lambdacalculus on The Neo-Babbage Files

Lambda Calculus and Lisp, part 2 (recursion excursion)

Benjamin.Slade@fakeEmailToMakeValidatorHappy.com (Benjamin Slade) — Thu, 20 Feb 2025 01:26:00 -0600

From the previous entry in this series, one of the things of note in discussing the nature of the connections between LISP and (the) lambda calculus was John McCarthy’s concern about recursion and higher-order functions.

A couple of excerpts from previous quotes from McCarthy on the subject to set the stage:

…And so, the way in which to [be able to handle function passing/higher order functions] was to borrow from Church’s Lambda Calculus, to borrow the lambda definition. Now, having borrowed this notation, one the myths concerning LISP that people think up or invent for themselves becomes apparent, and that is that LISP is somehow a realization of the lambda calculus, or that was the intention. The truth is that I didn’t understand the lambda calculus, really. In particular, I didn’t understand that you really could do conditional expressions in recursion in some sense in the pure lambda calculus.…

[McCarthy 1978a:190¹]

…Writing eval required inventing a notation representing LISP functions as LISP data, and such a notation was devised for the purposes of the paper with no thought that it would be used to express LISP programs in practice. Logical completeness required that the notation used to express functions used as functional arguments be extended to provide for recursive functions, and the LABEL notation was invented by Nathaniel Rochester for that purpose. D.M.R. Park pointed out that LABEL was logically unnecessary since the result could be achieved using only LAMBDA — by a construction analogous to Church’s Y-operator, albeit in a more complicated way.…

[McCarthy 1978a:179¹]

Examining Church’s Y Combinator will be something we return to (probably in a number of posts), but I’ll defer discussion of it for the moment.

For now, let’s consider recursion in lisps. We’ll be talking a lot of recursion in Emacs Lisp today in fact.

Self-reference, self-embedding

Recursion is a concept or process depends on a simpler or previous version of itself. It’s ubiquitous, including in the natural world: Wikipedia notes that “Shapes that seem to have been created by recursive processes sometimes appear in plants and animals, such as in branching structures in which one large part branches out into two or more similar smaller parts. One example is Romanesco broccoli.”

Figure 1: Romanesco broccoli (Brassica oleracea), from Wikipedia

Recursion is a thing I deal with a lot in my day job, as it is a feature of natural language, especially syntax and semantics. To provide a quick illustration — though this is not at all how modern generative syntax is done anymore — consider phrase structure grammar and phrase structure rules used by Noam Chomsky and his colleagues in the 1950s.

Sentences (and linguistics objects generally) have formal structure, and it is part of the productive/creative nature of language that we might envision this structure as involving abstract structure rules that can be expanded in different ways (and then have vocabulary filled in).

A phrase structure rule will generally have the form A → [B C], indicating that A may be rewritten or expanded into [B C]. (In the following, I’ll use round brackets ()'s to denote optional constituents.)

So, a subset of these rules for English might include something like (note that some things have multiple rules that can apply to them):

1. S → NP VP
2. NP → (Det) N
3. N → (AdjP)* N
4. VP → V
5. VP → V NP (NP)
6. VP → V CP
7. CP → (Comp) S
...

Code Snippet 1: a snippet of phrase structure grammar rules for English [Nb.: again, not prolog, but maybe the best fontlocking choice here]

(Where S is “sentence”; Det is a determiner (like “the”, “a”); NP is a noun phrase; Nₙ is a noun head; AdjP is an adjective phrase; VP is a verb phrase; V is a verb head; CP is a complementiser phrase; Comp is a complementiser (like “that”).)

So we can rewrite S as NP VP (1) and then rewrite NP VP as Det N VP (2, choosing an optional Det) and then Det N VP as Det N V (4) and then insert lexical items of the appropriate category into the ‘heads’ (the non-P elements). So we might choose “the” for Det and “cat” for N and “purrs” for V, and get the sentence “the cat purrs”.

But note that some of the rules allow for expansion into elements that contain expansions back into themselves. So rule (1) allows an S to exapnd into NP VP and rule (6) allows for a VP to expand into V CP and rule (7) allows for a CP to expand into an S. At which point we can apply rule (1) again to expand the new S into NP VP, and then repeat this process as many times as we like:

S
NP VP
N VP
N V CP
N V Comp S
N V Comp NP VP
N V Comp N VP
N V Comp N V CP
N V Comp N V Comp S
N V Comp N V Comp NP VP
N V Comp N V Comp N VP
N V Comp N V Comp N V CP
N V Comp N V Comp N V Comp S
N V Comp N V Comp N V Comp NP VP
N V Comp N V Comp N V Comp N VP
N V Comp N V Comp N V Comp N V CP
N V Comp N V Comp N V Comp N V Comp S
N V Comp N V Comp N V Comp N V Comp NP VP
...

Code Snippet 2: a recursive English sentence expansion [Nb.: again, not prolog, but maybe the best fontlocking choice here]

And it’s easy to imagine examples of what such a sentence could be like, e.g., “John said that Sita said that Bill said that Mary said that Ram said that Kim said that…". It won’t be infinitely long, but there’s no particular theoretical bound on how long it could be (memory processing and finite human lifespans will impose practical limits, of course).

On the formal semantics side, things are similar: consider that logical languages too (which are often used to formalise natural language semantics) allow for recursion. Propositional logic construction with Boolean operators too have no theoretical upper limits: we can write (t ↔ (p ∧ (q ∨ (r → (s ∧ ¬¬¬¬¬¬t))))),² and there’s nothing which prevents composing this bit of formalism with yet another bit, and so on.

And in mathematics and computer science, recursion is often a thing which suggests itself.

For instance, though there are other ways of calculating it, the Fibonacci sequence³, i.e., a sequence of numbers in which each element is the sum of the two elements that precede it (e.g. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…).

A natural way of writing an equation to calculate these is something like:

Fib(0) = 0 as base case 1.

Fib(1) = 1 as base case 2.

For all integers n > 1, Fib(n) = Fib(n − 1) + Fib(n − 2).

Rule 3 makes reference to itself. I.e., in order (by this method) to calculate Fib(6), you have to calculate Fib(5), for which you have to calculate Fib(4)), for which you have to calculate Fib(3), for which you have to calculate Fib(2), which you can then base on rules (1) & (2): you can add 0 and 1 (= Fib(0) and Fib(1)) together to get Fib(2), and then you can calculate Fib(3) by adding Fib(1) and Fib(2) and so on.

Cursed recursion

In early Lisp(s), despite the concern around recursion, writing recursive functions was/is not always pragmatically viable, because it can lead to stack overflows (potential infinities are hard in practice).

Scheme, a Lisp dialect originally created at MIT by Guy L. Steele, Jr. and Gerald Jay Sussman, was the first Lisp to implement tail call optimisation, which is a way of making significantly recursive functions viable, making tail calls similar in memory requirement to their equivalent loops.

Before talking (a little bit more) about tail recursion, let’s look at concrete examples of a tail recursive function and its loop equivalent. We’ve mentioned Fibonacci numbers already, so let’s try to write functions calculate these (in Emacs Lisp).

Following the mathematical abstraction for the Fibonacci sequence above, we could write a function like this:

;; -*- lexical-binding: t; -*-
(defun fib1 (n a b)
  "Calculate the first `n' Fibonacci numbers, recursively."
  (if (< n 1)
      a
    (cons a
          (fib1 (1- n) b (+ a b)))))

Code Snippet 3: a first go at a recursive elisp function for fibonacci numbers

I’ve tried to make this as simple as possible, we could make it nicer (say, with a wrapper function or some some of flet) so that we didn’t have to pass in the initial values. But, keeping it simple (for now): fib1 is a function taking three arguments: n, the quantity of Fibonacci numbers to return; a, the first number to start with; and b, the second number to start with.

Following the schema above, we’re going to pass 0 for a and 1 for b. Let’s get the first ten numbers of the sequence, and pass 10 for n:

(fib1 10 0 1)  ; (0 1 1 2 3 5 8 13 21 34 . 55)

I know, the output is a bit ugly because we’re just cons’ing the results and so it’s not a proper list, but we’re keeping things simple.

Let’s walk through how it works. The function first looks at n, if n is less than 1, it returns a (whatever it is). If n isn’t less than 1, we return the cons of a with the result of calling fib1 itself on “n minus 1” b and “a plus b”.

So if we start with n=3, a=0, b=1, that is, evaluate (fib1 3 0 1), the function would say, well, 3 isn’t less than 1, so I’m going to create a cons with 0 and the result of calling (fib1 2 1 1) (2 = “n minus 1”, because n is currently 3; 1 because b=1; and 1 because a + b = 0 + 1 = 1).

So at this point we have a cons that looks like this:

(0 . (fib1 2 1 1))  ;; [= (cons 0 (fib1 2 1 1))]

When we evaluate (fib1 2 1 1), n is still not less than 1, so we’re going to cons the current value of a (which is 1) with another call to fib1: (fib1 1 1 2) (1 = “n minus 1”, because n is currently 2; 1 because b=1; and 2 because a + b = 1 + 1 = 2).

So now we have:

(0 1 . (fib1 1 1 2))  ;; [= (cons 0 (cons 1 (fib1 1 1 2)))]

Now we have to evaluate (fib1 1 1 2). n is still not less than 1; so we create another cons of 1 (as a is currently 1) with yet another call of fib1: (fib1 0 2 3) (0 = “n minus 1”, because n is currently 1; 2 because b=2; and 3 because a + b = 1 + 2 = 3). And so now:

(0 1 1 . (fib1 0 2 3))  ;; [= (cons 0 (cons 1 (cons 1 (fib1 0 2 3))))]

And, finally, evaluating (fib1 0 2 3), now n is less than one, so we take the first branch of the conditional and just return a, which is 2. So the result of starting with (fib1 3 0 1) is:

(0 1 1 . 2)  ;; [= (cons 0 (cons 1 (cons 1 2)))]

And you can try this with other values of n, e.g., try evaluating (fib1 100 0 1) to get the first 100 members of the sequence.⁴

But, at least for me on Emacs 30.0.93, 529 is the limit. If we try (fib1 520 0 1), the debugger pops up with a 1622 line long error, which begins:

Debugger entered--Lisp error: (excessive-lisp-nesting 1622)
  cl-print--cons-tail(excessive-lisp-nesting (1601) #<buffer  *temp*>)
  #f(compiled-function (object stream) #<bytecode 0x491d55561699a09>)(#0 #<buffer  *temp*>)
  apply(#f(compiled-function (object stream) #<bytecode 0x491d55561699a09>) (#0 #<buffer  *temp*>))
  #f(compiled-function (&rest args) #<bytecode 0x1c31d892b8046a8b>)()
  #f(compiled-function (cl--cnm object stream) #<bytecode 0x7c361f66f109692>)(#f(compiled-function (&rest args) #<bytecode 0x1c31d892b8046a8b>) #0 #<buffer  *temp*>)
  apply(#f(compiled-function (cl--cnm object stream) #<bytecode 0x7c361f66f109692>) #f(compiled-function (&rest args) #<bytecode 0x1c31d892b8046a8b>) (#0 #<buffer  *temp*>))
  #f(compiled-function (object stream) #<bytecode 0x1f277a9a6dc403fa>)(#0 #<buffer  *temp*>)
  apply(#f(compiled-function (object stream) #<bytecode 0x1f277a9a6dc403fa>) #0 #<buffer  *temp*>)
  cl-print-object(#0 #<buffer  *temp*>)
  cl-prin1(#0 #<buffer  *temp*>)
  backtrace--print(#0 #<buffer  *temp*>)
  cl-print-to-string-with-limit(backtrace--print #0 5000)
  backtrace--print-to-string(#0 nil)
  backtrace-print-to-string(#0)
  debugger--insert-header((error #0 :backtrace-base eval-expression--debug))
  #f(compiled-function () #<bytecode 0x299e53d67d1ac62>)()
  backtrace-print()
  debugger-setup-buffer((error #0 :backtrace-base eval-expression--debug))
  debug(error #0 :backtrace-base eval-expression--debug)
  eval-expression--debug(#0)
  (if (< n 1) a (cons a (fib1 (1- n) b (+ a b))))
  fib1(0 259396630450514843945535792456880074043523940756078363514486570322782139633750401577338505233670220572153381665 419712564636128966418863068957011388899128076671547993021605479585858227224221424221791102364954108601240491394)
  (cons a (fib1 (1- n) b (+ a b)))
  (if (< n 1) a (cons a (fib1 (1- n) b (+ a b))))
  fib1(1 160315934185614122473327276500131314855604135915469629507118909263076087590471022644452597131283888029087109729 259396630450514843945535792456880074043523940756078363514486570322782139633750401577338505233670220572153381665)
  (cons a (fib1 (1- n) b (+ a b)))
  (if (< n 1) a (cons a (fib1 (1- n) b (+ a b))))
  fib1(2 99080696264900721472208515956748759187919804840608734007367661059706052043279378932885908102386332543066271936 160315934185614122473327276500131314855604135915469629507118909263076087590471022644452597131283888029087109729)
....

Code Snippet 4: beginning of excessive-lisp-nesting error for our fib1 function

Because we’ve built up a long, deeply-embedded list of conses and Emacs has a limit of how deep it’s willing/able to go (you can see above that we’ve almost made it to the end, just needing to calculate fib1(0), when Emacs decides it’s had enough).

In Scheme and elsewhere, self-recursive calls at the ends (“tails”) of functions can be optimised to avoid these sorts of stack overflows (excessive-lisp-nesting).⁵ Tail-call optimisation lets procedure calls in tail positions be treated a specialised GOTO statements, which can be efficiently processed:

…only in cases where structures are explicitly declared to be dynamically referenced should the compiler be forced to leave them on the stack in an otherwise tail-recursive situation. In general, procedure calls may be usefully thought of as GOTO statements which also pass parameters, and can be uniformly encoded as JUMP instructions. This is a simple, universal technique, to be contrasted with […] more powerful recursion-removal techniques…

[Steele 1977:155⁶]

But our fib1 function doesn’t do this, and we end up flooded the stack with too many conses.

Back to loops

Recursive functions are perhaps most idiomatic in Scheme (among lisps, I mean). Some implementations of Common Lisp can do tail-call optimisation, but loops are perhaps more common, and certainly in Emacs Lisp (for reasons you can see above), loops are usually what are used. And so we can write a new Fibonacci function with a loop. It won’t be nearly as pretty, but it’ll work better.

Here’s one possible implementation:

;; -*- lexical-binding: t; -*-
(defun fib2 (n a b)
  "Calculate the first `n' Fibonacci numbers,
in a loop."
  (let ((result nil))
    (dotimes (c n)
      (setq result (cons a result))
      (setq tempa b)
      (setq b (+ a b))
      (setq a tempa))
    (nreverse result)))

(fib2 10 0 1) ; (0 1 1 2 3 5 8 13 21 34)

(setq bigfib-50000 (fib2 50000 0 1)) ; this will work - we can get 50,000 numbers

Code Snippet 5: a second go at a fibonacci function, with looping

The result is prettier at least: a proper rather than an improper list (because we started by cons'ing onto an empty list). Our fib2 function itself isn’t as mathematically pleasing as our fib1 function, we end up with a lot of setq's (the nreverse at the end reverses our list, because the way we build up our list is by cons'ing the first results first, so they end up at the end until we flip them with nreverse). But it works well. If you try to (fib2 100000 0 1), it’ll fail, but not because of stack overflow, just because we end up with numbers that are too big for Emacs. But you can certain get the over 50,000 members of the Fibonacci sequence, which is much better than fib1's limit of 529.

And dotimes is just one loop procedure available. (See cl-loop for a more powerful one.)

Optimal Tail with Emacs

Ok, so, practically, we should probably generally prefer loops over tail-recursive functions in Emacs. But, what if we just like the latter more?⁷ Are there any other possibilities?

Wilfred Hughes has an emacs package tco.el which implements a special macro for writing tail-recursive functions.⁸ It works by replacing each self-call with a thunk, and wrapping the function body in a loop that repeatedly evaluates the thunk. Thus a function foo defined with the defun-tco macro:

(defun-tco foo (...)
  (...)
  (foo (...)))

would be re-written as:

(defun foo (...)
   (flet (foo-thunk (...)
               (...)
               (lambda () (foo-thunk (...))))
     (let ((result (apply foo-thunk (...))))
       (while (functionp result)
         (setq result (funcall result)))
       result)))

And this delays evaluation in such a way as to avoid stack overflows. Unfortunately, at least currently for me (Emacs 30.0.93 again), tco.el seems to have some issues.

In Emacs 28.1, cl-labels (one of the ways of sort of doing let's for functions) gained some limited tail-call optimisation (as did named-let, which uses cl-labels).

;; -*- lexical-binding: t; -*-
(defun fib3 (n)
  "Calculate the first `n' Fibonacci numbers,
recursively, with limited tail-call optimisation
through `cl-labels'?!"
  (cl-labels ((fib* (n a b)
                (if (< n 1)
                    a
                  (cons a
                        (fib* (1- n)
                              b
                              (+ a b))))))
    (fib* n 0 1)))

(setq bigfib3 (fib3 397)) ; 396 highest that works

Code Snippet 6: a third go at a fibonacci function, with cl-labels

At least the way I’ve written it, it seems to suffer an overflow even sooner (at 397 rather than 529 as for our fib1), because it has to come back and do the cons after the tail-call — so fib* isn’t actually in the tail. (We can fix this in at least one way, which we’ll do in fib5.)

We could try to write an accumulator as a hack, where we try to do ours conses one at a time and pass along the results, but this fares no better than our fib1:

;; -*- lexical-binding: t; -*-
(defun fib4 (n a b accum)
  "Calculate the first `n' Fibonacci numbers, recursively,
but collect conses as we go and keep track of the length of
the `accum' cp. against `n'."
  (let* ((accum (cons a accum))
         (accum-lng (length accum)))
    (if (< n accum-lng)
        (nreverse accum)
      (fib4 n b (+ b a) accum))))

(setq bigfib4-529 (fib4 529 0 1 nil)) ; last good
(setq bigfib4-530 (fib4 530 0 1 nil)) ; overflows

Code Snippet 7: a fourth go at a fibonacci function, with an accumulator

If we combine cl-labels and the accumulator trick, however, we do seem to be able to escape stack overflows, because now we’ve got fib* properly in the tail:

;; -*- lexical-binding: t; -*-
(defun fib5 (n)
  "Calculate the first `n' Fibonacci numbers, recursively,
using both cl-labels and the accumulator trick."
  (cl-labels ((fib* (a b accum)
                   (let* ((accum (cons a accum))
                         (accum-lng (length accum)))
                     (if (< n accum-lng)
                         (nreverse accum)
                         (fib* b (+ b a) accum)))))
            (fib* 0 1 nil)))

(setq bigfib5-10000 (fib5 10000)) ; ok
(setq bigfib5-50000 (fib5 50000)) ; very slow, but ok

Code Snippet 8: a fifth go at a fibonacci function, with cl-labels and an accumulator

Now we’re back in the realms of what our fib2 non-recursive loop-style function could do. Although (setq bigfib5-50000 (fib5 50000)) calculates very slowly (worse than our looping fib2), so that’s not ideal.

Stream of Conses-ness

But here’s another possibility: Nicholas Pettton‘s stream package for emacs, where “streams” are delayed evaluations of cons cells.

;; -*- lexical-binding: t; -*-
(defun fib6 (n)
  "Return a list of the first `n' Fibonacci numbers,
implemented as stream of (delayed evaluation) conses."
  (cl-labels ((fibonacci-populate (a b)
                (stream-cons a (fibonacci-populate b (+ a b)))))
    (let ((fibonacci-stream
           (fibonacci-populate 0 1))
          (fibs nil))
      (dotimes (c n)
        (setq fibs (cons (stream-pop fibonacci-stream) fibs)))
      (nreverse fibs))))

(setq fib6-10k (fib6 10000)) ; ok
(setq fib6-50k (fib6 50000)) ; little slow, but works
(setq fib6-100k (fib6 100000)) ; little slow & overflow error

Code Snippet 9: a sixth go at a fibonacci function, with delayed evaluation conses

This works well. (fib6 50000) still turns out to run a bit slower than our (fib2 50000), so loops are still probably more efficient, but streams are pretty interesting. They can can used to represent infinite sequences. So here, above, fibonacci-stream (set by (fibonacci-populate 0 1)) is actually an infinite stream of Fibonaccis numbers, but lazily evaluated, so we just get the next one each time we call stream-pop on our fibonacci-stream local variable. (What happens is that stream-pop takes the car of fibonacci-stream, evaluates and returns it, and then sets fibonacci-stream to be its cdr (i.e., popping off and “discarding” the first element; which which captured in our fibs collector.))

Cascades of Fibonacci numbers

Oh, incidentally and irrelevantly, if you inspect the contents of your fib6-50k, it’s very aesthetically pleasing, a cascade of numbers:

Figure 2: fibonacci numbers burst forth from their seeds and spill out into the buffer

`excessive-lisp-nesting`: (overflow-error)

I had hoped to get to the Y Combinator today (and think I might have suggested a promise of that), for that’s where things really get interesting. And we need to get back to lambda calculus, of course. But we may be near the limits of excessive lisp nesting ourselves here.

However, the recursion discussion here has set the stage for the Y Combinator, which we’ve already talked a couple of times, especially in connection to John McCarthy’s claims about “not really understanding” lambda calculus and the fact that these really centre on his not seeing how one could get recursion without direct Self-reference (and thus the need for LABEL) because of not knowing about the Y Combinator.

Figure 3: a close-up of a section of Michał “phoe” Herda’s hand-illuminated Y Combinator Codex showing part of a Fibonacci defun

And, the Y Combinator ties in with all sorts of other curious things. Paradoxes, types, calligraphy.

[Thus, I ended up with an excursus on this excursus as the next post. I’ll put a link here to the proper third part when it’s up.]

McCarthy, John. 1978a. History of Lisp. In History of programming languages, ed. Richard L. Wexelblat, 173–185. New York: Association for Computing Machinery. https://dl.acm.org/doi/10.1145/800025.1198360 ↩︎
“t if and only if p and q or if r then s and not not not not not not t” ↩︎
A number of Indian philosophers, at least as far back as Virahāṅka (ca. AD 600–800), gave formulations for what is usually called the Fibonacci sequence. See Singh, P. (1985). The so-called fibonacci numbers in ancient and medieval India. Historia Mathematica, 12(3), 229–244. https://doi.org/10.1016/0315-0860(85)90021-7 [pdf] ↩︎
You might actually want to do something like (setq my-fib1-100 (fib1 100 0 1)) to put the result into a variable, because the echo area at the bottom of Emacs isn’t big enough for all of the numbers. (Oh, to eval things in an Emacs buffer, put your cursor/point at the end of the expression and press C-x C-e. But don’t do that for this one. If you want Emacs to just stick the results in directly into the buffer rather than echoing them, press C-u C-x C-e. But don’t do that here either, because Emacs will still end up printing an ellipsis because it thinks it’s too long.) And then press C-h v and type my-fib1-100 and enter to see the result. ↩︎
See further here, for instance, for more about tail calls and tail call optimisation. ↩︎
Steele, Guy L., Jr. 1977. Debunking the “expensive procedure call” myth or, procedure call implementations considered harmful or, LAMBDA: The Ultimate GOTO. ACM ‘77: Proceedings of the 1977 annual conference, 153–162. [https://dl.acm.org/doi/10.1145/800179.810196] ↩︎
If you’ve read any of Paul Graham’s Common Lisp books (e.g., On Lisp) or any of the Little Schemer books (the latter with Duane Bibby‘s lovely artwork in them), you may be disposed towards using recursion rather than loops.

Figure 4: close-up of Duane Bibby’s cover illustration for “The Little Schemer”

↩︎
One of my emacs packages, Equake, actually used to use tco.el (until commit #0ab0801 [2020-08-24 Mon]). ↩︎

Lambda Calculus and Lisp, part 1

Benjamin.Slade@fakeEmailToMakeValidatorHappy.com (Benjamin Slade) — Tue, 18 Feb 2025 03:05:00 -0600

The first of a series of envisioned blog posts on lambda calculus, and Lisp. It’s unclear exactly where to start: there is a whole heap of interesting issues, both theoretical and in terms of concrete implementations, which tangle and interconnect.

A particular application of lambda calculus is a very salient part of my “day job” as a formal semanticist of natural language. And my interests in Emacs and lisp(s) feel like they tie in here as well—though that’s a question in itself which is probably as good of a starting point into this (planned) series of posts as any.

There is much to explore: origins of John McCarthy‘s Lisp and Alonzo Church‘s lambda calculus; encodings of the simple made complex by restriction to a limited set of tools; recursion, fixed points, and paradoxes; infinities, philosophy, and engineering. But much of this requires stage setting.

And finding an exact entry point is yet tricky. But perhaps we start with λ: the divining rod, the wizard’s crooked staff, as it is the key component of much magic of a sort.

Lisp: LAMBDA the Ultimate?

We’ll start on the programming side, before turning to more philosophical or mathematical abstractions, with lambda.

It is now not only Lisps which contain lambda as a keyword, many/most programming languages have lambda as a keyword, usually for the introduction of an anonymous function. That is, an unnamed function, sometimes for one-off use.

But in McCarthy’s original formulation of LISP in 1958, LAMBDA was used as the basis for the implementation of functions generally:

Let f be an expression that stands for a function of two integer variables. It should make sense to write f (3, 4) and the value of this expression should be determined. The expression y^2 + x does not meet this requirement; y^2 + x(3, 4) is not a conventional notation, and if we attempted to define it we would be uncertain whether its value would turn out to be 13 or 19. Church calls an expression like y2 + x, a form. A form can be converted into a function if we can determine the correspondence between the variables occurring in the form and the ordered list of arguments of the desired function. This is accomplished by Church’s λ-notation. [p.6]

….

{λ[[x_1;…; x_n]; 𝓔]}∗ is (LAMBDA, (x∗_1,…, x∗_n), 𝓔∗). [p.16]

[McCarthy 1960:6, 16¹]

As in lambda calculus, LAMBDA binds variables, and replaces any occurrences of them in the scope of the operator with whatever it receives as arguments.

So the expression:

(LAMBDA (x y z) (+ (* y x) (* z x)))

if given the arguments 5, 2, 3, would replace x's with 5; y's with 2, and z's with 3:

(+ (* 2 5) (* 3 5))  ;; = (+ 10 15) = 25

The illusion of a blue-suffused platonic universe of `car`'s

The origin of lambda keywords in LISP, and the origin of LISP’s LAMBDA in lambda calculus has suggested the idea that LISP was something like an implementation of lambda calculus as a programming language, and the certain mysticism² attaching to both suggests perhaps a tighter surface association than there is direct evidence for.

Figure 1: xkcd 224 [see https://www.explainxkcd.com/wiki/index.php/224:_Lisp]

This is the topic of a 2019 blog post based on his talk for the Heart of Clojure conference in Daniel Szmulewicz expands on the theme “Lisp is not a realization of the Lambda Calculus”.³ One of the points Szmulewicz draws attention to is McCarthy’s own words:

…one of the myths concerning LISP that people think up or invent for themselves becomes apparent, and that is that LISP is somehow a realization of the lambda calculus, or that was the intention. The truth is that I didn’t understand the lambda calculus, really.

[McCarthy 1978b:190⁴]

In the paper version of the talk, McCarthy makes a similar point, but it’s worth looking at it in the larger context:

…how do you talk about the sum of the derivatives, and in programming it, there were clearly two kinds of programs that could be written.

One is where you have a sum of a fixed number of terms, like just two, where you regard a sum as a binary operation. And then you could write down the formula easy enough. But the other was where you have a sum of an indefinite number of terms, and you’d really like to make that work too. To make that work, what you want to be able to talk about is doing the same operation on all the elements of a lit. You want to be able to get a new list whose elements are obtained from the old list by just taking each element and doing a certain operation to it.

In order to describe that, one has to have a notation for functions. So one could write this function called mapcar. This says, “Apply the function f to all the elements of the list.” If the list is null then you’re going to get a NIL here. Otherwise you are going to apply the function to the first element of the list and put that onto a front of a list which is obtained by doing the same operation again to the rest of the list. So that’s mapcar. It wasn’t called mapcar then. It was called maplist, but maplist is something different, which I will describe in just a moment.

That was fine for that recursive definition of applying a function to everything on the list. No new ideas were required. But then, how do you write these functions?

And so, the way in which to do that was to borrow from Church’s Lambda Calculus, to borrow the lambda definition. Now, having borrowed this notation, one the myths concerning LISP that people think up or invent for themselves becomes apparent, and that is that LISP is somehow a realization of the lambda calculus, or that was the intention. The truth is that I didn’t understand the lambda calculus, really. In particular, I didn’t understand that you really could do conditional expressions in recursion in some sense in the pure lambda calculus. So, it wasn’t an attempt to make the lambda calculus practical, although if someone had started out with that intention, he might have ended up with something like LISP.

[McCarthy 1978a:189–190⁵]

The Discovery of LISP

Two bits from the end of this I want to highlight. The first, well, it’ll come up in future posts, and probably later in this post itself, and it has to do with recursion:

“I didn’t understand that you really could do conditional expressions in recursion in some sense in the pure lambda calculus”

And the second is that McCarthy hedges his “LISP as a realisation of the lambda calculus is myth” stance slightly:

“So, it wasn’t an attempt to make the lambda calculus practical, although if someone had started out with that intention, he might have ended up with something like LISP."

This does fit rather well with (and perhaps suggested) the framing that Paul Graham does of McCarthy as the “discoverer” of Lisp — like Euclid of geometry — rather than its inventor in his “The Roots of Lisp” paper:

In 1960, John McCarthy published a remarkable paper in which he did for programming something like what Euclid did for geometry. He showed how, given a handful of simple operators and a notation for functions, you can build a whole programming language. He called this language Lisp, for “List Processing,” because one of his key ideas was to use a simple data structure called a list for both code and data.

It’s worth understanding what McCarthy discovered, not just as a landmark in the history of computers, but as a model for what programming is tending to become in our own time. …. In this article I’m going to try to explain in the simplest possible terms what McCarthy discovered. The point is not just to learn about an interesting theoretical result someone figured out forty years ago, but to show where languages are heading. The unusual thing about Lisp—in fact, the defining quality of Lisp—is that it can be written in itself.

[Graham 2002:1⁶ (emphasis mine)]

Graham’s paper itself, as well as some of Graham’s other publications/postings (e.g., talking about Lisp as a sort of “secret weapon” in comparison to “blub languages”)⁷ is perhaps another contributor to the mystique of Lisp. (With the flip side of “Lisp as the Cleveriest Hacker’s Secret Weapon” enchanted coin being “the Curse of Lisp”⁸.)

There are other components of the history of LISP, which are suggestive of discovery rather than invention. McCarthy’s initial aim for LISP was more akin that of the Turing Machine: as a formal abstraction describing a mathematical model whose components were simple and few but yet was capable of performing any (and all) arbitrary computational operation:

One mathematical consideration that influenced LISP was to express programs as applicative expressions built up from variables and constants using functions. I considered it important to make these expressions obey the usual mathematical laws allowing replacement of expressions by expressions giving the same value. The motive was to allow proofs of properties of programs using ordinary mathematical methods. This is only possible to the extent that side effects can be avoided. Unfortunately, side effects are often a great convenience when computational efficiency is important, and “functions” with side effects are present in LISP. However, the so-called pure LISP is free of side effects, and Cartwright (1976) and Cartwright and McCarthy (1978) show how to represent pure LISP programs by sentences and schemata in first-order logic and prove their properties. This is an additional vindication of the striving for mathematical neatness, because it is now easier to prove that pure LISP programs meet their specifications than it is for any other programming language in extensive use. (Fans of other programming languages are challenged to write a program to concatenate lists and prove that the operation is associative.)

Another way to show that LISP was neater than Turing machines was to write a universal LISP function and show that is briefer and more comprehensible than the description of a universal Turing machine. This was the LISP function eval[e,a], which computers the value of a LISP expression e, the second argument a being a list of assignments of values to variables. (a is needed to make the recursion work.) Writing eval required inventing a notation representing LISP functions as LISP data, and such a notation was devised for the purposes of the paper with no thought that it would be used to express LISP programs in practice. Logical completeness required that the notation used to express functions used as functional arguments be extended to provide for recursive functions, and the LABEL notation was invented by Nathaniel Rochester for that purpose. D.M.R. Park pointed out that LABEL was logically unnecessary since the result could be achieved using only LAMBDA — by a construction analogous to Church’s Y-operator, albeit in a more complicated way.

S.R. Russell noticed that eval could serve as an interpreter for LISP, promptly hand coded it, and we now had a programming language with an interpreter.

[McCarthy 1978:179⁵]

(There’s a lot going on in this passage, and its context.

Notions of reasons to avoid side-effects (an ideal of “functional programming”, emphasised in Haskell and Clojure (another lisp)): thus the “functional” mode of lambda calculus rather than the “everything is state” mode of Turing machines and their infinitely long memory tapes.

Recursion again (which we’ll get to, repeatedly).

And Alonzo Church, the originator (discoverer?)^{see “X” below} of lambda calculus, who we’ll talk more about soon, and who was also Alan Turing’s PhD supervisor at Princeton.

But, first: let’s turn back to the topic of the concrete instantiation of Lisp on physical hardware by Stephen Russell.)

Figure 2: cover of Byte Magazine August 1979 [full mag here]

Elsewhere, McCarthy makes clear that he hadn’t thought at that point of LISP being instantiatable but as a theoretical exploration of computing at an abstract level. But, instead, the theoretical description translated fairly easily and directly to a runnable program, a LISP interpreter:

This EVAL was written and published in the paper and Steve Russell said, “look, why don’t I program this EVAL” … and I said to him, “ho, ho, you’re confusing theory with practice, this EVAL is intended for reading, not for computing”. But he went ahead and did it. That is, he compiled the EVAL in my paper into IBM 704 machine code, fixing bugs, and then advertised this as a Lisp interpreter, which it certainly was. So at that point Lisp had essentially the form that it has today…

[McCarthy 1974a:307⁹]

“No compute. Only read.” “Hell, EVAL that.”

This is the EVAL code “in the paper” referred to above:

eval[e; a] = [
   atom [e] → assoc [e; a];

   atom [car [e]] → [
        eq [car [e]; QUOTE] → cadr [e];
        eq [car [e]; ATOM] → atom [eval [cadr [e]; a]];
        eq [car [e]; EQ] → [eval [cadr [e]; a] = eval [caddr [e]; a]];
        eq [car [e]; COND] → evcon [cdr [e]; a];
        eq [car [e]; CAR] → car [eval [cadr [e]; a]];
        eq [car [e]; CDR] → cdr [eval [cadr [e]; a]];
        eq [car [e]; CONS] → cons [eval [cadr [e]; a]; eval [caddr [e]; a]];
        T → eval [cons [assoc [car [e]; a]; evlis [cdr [e]; a]]; a]];

   eq [caar [e]; LABEL] → eval [cons [caddar [e]; cdr [e]];
                                cons [list [cadar [e]; car [e]; a]]];

   eq [caar [e]; LAMBDA] → eval [caddar [e];
                                 append [pair [cadar [e];
                                               evlis [cdr [e]; a]; a]]]

evcon[c; a] = [eval[caar[c]; a] → eval[cadar[c]; a]; T → evcon[cdr[c]; a]]

evlis[m; a] = [null[m] → NIL; T → cons[eval[car[m]; a]; evlis[cdr[m]; a]]]

Code Snippet 1: McCarthy 1960, p.17 (see fn. [1]) [Nb.: not actually `prolog' but highlights better as]

Translated into slightly more familiar Lisp style (with added named-function-making label's), it is:¹⁰

(label eval
       (lambda (e a)
         (cond
           ((atom e) (assoc e a))
           ((atom (car e))
            (cond
              ((eq (car e) 'quote) (cadr e))
              ((eq (car e) 'atom)  (atom  (eval (cadr e) a)))
              ((eq (car e) 'eq)    (eq    (eval (cadr e) a)
                                          (eval (caddr e) a)))
              ((eq (car e) 'car)   (car   (eval (cadr e) a)))
              ((eq (car e) 'cons)  (cons  (eval (cadr e) a)
                                          (eval (caddr e) a)))
              ((eq (car e) 'cond)  (evcon (cdr e) a))
              ('t                  (eval (cons (assoc (car e) a)
                                               (cdr e))
                                         a))))
           ((eq (caar e) 'label)   (eval (cons (caddar e) (cdr e))
                                         (cons (list (cadar e) (car e)) a)))
           ((eq (caar e) 'lambda)  (eval (caddar e)
                                         (append (pair (cadar e)
                                                       (evlis (cdr e) a))
                                                 a))))))

(label evcon
       (lambda (c a)
         (cond ((eval (caar c) a)
                (eval (cadar c) a))
               ('t
                (evcon (cdr c) a)))))

(label evlis
       (lambda (m a)
         (cond ((null m) '())
               ('t (cons (eval  (car m) a)
                         (evlis (cdr m) a))))))

Code Snippet 2: EVAL in more recognisable lisp form

The above, given the few additional definitions for convenience immediately following, is a full LISP interpreter.¹¹

(label null
       (lambda (x)
         (eq x '())))

(label and
       (lambda (x y)
         (cond (x (cond (y 't) ('t '())))
               ('t '()))))

(label not
       (lambda (x)
         (cond (x '())
               ('t 't))))

(label append
       (lambda (x y)
         (cond ((null x) y)
               ('t (cons (car x)
                         (append (cdr x) y))))))

(label pair
       (lambda (x y)
         (cond ((and (null x) (null y)) '())
               ((and (not (atom x)) (not (atom y)))
                (cons (list (car x) (car y))
                      (pair (cdr x) (cdr y)))))))

(label assoc
       (lambda (x y)
         (cond ((eq (caar y) x) (cadar y))
               ('t (assoc x (cdr y))))))

Code Snippet 3: additional convenience functions for EVAL

The brevity of the code combined with the details of the story of the implementation: seemingly the theoretical code works without translation, suggesting a sort of natural discovery. But, the nature of even the theoretical, pre-implementation code did not exist in some Platonic heaven of mathematics, as can be seen by the nature of some of the operations, particularly car and cdr (often in modern Lisps, especially Scheme and Scheme-influenced Lisps, rendered instead more transparently as first and rest.)

The Non-Platonic mechanics of 1950s IBM mainframes

LISP was designed with the IBM 704-style architecture in mind, and car and cdr make some reference to particular details of the hardware, where the IBM 704 had “address” and “decrement” fields in memory index registers (≈locations in physical RAM), and car referenced the “address” field of the register (so CAR = “C-ontents of the A-ddress part of the R-egister”) and cdr the “decrement” field of the register (CDR = “C-ontents of the D-ecrement part of the R-egister”).¹²

Figure 3: my other car is a cdr

Lists in Lisp are singly-linked lists, with each node in the list having a “value” field and a “next” field, which points to the next node; these “value” and “next” fields correspond to the car and cdr operations. So, if we have a list like (a b c), the first node in the list has a “value” field of a and a “next” field pointing at another node, with this second node actually itself being — not b — but rather the list (b c). A proper list in Lisp is nil-terminated: that is, the last item in the list is actually nil (which is the empty list '()).

The operation cons above (for CONstructor) is a pair-forming operation (where the pairs correspond to the “value” and “next” fields), returning what are variously called (in different lisps) “conses” or “pairs”. Not all conses/pairs are lists (at least in most Lisps), since a proper list in Lisp is nil-terminated.

The operation (cons 'a 'b) will return a cons'ed object with essentially a single node, where the “value” field will be a and the “next” field will be b. Lisps will usually print such non-list conses (sometimes called “improper lists”) as “dotted lists”, i.e., (cons 'a 'b) will print out as (a . b).

The list (a b c) is actually the result of doing (cons a (cons b (cons c nil))) and would be printed in dotted-pair notation as (a . (b . (c . nil))) [which is equivalent to (a . (b . (c . ()))), since nil is the empty list].

Unsurprisingly, a lot of early/traditional Lisp programming involves manipulations of lists. But all modern Lisps implement other types of data structures as well, including vectors/arrays, hash tables, objects, and so on.

But, on the main topic of (eq 'lisp 'lambdacalculus), others have also pointed out the concrete hardware-connections of LISP from early days, telling against the “Lisp-as-pure-formal-~~invention~~-discovery” or “Lisp as (semi-)direct implementation of lambda calculus” notions:

Lisp was intended to be implemented on a computer from day 0. For their IBM 704. Actually Lisp is based on earlier programming experience. From 56 onwards John McCarthy implemented Lisp ideas in Fortran & FLPL. Then 58 the implementation of Lisp was started. 59 a first runnable version was there. 1960 there was the Lisp 1 implementation. The widely known paper on the Lisp implementation and recursive function theory was published in 1960. But his original prime motivation was not to have a notation for recursive function theory, it was to have a list processing programming language for their IBM 704 for AI research.

Lisp as designed by McCarthy was very different from lambda calculus.

[going on to point to the Stoyan (1984) Early LISP history (1956 - 1959) paper.]

— lispm‘s comment on r/lisp thread on this topic

There’s obviously much more to explore for the early history of Lisp and the nature of its connections to lambda calculus, but this much at least should give a general sense of the distinctions/divergences between Lisp and lambda calculus, while not ignoring important interconnections between them.

Reaching the `'()` of the line

And so while it’s tempting to delve off into other interesting features (homoiconicity!) of LISP/Lisps and their history and dialects (the Common Lisp of Endor; Schemes, Rackets, Chickens and Guile (oh my!), Clojure, Fennel and others), I’d wanted to get to lambda calculus proper much earlier in this piece already, and so we’ll set our (lispy) parens down for a moment, and trade in our LAMBDA for a λ.

Cattle-prodding functions: the lambda calculus

The aforementioned Alonzo Church¹³, a Princeton mathematician who supervised 31 doctoral students during his career, influencing others important researchers (including Haskell Curry [for whom the programming language Haskell is named; as well as the operating of currying]), developed (the) lambda calculus as part of his research into the foundations of mathematics.

Lambda calculus is Turing complete, thus equivalent in computation power to a Turing machine and a universal model of computation.

There are very few bits of machinery in basic untyped lambda calculus. (A reason for which it seemed to be attractive to McCarthy as a touchstone.)

Lambda calculus has variables and lambdas; function application; a reduction operation (which may follow function application); and a convenience variable-renaming operation.

More specifically, we have:

variables, like x, which are characters or strings representing “a parameter”.
lambda abstraction: essentially just the definition of a function, specifying its input (by a bound variable, say, λx) and returning an output (say, M). E.g., the expression λx.M will take an input, and replace any and all instances of x in the body M¹⁴ with whatever the input was. (The . separates the lambda and specification of bound variable (here x) (“input taker”) from the body (the “output”).)
function application: a representation like (M N), the function M applies to N, where both M and N are some sort of lambda terms.
β-reduction (beta reduction): bound variables inside body of the expression are replaced by inputs “taken” by the lambda expressions. The basic form: ((λx.M) N) → (M[x := N]). (That is, an expression (λx.M) combining with an expression N returns (M) where where all instances of x inside of M are replaced with N.)

(Setting aside rule 4 for the moment.)

For example, we might have an expression:

λf.λx.(f x)

This would be an expression which combines, one at a time, with two inputs, the lambdas operating from left to right, and then applying the f input to the x input.

To see this in full working order, we need to specific one of the two remaining operations (both reduction operations):

So, turning to rule 4 for β-reduction, taking our expression from above and providing it with inputs, and walking through the (two) application+β-reduction steps one at a time:

((λf.λx.(f x)) b a) =

((λx.(b x)) a) =

(b a)

That is, first, the leftmost lambda, λf, “takes” the leftmost argument b (in “taking” an argument, it “discharges” and disappears) and the (single) bound instance of the variable f in the body is replaced by b. Then, the same thing happens with the remaining lambda, λx, and the remaining argument, a. The result is a function where b applies to a. (Though since in this case there are no more lambdas, nothing more happens.)

Since (b a) looks somewhat unexciting/opaque, we can imagine a sort of hybrid proper untyped lambda calculus/Lisp hybrid language — let’s call this toy language of ours ΛΙΣΠ — and illustrate what things might look like there (assuming here that numbers are numbers and #'* is a lispy prefix multiplication function):

((λf.λx.λy.(f x y)) #'* 6 7) =

((λx.λy.(* x y)) 6 7) =

((λy.(* 6 y)) 7) =

(* 6 7) =

42

This isn’t how mathematics works in classical untyped lambda calculus — because we only have the 4 rules/entities enumerated above [plus a variable-clash reduction operation called α-reduction]¹⁵ and nothing else¹⁶: no integers, no stipulated mathematical operations, no car or cdr or cons or eq or anything — we can do all of these things in lambda calculus with the tools we have, and we’ll explore that in another post, but for now I just wanted to show you the toy ΛΙΣΠ language snippet as I find something that feels a bit more familiar and concrete can be helpful for understanding the notional unpinnings of what’s going on in lambda calculus.

Ok, so there’s no integers or cars, but what’s all this about cattle prods?

Well, why is λ / LAMBDA the “function”-making operator? Maybe just eeny-meeny-miney-moe amongst Greek letters, but at least at one point Church explained that [A. Church, 7 July 1964. Unpublished letter to Harald Dickson, §2] that it came from the notation “x̂” used for class-abstraction by Whitehead and Russell in their Principia Mathematica (which we’re refer to later on), by first modifying “x̂” to “^⁣x” to (and then for better visibility to “∧x”) to distinguish function-abstraction from class-abstraction, and then changing “∧” (which is similar to uppercase Greek “Λ”) to (lowercase Greek) “λ” for ease of printing (and presumably to avoid confusion with other mathematical uses of “∧”, e.g., logical AND).¹⁷ [So maybe “x̂” ⇒ “^⁣x” → “∧x” → “λx”.]

The Greek lambda (uppercase Λ, lowercase λ) as an ortheme itself derives ultimately from a Semitic abjad, specifically from the Phoenician lāmd 𐤋, which (like most letters began as a pictogram of sorts) is considered to originate from something like an ox-goad, i.e., a cattle prod, or else a shepherd’s crook, i.e., a pastoral staff. (The reconstructed Proto-Semitic word *lamed- means a “goad”.)¹⁸

Are We in Platonic Heaven Yet?

Lambda calculus, not being tied to any particular hardware and being a true formula abstraction, feels like something that might have a better claim to being something like a property of the universe that one might discover (I admit I find it hard not to feel something of the sort — but then I’ve used lambda calculus for work on natural language within a framework that wants to assign some sort of reality to our formalisations, so it’s hard not to be pulled in this direction), however, Church says (of his own formalism):

We do not attach any character of uniqueness or absolute truth to any particular system of logic. The entities of formal logic are abstractions, invented because of their use in describing and systematizing facts of experience or observation, and their properties, determined in rough outline by this intended use, depend for their exact character on the arbitrary choice of the inventor.

[Alonzo Church 1932:348¹⁹]

(This seems part of Church’s constructivist philosophy, in common with Frege.)²⁰

What’s `next`? : `λy.(equal? (cdr L) y)`

We’ve pulled at a lot of disparate threads here, trying to explore the nature of the connections between Lisp and (the) lambda calculus. Neither the idea that Lisp is a direct instantiation of lambda calculus nor the idea that McCarthy was largely ignorant of properties of lambda calculus are quite right. But that there are an interesting interplay of connections.

But. This is really to set the stage for me to talk about things I’m interested in which draw on different aspects of Lisp(s) and lambda calculus and formal or applied applications to do with one or the other.

I was going to talk about Montague Grammar, because it’s fascinating (and it’s my day job), for which lambda calculus is a crucial component.²¹ But we’re at length now, so it should be another day.

What I do want to look at next is a combination of Lisp and lambda calculus, in various ways, starting with attempts to implement aspects of lambda calculus in Emacs Lisp, and the challenges therein.

[fingers crossed that (next 'blog) does not eval to undefined.]

(Update: (eval (next 'blog)): Part 2: Recursion Excursion.)

McCarthy, John. 1960. Recursive functions of symbolic expressions and their computation by machine. Communications of the Association for Computing Machinery 3(4):184-195. [available at the Wayback Archive; page nos. refer to those of the PDF at the link, a reformatted version in LaTeX, not those of the original publication.] ↩︎
On Lisp’s mystical aura, see How Lisp Became God’s Own Programming Language (and the associated Hacker News discussion). ↩︎
“Lisp ≠ Lambda Calculus” | Daniel Szmulewicz: Perfumed nightmare (A blog for the somnambulisp) ↩︎
McCarthy, John. 1978b. Transcript of presentation. History of Lisp. In History of programming languages, ed. Richard L. Wexelblat, 185–191. New York: Association for Computing Machinery. https://dl.acm.org/doi/10.1145/800025.1198361 ↩︎
McCarthy, John. 1978a. History of Lisp. In History of programming languages, ed. Richard L. Wexelblat, 173–185. New York: Association for Computing Machinery. https://dl.acm.org/doi/10.1145/800025.1198360 ↩︎
Graham, Paul. 2002. The Roots of Lisp. Ms., https://paulgraham.com/rootsoflisp.html. [PDF version] ↩︎
See, e.g., Graham’s 2001 “Beating the Averages”. ↩︎
On the “Curse of Lisp”, see for instance Rudolf Winestock’s “The Lisp Curse”; but also this 2021 discussion on r/clojure, where it’s mentioned that Winestock said in 2017 in a comment on (one of the innumerable) Hacker News threads on the topic “I wrote that essay five years ago. Nowadays, just use Clojure or Racket and ignore what I’ve written.” ↩︎
J. McCarthy: LISP History. Talk at MIT, Spring or Summer 1974 (Written from tape 7/10/75, unpublished: cited in Stoyan, Herbert. 1984. Early LISP history (1956 - 1959). LFP ‘84: Proceedings of the 1984 ACM Symposium on LISP and functional programming, eds. Robert S. Boyer, Edward S. Schneider, Guy L. Steele, 299–310. New York: Association for Computing Machinery. [https://doi.org/10.1145/800055.802047]) (Quote from p.307.) ↩︎
Descriptively:
1. (quote x) returns (the symbol) x (rather than the value of x).
2. (atom x) returns t if the value of x is an atom or else '().
3. (eq x y) returns t if the values of x and y are the same atom, or both the empty list; otherwise returns ().
4. (car x) expects the value of x to be cons expression of some sort, like a list, and returns its first element.
5. (cdr x) expects the value of x to be cons expression of some sort, like a list, and returns everything after the first element.
6. (cons x y) returns an object composed of a link between x and y; this is how lists are formed, if the second argument of the innermost cons is ().
7. (cond (p₁ e₁)…(pₙ eₙ)) evaluates the p expressions in order until one returns t, at which point it returns the co-indexed e.
8. ((lambda (p₁…pₙ) e) a₁…aₙ) evaluates each aᵢ expression and replaces the occurrence of all occurrences of each pᵢ in e with the value of the corresponding aᵢ expression, and then evaluates e.
9. (label f (lambda (p₁…pₙ) e)) makes all occurrences of the symbol f behave like the following (lambda (p₁…pₙ) e) expression, including inside of e.
↩︎

With just tad bits more sugar to reduce car / cdr recursion spam:

(label caar
       (lambda (x)
         (car (car x))))

(label cadr
       (lambda (x)
         (car (cdr x))))

(label caddr
       (lambda (x)
         (car (cdr (cdr x)))))

(label cadar
       (lambda (x)
         (car (cdr (car x)))))

(label caddar
       (lambda (x)
         (car (cdr (cdr (car x))))))

↩︎

See McCarthy, John, Paul W. Abrahams, Daniel J. Edwards, Timothy
1. Hart, and Michael I. Levin. 1962. /LISP 1.5 Programmer’s
Manual/. Cambridge, Mass.: M.I.T. Press [pdf], at p.36f and also McCarthy (1960:26–7), as well as the Wikipedia page on CAR and CDR. ↩︎
Church is also well-known for his proof that the Entscheidungsproblem is undecidable (Church’s theorem), the Church-Turing thesis, the Frege-Church ontology, amongst much else. I don’t know any particularly interesting details of his personal life (he was no Richard Montague, about whom more later); he was apparently a lifelong Presbyterian. ↩︎
The M here can be standing in for something more complex, e.g., the whole expression might really be λx.(a (b d)), where M here is a place-holder for (a (b d)). ↩︎
The α-conversion (fifth) rule is: bound variables can be substituted with different bound variables. So, λx.M[x] can be α-converted to λy.M[y]. Bound variable names have no significance except as place-holders, so these two expressions are equivalent. The reason for doing this is to avoid name-clashes (see Two Hard Things?) when combining expressions.

E.g., if we wanted to combine λxλy.(x y) with itself, e.g. ((λxλy.(x y)) (λxλy.(x y))) (this is of the function application form (M N)), we could do α-conversion so we don’t end up with accidental variable capture.

If we didn’t this would happen:

((λxλy.(x y)) (λxλy.(x y))) =

((λy.((λxλy.(x y)) y)) =

(λy.(λy.(y y))

Since variables are arbitrary names, whose point is distinctionness, we need instead to α-convert on one of the expressions:

λxλy.(x y) = (by α-conversion)

λaλb.(a b)

And then we can combine as we tried to before, and correctly arrive at:

((λxλy.(x y)) (λaλb.(a b))) =

(λy.((λaλb.(a b)) y)) =

λy.(λb.(y b))

(We’ll talk more about α-conversion and its implement in a later post.) ↩︎
Well, there’s also (optional/debated) sixth rule: η-conversion, which touches on some more philosophical concerns (though ones which will eventually concern us too), to do with extensionality (vs. intensionality) [essentionally Frege’s Sinn und Bedeutung].

η-conversion says that in a case where (f x) = (g x) for all possible values of x, then f = g. It’s not always something which is implemented, and it raises questions around extensionality vs intensionality vs hyperintensionality we’ll probably touch on a later point.

But, for a taste, by η-conversion (because it’s about extensionality, roughly what’s true in the world rather than conceptually), the following two expressions are identical (G has an extra component involving an identity function):

F = λx.x

G = λx.(λy.y)x

Do they conceptuality (intensionally) count as the same function, F and G? No, perhaps, but their extensions are the same (they’re always produce the same results), and so F counts as the α-reduction of G. ↩︎
On the Church λ story, see Cardone, Felice & Hindley,
1. Roger, 2006. History of Lambda-calculus and Combinatory Logic. In
Gabbay and Woods (eds.), Handbook of the History of Logic, vol. 5. Elsevier. [pdf] On uses of ∧ as notation in formal theories, see here. ↩︎
See Wikipedia on Lamedh. ↩︎
Church, Alonzo. 1932. A set of postulates for the foundation of logic. Annals of mathematics 33(2):346-366. [pdf] ↩︎
Thanks to Dr E.J. Slade for discussion on this point (Church’s constructivism, and the connection with Frege’s positions). ↩︎
And one I’ve worked on implementing in a Lisp at various points, e.g., Frege, the beginnings of a Racket-based DSL for natural language semantics. ↩︎