Table of Contents
Setup
The code examples are written in a lisp I'm working on.
To try it out, clone the git repo, make sure rust is installed and run
cargo run repl
to get an interactive REPL or
cargo run run example.scm
to evaluate a file.
Basic Syntax

Booleans:
true
andfalse

The empty List:
nil
or'()
or(list)
(they are pretty much equivalent) 
(def a 1)
defines a variable nameda
with a value of1
. 
(set! a 2)
overwritesa
with2

(fn (n div) (
0 (% n div))= creates an anonymous function = with two argumentsn
anddiv
that checks ifn
is divisible bydiv
. 
(defn name (arg1 arg2) body)
is syntax sugar for(def name (fn (arg1 arg2) body))
and can be used to define named functions 
(cons 1 2)
creates a pair of two values 
(if test consequent alternative)
is equivalent toif (test) { consequent } else {alternative }
in other languages 
(do expr1 expr2 ...)
evaluates each of the arguments and returns the result of the last one. It can be used to do more than one thing inside theconsequent
of anif
etc. 
(inc n)
and(dec n)
are equiv. to(+ n 1)
and( n 1)

fst
andrst
("first" and "rest") are equivalent tocar
andcdr
in languages like Chicken Scheme and can be used to access the first and second element of acons
pair 
print
andprintln
print values on the screen, without or with a newline
Special Forms
Given an expression like
(println (+ 1 2))
the 'normal' way (in a
language that is not lazily evaluated) to treat it would be to evaluate
all elements (
println
to a function reference,
(+ 1 2)
to
3
) and
then apply the rest of the elements to the first.
This is good enough most times, but consider this example:
(defn unless (test consequent alternative) (if test alternative consequent))
Looks pretty harmless, but what happens if we try to use it like this?
(def n 5) (unless (= 0 (% n 2)) (println "n is odd") (println "n is even"))
The result will be something like
n is odd n is even => undefined (ignored by the repl)
With the 'normal' way of evaluating expressions, the first step would be
to evaluate
(= 0 (% n 2))
to
true
,
(println "n is odd")
to
undefined
(because it does not return anything useful) and
(println "n is even")
to
undefined
, too.
The remaining expression is
(if true undefined undefined)
and that
evaluates to
undefined
.
To avoid this,
unless
(and some other syntax constructs like
if
,
delay
) must be implemented as
special forms
.
Simplified this means they are called with 'raw', unevaluated arguments and can choose when (and if) to evaluate them themselves.
An implementation of
unless
as special form would need to evaluate the
test first and then, depending on the result, evaluate either the
consequent or the alternative.
Promises
delay
and
force
can be used to delay the evaluation of an
expression.
A naive implementation would be to
(defn delay (body) (fn () body))
(just wrap an anonymous function around the body) and
(defn force (promise) (promise))
(call the underlying lambda).
Because of the reasons described in the section above, this would not
work as intended,
delay
needs to be a special form.
One other idiosyncracy is that once a promise was forced, it remembers its value.
(def a 1) (def promise (delay a)) (set! a 2) (force a) ; => 2 (set! a 3) (force a) ; => 2
Lists and Streams
In Lisp,
lists
are built by nesting
cons
.
(list 1 2 3)
is just syntactic sugar for
(cons 1 (cons 2 (cons 3 '())))
.
Streams are lazy lists, their elements are computed on demand they can be infinitely long.
Let's start with a simple example:
(def ones (cons 1 ones))
This doesn't work because at the time the body
(cons 1 ones)
is being
evaluated,
ones
is not yet defined in the parent environment.
On the other hand,
(def ones (cons 1 (delay ones)))
works perfectly fine, because the evaluation of
ones
in the
cons
is
being delayed until we actually access it (and evaluating
def
is
completed).
(fst ones) ; => 1 (rst ones) ; => promise(?), a promise that has not yet been evaluated (force (rst ones) ; => (1 . promise(?))
To keep the code as simple as possible, I added a special form
(streamcons foo bar)
that is equivalent to
(cons foo (delay bar)
and a method
(streamrst stream)
^{
1
}
that forces the
rst
of
stream
.
(def ones (streamcons 1 ones)) (streamrst ones) ; => (1 . promise(?))
Working with Streams
Before we continue, it would be nice to have some way to display streams (up to some fixed, finite length).
(defn streamprint (limit stream) (if (> limit 0) (do (println (fst stream)) (streamprint (dec limit) (streamrst stream)))))
streamprint
checks if the limit is greater than
0
, prints the
fst
of the stream and calls itself recursively with
(dec limit)
(
limit  1
) and the forced
rst
(
streamrst
) of the stream.
(streamprint 5 ones) ; 1 ; 1 ; 1 ; 1
To do some more interesting things, we need ways to combine and manipulate streams.
streammap
creates a new stream with the results of applying
fun
to
the elements of
stream
.
(defn streammap (fun stream) (streamcons (fun (fst stream)) (streammap fun (streamrst stream))))
(streamprint 4 (streammap inc ones)) ; 2 ; 2 ; 2 ; 2
(defn streamcombine (fun s1 s2) (streamcons (fun s1 s2) (streamcombine fun (streamrst s1) (streamrst s2)))) (defn streamadd (s1 s2) (streamcombine + s1 s2))
(streamprint 4 (streamadd ones ones)) ; 2 ; 2 ; 2 ; 2
Still pretty boring, how about a stream of natural numbers (excluding 0)?
(def naturalnumbers (streamcons 1 (streamadd naturalnumbers ones))) (streamprint 4 naturalnumbers) ; 1 ; 2 ; 3 ; 4 ; 5
This might be a little hard to understand, especially if you have never worked with streams or lazy evaluation before.
naturalnumbers = 1, (naturalnumbers[0] + 1), (naturalnumbers[1] + 1), ...
An alternative implementation that might be easier to understand:
(defn naturalnumbersfrom (n) (streamcons n (naturalnumbersfrom (inc n)))) (def naturalnumbers (naturalnumbersfrom 1))
Fibonacci Numbers
The Fibonacci Numbers are defined as

(fib 0) = 0

(fib 1) = 1

(fib n) = (+ (fib ( n 1)) (fib ( n 2)))
This is equivalent to adding the
fib
stream its
rst
, a version of
itself that is shifted by one.
0 1 1 2 3 5 8 13 ... + _ 0 1 1 2 3 5 8 ... = 0 1 2 3 5 8 13 21 ...
(def fibs (consstream 0 (consstream 1 (streamadd fibs (streamrst fibs))))) (streamprint 10 fibs) ; 0 ; 1 ; 1 ; 2 ; 3 ; 5 ; 8 ; 13 ; 21 ; 34
Filtering Streams
(defn streamfilter (pred stream) (if (pred (fst stream)) (streamcons (fst stream) (streamfilter pred (streamrst stream))) (streamfilter pred (streamrst stream))))
Filtering over streams is more involed than
map
.
If the result of
(pred (fst stream))
is true, we construct a new
stream with
(fst stream)
and the
delayed
result of filtering the
rest of the stream. Otherwise, we call
streamfilter
again with the
rest of the stream, skipping
fst
.
To better understand what is happening when forcing elements of the filtered stream, let's build a stream that displays a message when one of its elements is forced.
(def debugstream (streammap (fn (n) (do (print "Forcing ") (println n) n)) (naturalnumbersfrom 0))) ; Forcing 0
Only one line is being printed, the rest of the new stream will only be evaluated when we need it.
(defn ismultipleof (div) (fn (n) (= 0 (% n div)))) (def multiplesoffive (streamfilter (ismultipleof 5) debugstream)) (fst multiplesoffive) ; => 0
No new lines are being printed, the first element
0
is already a
multiple of
5
.
When we try to output more elements of the stream, something strange happens:
(println (fst (streamrst multiplesoffive))) ; Forcing 1 ; Forcing 2 ; Forcing 3 ; Forcing 4 ; Forcing 5 5
streamfilter
starts forcing elements of the stream until it finds the
next one that satisfies
pred
, in this case
5
.
An Infinite Prime Sieve
The Sieve of Erathosthenes works by taking a list of numbers (starting with 2), going to the first element, removing all its multiples from the list (4, 6, 8, …) and then repeating these steps with the next elements in the list (first 3, then 5) over and over again.
Once this process has reached the end of the list, only prime numbers remain.
Based on
(naturalnumbersfrom 2)
we can use the same algorithm to
create a (infinite) stream of primes!
Start with removing all multiples of the first element from a stream:
(defn isnotmultipleof (div) (fn (n) (!= 0 (% n div)))) (defn removemultiplesoffirst (stream) (streamcons (fst stream) (streamfilter (isnotmultipleof (fst stream)) (streamrst stream)))) (streamprint 6 (removemultiplesoffirst (naturalnumbersfrom 2))) ; 2 ; 3 ; 5 ; 7 ; 9 ; 11 ; 13 ; 15 ; 17 ; 19
We are nearly there, the first few numbers are already primes but
9
is
a multiple of
3
,
15
is a multiple of
5
, …
What is missing is the next step of the algorithm, > … and then repeating these steps with the next elements in the list > (first 3, then 5) over and over again.
(defn removemultiplesoffirst (stream) (streamcons (fst stream) (removemultiplesoffirst (streamfilter (isnotmultipleof (fst stream)) (streamrst stream))))) (def primes (removemultiplesoffirst (naturalnumbersfrom 2))) (streamprint 10 primes) ; 2 ; 3 ; 5 ; 7 ; 11 ; 13 ; 17 ; 19 ; 23 ; 29
The change in the code was pretty minor but now
removemultiplesoffirst
applies itself recursively to the new,
filtered stream, completing the prime sieve.
To make sure, we can define a method that returns the nth ^{ 2 } element of a stream and check agains a list of primes .
(defn streamnth (n stream) (if (= n 0) (fst stream) (streamnth (dec n) (streamrst stream)))) (streamnth 49 primes) ; => 229, which is in fact the 50th prime
Footnotes
There is no need to have something like
streamfst
, the
fst
of the stream is not a promise.
(streamnth 49 primes)
is the 50th element of the stream
because it is 0indexed