| > Alonzo Church developed the lambda calculus in 1929. His first publication that showed the elements of the lambda calculus was the 1932 paper "A set of postulates for the foundation of logic", as I cited in my recent paper [1]. It's quite possible he worked on it prior to 1932, but I don't know of any credible evidence on that (would be very interested to learn about any). > Wait! How the heck is this a "programming" language?
> At first glance, this simple language seems to lack both recursion and iteration, not to mention numbers, booleans, conditionals, data structures and all the rest. How can this language possibly be general-purpose? What most stops lambda calculus from being a programming language is that it doesn't directly support I/O. However, one can adopt some very simple conventions for representing bits, lists of bits (bytes), and lists of bytes, and for letting a lambda term operate on these [2] which make the so-called Binary Lambda Calculus (BLC) a programming language. And a very expressive language it is too: a BLC self interpreter [4] can be as small as the 170-bits 01000110100001000
00001100000010111
00110000111111100
00101110011111110
00000111100000010
11101110011011110
01111111100001111
11110000101111010
01110100101111101
00101101010011010
which encodes the term
(λ11)(λ(λλλ1(λλ2(1(λ6(λ2(6(λλ3(λλ23(14))))(7(λ7(λ31(21)))))))(41(111))))(11))
in De Bruijn notatation, with lambda diagram [3] ┬─┬ ────────────────────────────────────────────┬─┬
└─┤ ──────┬───────────────┬──────────────────── ├─┘
│ ──────┼───┬───────────┼─┬─────────┬──────── │
│ ┬─────┼───┼───────────┼─┼─────────┼──────── │
│ │ ┬───┼───┼───────────┼─┼─────────┼──────── │
│ │ ┼─┬─┼───┼───────────┼─┼─────────┼─┬─┬─┬─┬ │
│ │ │ │ ┼─┬─┼───────────┼─┼──────── └─┤ └─┤ │ │
│ │ │ │ │ ┼─┼─┬─────────┼─┼─┬────── │ ├─┘ │
│ │ │ │ │ │ │ ┼───────┬ │ ┼─┼───┬── ├───┘ │
│ │ │ │ │ │ │ ┼───┬───┼ │ │ ┼─┬─┼─┬ │ │
│ │ │ │ │ │ │ │ ┬─┼───┼ │ │ └─┤ ├─┘ │ │
│ │ │ │ │ │ │ │ ┼─┼─┬─┼ │ │ ├─┘ │ │
│ │ │ │ │ │ │ │ └─┤ ├─┘ │ ├───┘ │ │
│ │ │ │ │ │ │ │ ├─┘ ├─┘ │ │
│ │ │ │ │ │ │ ├───┘ │ │ │
│ │ │ │ │ │ ├─┘ │ │ │
│ │ │ │ │ └─┤ │ │ │
│ │ │ │ │ ├───────────┘ │ │
│ │ │ │ ├───┘ │ │
│ │ │ ├─┘ │ │
│ │ └─┤ │ │
│ │ ├───────────────────────────────┘ │
│ └───┤ │
│ ├───────────────────────────────────────┘
└─────┘
10 times smaller than the 7 lines of R5RS Scheme (define (eval e env) (cond
((symbol? e) (cadr (assq e env)))
((eq? (car e) 'λ) (cons e env))
(else (apply (eval (car e) env) (eval (cadr e) env)))))
(define (apply f x)
(eval (cddr (car f)) (cons (list (cadr (car f)) x) (cdr f))))
(display (eval (read) '())) (newline)
> This code will read a program from stdin, parse it, evaluate it and print the result.Except that it leaves all the actual parsing to the "read" library function. In contrast, the BLC code does all parsing itself.
One of the neatest tricks is how it represents the environment as a list built with cons' = \x\y\zx\zy. zx x (zy y)
which allows a list of bits like "1110" (the code for de Bruijn index 3) to index the environment by simply applying it to the environment.[1] https://www.mdpi.com/1099-4300/28/5/494 "The Largest Number Representable in 64 Bits" [2] https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d8... [3] https://tromp.github.io/cl/diagrams.html [4] https://github.com/tromp/AIT/blob/master/ait/int.lam |