write our sum function as follows:
(define sum-to-n (lambda (r) (lambda (n) (if (Z n) zero (add n (r (pre n)))))))
We can also write a factorial function just for fun:
(define factorial (lambda (fact) (lambda (n) (if (Z n) one (mult n (fact (pre n)))))))
And with that we are able to implement recursion!
Even though building functionality already available in Racket using anonymous functions is not a particularly practical exercise, I still find it interesting and fun. The Church-Turing thesis implies that lambda calculus is capable of computing anything a Turing machine can, and so it should be possible to implement any language feature using lambda calculus. Although this post is more informal and exploratory than anything, I hope it is a good introduction to the world of lambda calculus.