A Scheme Primer

In all the world of computer programming, there are few languages as simple, clean, comprehensive, powerful, and extensible as Scheme. The introduction to the R5RS edition of Scheme's standardization¹ explains its philosophy well:

Programming languages should be designed not by piling feature on top of feature, but by removing the weaknesses and restrictions that make additional features appear necessary.

This minimalism means that the foundations of Scheme are easy to learn. The R5RS introduction continues with:

Scheme demonstrates that a very small number of rules for forming expressions, with no restrictions on how they are composed, suffice to form a practical and efficient programming language that is flexible enough to support most of the major programming paradigms in use today.

With just a few rules and an incredibly simple syntax, Scheme manages to be able to handle any language paradigm you can throw at it. Its minimal base and strong support for extensibility means that it is beloved by and frequently used as a foundation for academic language research.

But a strong reputation in one domain can also be a weak reputation in another. Scheme's association with being an academic language has also lead it to be frequently interpreted as too difficult for the "average" programmer to adopt.

In reality, there is nothing difficult about Scheme, and any programmer can learn it in a very short amount of time (even, maybe especially, children and non-programmers).²

At the Spritely Institute, we decided to base our core technology, Spritely Goblins, on Scheme. We found that while there were excellent in-depth writings on Scheme, and some simple and short Scheme tutorials, what was lacking was a middle-of-the-road introduction. The following is somewhere between a brief and comprehensive overview of Scheme. A shallow read of the following text is sufficient to begin being productive with Scheme, but the enthusiastic reader will find much depth (especially by reading the footnotes).

We begin with no assumption of programming experience, but such experience will help speed the reader through some of the early chapters. The further we go in the tutorial, the more advanced topics become. We will end with a real whammy: how to write a Scheme interpreter in Scheme in a mere 30 single lines of code.

You will need to choose a Scheme implementation to work with, as well as an editor. There are many choices for each, but we will narrow our suggestions to two paths:

• Guile Scheme + GNU Emacs + Geiser: this is what this tutorial was written using, and is a powerful option (which also opens the door to working with Guix, one of the most interesting Scheme projects out there). However, it is also a path with a considerable learning curve.
• Racket, which comes with a built-in IDE called DrRacket. This is an easy path to get started with.

Some code examples are preceded by REPL>. REPL stands for "Read Eval Print Loop", which here means an interactive scheme prompt to experiment with entering expressions.

Here's the familiar "hello world", written in Scheme:

(display "Hello world!\n")

This prints "Hello world!" to the screen. (The "\n" represents a "newline", like if you pressed enter after typing some text in a word processor.)

If you are familiar with other programming languages, this might look a little bit familiar and a little bit different. In most other programming languages, this might look like:

display("Hello world!\n")

In this sense, calling functions in Scheme (and other Lisps like it) is not too different than other languages, except that the function name goes inside the parentheses.

Unlike in some other languages, math expressions like + and - are prefix functions just like any other function, and so they go first:

(+ 1 2)         ; => 3
(/ 10 2)        ; => 5
(/ 2 3)         ; => 2/3 

Most of these can accept multiple arguments:

(+ 1 8 10)   ; equivalent to "1 + 8 + 10" in infix notation

Procedures can also be nested, and we can use the "substitution method" to see how they simplify:

(* (- 8 (/ 30 5)) 21)   ; beginning expression
(* (- 8 6) 21)          ; simplify: (/ 30 5) => 6
(* 2 21)                ; simplify: (- 8 6)  => 2
42                      ; simplify: (* 2 21) => 42

A variety of types are supported. For example, here are some math types:

42          ; integer
98.6        ; floating point
2/3         ; fractions, or "rational" numbers
-42         ; these can all also be negative

Since Scheme supports both "exact" numbers like integers and fractions, and does not have any restriction on number size, it is very good for more precise scientific and mathematical computing. The floating point representation is considered "inexact", and throws away precision for speed.

Here are some more types:

#t                             ; boolean representing "true"
#f                             ; boolean representing "false"
"Pangalactic Gargleblaster"    ; string (text)
'foo                           ; a quoted symbol
'(1 2 3)                       ; a quoted list (of numbers, in this case)
(lambda (x) (* x 2))           ; procedure (we'll come back to this)
'(lambda (x) (* x 2))          ; a list of lists, symbols, and numbers

Symbols are maybe the strangest type if you've come from non-Lisp programming languages (with some exceptions). While symbols look kind of like strings, they represent something more programmatic. (In Goblins' methods syntax, we use symbols to represent method names.)

The quote symbol ' prevents the following symbol or expression from being evaluated. If a list itself is quoted, as in the quoted lambda expression above, the symbols inside are also not evaluated.

We will devote some time to discussing lists in Lists and "cons". The combination of lists and symbols is featured very prominently in many Lisps, including Scheme, because they lie at the heart of Lisp's extensibility: code which can write code. We will see how to take advantage of this power in On the extensibility of Scheme (and Lisps in general).

We can assign values to variables using define:

REPL> (define name "Jane")
REPL> (string-append "Hello " name "!")
; => "Hello Jane!"

However, if what follows define is wrapped in parentheses, Scheme interprets this as a procedure definition:

(define (greet name)
  (string-append "Hello " name "!"))

Now that we have named this procedure we can invoke it:

REPL> (greet "Samantha")
; => "Hello Samantha!"

Note that Scheme has implicit return. By being the last expression in the procedure, the result of the string-append is automatically returned to its caller.

This second syntax for define is actually just syntactic sugar. These two definitions of greet are exactly the same:

(define (greet name)
  (string-append "Hello " name "!"))

(define greet
  (lambda (name)
    (string-append "Hello " name "!")))

lambda is the name for an "anonymous procedure" (ie, no name provided). While we have given this the name greet, the procedure would be usable without it:

REPL> ((lambda (name)
         (string-append "Hello " name "!"))
       "Horace")
; => "Hello Horace!"

There is also another way to name things aside from define, which is let, which allows for a sequence of bound variables and then a body which is evaluated with those bindings. let has the form:

(let ((<VARIABLE-NAME> <VALUE-EXPRESSION>) ...)
  <BODY> ...)

(The ... in the above example represents that its previous expression can be repeated multiple times.)

Here is an example of let in use:

REPL> (let ((name "Horace"))
        (string-append "Hello " name "!"))
; => "Hello Horace!"

Clever readers may notice that this looks very similar to the previous example, and in fact, let is syntax sugar for a lambda which is immediately applied with arguments. The two previous code examples are fully equivalent:

REPL> (let ((name "Horace"))
        (string-append "Hello " name "!"))
; => "Hello Horace!"
REPL> ((lambda (name)
         (string-append "Hello " name "!"))
       "Horace")
; => "Hello Horace!"

let* is like let, but allows bindings to refer to previous bindings within the expression:³

REPL> (let* ((name "Horace")
             (greeting
              (string-append "Hello " name "!\n")))
        (display greeting))   ; print greeting to screen
; prints: Hello Horace!

It is possible to manually apply a list of arguments to a procedure using apply. for example, to sum a list of numbers, we can use apply and + in combination:

REPL> (apply + '(1 2 5))
; => 8

As the inverse of this, it is possible to capture a variable-length set of arguments using "dot notation".⁴ Here we show this off while also demonstrating Guile's format (which when called with #f as its first argument returns a formatted string as a value, and when called with #t as its first argument prints to the screen, the latter of which is what we want here):

REPL> (define (chatty-add chatty-name . nums)
        (format #t "<~a> If you add those together you get ~a!\n"
                chatty-name (apply + nums)))
REPL> (chatty-add "Chester" 2 4 8 6)
; Prints:
;   <Chester> If you add those together you get 20!

While not standard in Scheme, many Scheme implementations also support optional and keyword arguments. Guile implements this abstraction as define*:

REPL> (define* (shopkeeper thing-to-buy
                           #:optional (how-many 1)
                           (cost 20)
                           #:key (shopkeeper "Sammy")
                           (store "Plentiful Great Produce"))
        (format #t "You walk into ~a, grab something from the shelves,\n"
                store)
        (display "and walk up to the counter.\n\n")
        (format #t "~a looks at you and says, "
                shopkeeper)
        (format #t "'~a ~a, eh? That'll be ~a coins!'\n"
                how-many thing-to-buy
                (* cost how-many)))
REPL> (shopkeeper "apples")
; Prints:
;   You walk into Plentiful Great Produce, grab something from the shelves,
;   and walk up to the counter.
;
;   Sammy looks at you and says, '1 apples, eh? That'll be 20 coins!'
REPL> (shopkeeper "bananas" 10 28)
; Prints:
;   You walk into Plentiful Great Produce, grab something from the shelves,
;   and walk up to the counter.
;
;   Sammy looks at you and says, '10 bananas, eh? That'll be 280 coins!'
REPL> (shopkeeper "screws" 3 2
                  #:shopkeeper "Horace"
                  #:store "Horace's Hardware")
; Prints:
;   You walk into Horace's Hardware, grab something from the shelves,
;   and walk up to the counter.
;
;   Horace looks at you and says, '3 screws, eh? That'll be 6 coins!'

Finally, Scheme's procedures can do something else interesting: they can return multiple values using… values! As a particularly silly example, perhaps we would like to compare what it's like to both add and multiply two numbers:

REPL> (define (add-and-multiply x y)
        (values (+ x y)
                (* x y)))
REPL> (add-and-multiply 2 8)
; => 10
; => 16
REPL> (define-values (added multiplied)
        (add-and-multiply 3 10))
REPL> added
; => 13
REPL> multiplied
; => 30

As you can see, we can capture said values with define-values, as shown above. (let-values and call-with-values can also be used, but that's enough new syntax for this section!)

Sometimes we would like to test whether or not something is true. For instance, we can see whether or not an object is a string by using the string?:⁵

REPL> (string? "apple")
; => #t
REPL> (string? 128)
; => #f
REPL> (string? 'apple)
; => #f

(Remember that #t represents "true" and #f represents "false".)

We can use this in combination with if, which has the form:

(if <TEST>
    <CONSEQUENT>
    [<ALTERNATE>])

(The square brackets around <ALTERNATE> means that it is optional.)⁶

So, we could write a silly function that excitedly reports on whether or not an object is a string or not:

REPL> (define (string-enthusiast obj)
        (if (string? obj)
            "Oh my gosh you gave me A STRING!!!"
            "That WASN'T A STRING AT ALL!! MORE STRINGS PLEASE!"))
REPL> (string-enthusiast "carrot")
; => "Oh my gosh you gave me A STRING!!!"
REPL> (string-enthusiast 529)
; => "That WASN'T A STRING AT ALL!! MORE STRINGS PLEASE!"

As we can see, unlike in some other popular languages, if also returns the value of evaluating whichever branch is chosen based on <TEST>.

Scheme also ships with some mathematical comparison tests. > and < stand for "greater than" and "less than" respectively, and >= and <= stand for "greater than or equal to" and "less than or equal to", while = checks for numerical equality:⁷

REPL> (> 8 9)
; => #f
REPL> (< 8 9)
; => #t
REPL> (> 8 8)
; => #f
REPL> (>= 8 8)
; => #t

If we wanted to test for multiple possibilities, we could use nested if statements:

REPL> (define (goldilocks n smallest-ok biggest-ok)
        (if (< n smallest-ok)
            "Too small!"
            (if (> n biggest-ok)
                "Too big!"
                "Just right!")))
REPL> (goldilocks 3 10 20)
; => "Too small!"
REPL> (goldilocks 33 10 20)
; => "Too big!"
REPL> (goldilocks 12 10 20)
; => "Just right!"

However, there is a much nicer syntax named cond which we can use instead which has the following form:⁸

(cond
 (<TEST>
  <THEN-BODY> ...) ...
 [(else <ELSE-BODY> ...)])

Compare how much nicer our goldilocks procedure looks with cond instead of nested if statements:

;; Nested "if" version
(define (goldilocks n smallest-ok biggest-ok)
  (if (< n smallest-ok)
      "Too small!"
      (if (> n biggest-ok)
          "Too big!"
          "Just right!")))

;; "cond" version
(define (goldilocks n smallest-ok biggest-ok)
  (cond
   ((< n smallest-ok)
    "Too small!")
   ((> n biggest-ok)
    "Too big!")
   (else
    "Just right!")))

Scheme also provides some different ways to compare whether or not two objects are the same thing. The shortest, simplest (but not comprehensive) summary of the zoo of equality predicates is that equal? compares based on content equivalence, whereas eq? compares based on object identity (as defined by the language's runtime).⁹ For example, list constructs a fresh list with a new identity every time, so the following are equal? but not eq?:

REPL> (define a-list (list 1 2 3))
REPL> (define b-list (list 1 2 3))
REPL> (equal? a-list a-list)
; => #t
REPL> (eq? a-list a-list)
; => #t
REPL> (equal? a-list b-list)
; => #t
REPL> (eq? a-list b-list)
; => #f

Finally, in Scheme, anything that's not #f is considered true. This is sometimes used with something like member, which looks for matching elements and returns the remaining list if anything is found, and #f otherwise:

REPL> (member 'b '(a b c))
; => (b c)
REPL> (member 'z '(a b c))
; => #f
REPL> (define (fruit-sleuth fruit basket)
        (if (member fruit basket)
            "Found the fruit you're looking for!"
            "No fruit found! Gadzooks!"))
REPL> (define fruit-basket '(apple banana citron))
REPL> (fruit-sleuth 'banana fruit-basket)
; => "Found the fruit you're looking for!"
REPL> (fruit-sleuth 'pineapple fruit-basket)
; => "No fruit found! Gadzooks!"

"My other CAR is a CDR"
  — Bumper sticker of a Lisp enthusiast

For structured data, Scheme supports lists, which can contain any other type.¹⁰ Here are two ways to write the same list:

REPL> (list 1 2 "cat" 33.8 'foo)
; => (1 2 "cat" 33.8 foo)
REPL> '(1 2 "cat" 33.8 foo)
; => (1 2 "cat" 33.8 foo)

One difference between the two above is that in the latter quoted example, the symbol foo did not need to be quoted, since the outer list's quoting excused everything within from being evaluated.

Additionally, quotes within an expression are cumulative. Notice that in the latter example, because the entire list is quoted, the list member foo does not need to be. Adding an additional quote before foo is legal, but may make it harder to work with the list later by needing to 'unquote' or evaluate it many times to get the actual symbol back.

REPL> '(foo 'bar '''baz)
; => (foo (quote bar) (quote (quote (quote baz))))

There is a "special" list known as "the empty list", which is a list with no elements, simply designated '() (also known as nil, and which is the only object which will return #t in response to the predicate null? in standard Scheme). Lists in Scheme are actually "linked lists", which are combinations of pairs called "cons cells" that terminate in the empty list:

REPL> '()
; => ()
REPL> (cons 'a '())
; => (a)
REPL> (cons 'a (cons 'b (cons 'c '())))
; => (a b c)

The latter of which is equivalent to either:

REPL> (list 'a 'b 'c)
; => (a b c)
REPL> '(a b c)
; => (a b c)

For very historical reasons,¹¹ accessing the first element of a cons cell is done with car and the second element of a cons cell with cdr (pronounced "could-er"):¹²

REPL> (car '(a b c))
; => a
REPL> (cdr '(a b c))
; => (b c)
REPL> (car (cdr '(a b c)))
; => b

The second member of cons does not have to be another cons cell or the empty list. If not, it is considered a "dotted list", and has an unusual-for-lisp infix syntax:

REPL> (cons 'a 'b)
; => (a . b)

Notice how this is structurally different from the following:

REPL> (cons 'a (cons 'b '()))
; => (a b)

It's easy to get caught up on piecing apart cons cells (arguably schemers do far too often, but cons is also elegantly powerful).¹³

In a sense, this subsection is a digression. We intentionally do not use cons too much in this paper, and we have entirely kept car and cdr out of the main text. This may lead to the question, why contain this subsection on lists at all?

The reason is that we are building up to something we will explore further shortly, the extensibility of Scheme. Scheme is written in its core data types, and is modifiable as such. We will get to this more shortly, but as an example, we can quote any expression, transforming code into data:

REPL> (+ 1 2 (- 8 4))
; => 7
REPL> '(+ 1 2 (- 8 4))
; => (+ 1 2 (- 8 4))
REPL> (let ((name "Horace"))
        (string-append "Hello " name "!"))
; => "Hello Horace!"
REPL> '(let ((name "Horace"))
         (string-append "Hello " name "!"))
; => (let ((name "Horace")) (string-append "Hello " name "!"))

This last example is especially curious: we finally see the reason for symbols in Scheme to be important, as the function and syntax names become captured as symbols upon being quoted. In this sense, Lisp (including Scheme) is written in Lisp: there is little distinction between the representation the programmer sees and the representation the compiler sees, as see in On the extensibility of Scheme (and Lisps in general).

By the way, the apostrophe quote is just a shorthand for (quote <EXPR>):

;; these two are the same
'foo
(quote foo)

;; and these two are the same
'(lambda (x) (* x 2))
(quote (lambda (x) (* x 2)))

Lists can also be used as an associative mapping between keys and values, called alists (association lists). A variety of procedures for convenient lookup exist, such as assoc, which returns the pair if found or #f if not:

REPL> (define animal-noises
        '((cat . meow)
          (dog . woof)
          (sheep . baa)))
REPL> (assoc 'cat animal-noises)
; => (cat . meow)
REPL> (assoc 'alien animal-noises)
; => #f

Association lists are easy to implement, look nice enough in Scheme's printed representation, and are easy to use with functional programming. (Want to add more to an alist? Just cons on another cons cell!) This means they tend to be popular with schemers. However, they are not always efficient. While assoc is fine for small alists, an alist that is one thousand elements long will take one thousand steps to find a key-value pair buried at its bottom. Other datastructures, such as hashmaps which provide constant-time average lookups, are commonly provided in many Scheme implementations, and are sometimes a better choice.

Aside from quote, it is also possible to use quasiquote, which uses the backtick to begin a quasiquote, and the comma to unquote. In this way we can move quickly between the world of data and code. For example, using a somewhat apocryphal metric for converting cat years to human years:

REPL> (define (cat-years years)
        (cond
         ((<= years 1)       ; first year equivalent to 15
          (* years 15))
         ((<= years 2)
          (+ 15 (* 9 (- years 1))))   ;      second year 9
         (else
          (+ 24 (* 4 (- years 2)))))) ; years after that 4
REPL> (define (cat-entry name age)
        `(cat (name ,name)
              (age  ,age)
              (cat-years-age ,(cat-years age))))
REPL> (cat-entry "Missy Rose" 16)
; => (cat (name "Missy Rose")
;         (age 16)
;         (cat-years-age 80))
REPL> (cat-entry "Kelsey" 22)
; => (cat (name "Kelsey")
;         (age 21)
;         (cat-years-age 104))

Wow! Those are some old cats!

Recall our earlier definition and use of goldilocks:

REPL> (define (goldilocks n smallest-ok biggest-ok)
        (cond
         ((< n smallest-ok)
          "Too small!")
         ((> n biggest-ok)
          "Too big!")
         (else
          "Just right!")))
REPL> (goldilocks 3 10 20)
; => "Too small!"
REPL> (goldilocks 33 10 20)
; => "Too big!"
REPL> (goldilocks 12 10 20)
; => "Just right!"

Entering the same values for smallest-ok and biggest-ok over and over again is tedious. Goldilocks' range of preferences are unlikely to change from invocation to invocation. Is there a way we could produce a version of Goldilocks with a kind of memory so we only have to pass in smallest-ok and biggest-ok once but still test against multiple versions of n? Indeed there is… closures to the rescue!

(define (make-goldilocks smallest-ok biggest-ok)
  (define (goldilocks n)   ; make a procedure which encloses
    (cond                  ;  smallest-ok and biggest-ok so
     ((< n smallest-ok)    ;  that only the n argument needs
      "Too small!")        ;  to be passed in  
     ((> n biggest-ok)
      "Too big!")
     (else
      "Just right!")))
  goldilocks)              ; return goldilocks procedure

We can now invoke make-goldilocks, which returns the enclosed goldilocks procedure¹⁴.

REPL> (make-goldilocks 10 30)
; => #<procedure goldilocks (n)>

Now we can call the inner goldilocks over and over again.

REPL> (define goldi
        (make-goldilocks 10 30))
REPL> (goldi 7)
; => "Too small!"
REPL> (goldi 256)
; => "Too big!"
REPL> (goldi 22)
; => "Just right!"

The outer procedure "closes over" the inner procedure, giving it access to (and a memory of) smallest-ok and biggest-ok.

Notably, this is the same pattern Goblins uses to implement constructors for its objects: the outer procedure is the constructor, the inner procedure is the behavior of the object. (The primary difference is indeed that Goblins objects spawned with spawn get a bcom capability which they can use to change their behavior!)

Beautifully, we can also build our own cons cells out of pure abstraction using this same technique.

REPL> (define (abstract-cons car-data cdr-data)
        (lambda (method)
          (cond
           ((eq? method 'car)
            car-data)
           ((eq? method 'cdr)
            cdr-data)
           (else (error "Unknown method:" method)))))
REPL> (define our-cons (abstract-cons 'foo 'bar))
REPL> (our-cons 'car)
; => foo
REPL> (our-cons 'cdr)
; => bar

In this sense, closures are also data structures built from the code flow of the program itself.

Closures are a property of lexical scoping. We take advantage of this in Goblins: the capabilities an object has access to is merely the capabilities it has within its behavior's scope.

Much of programming involves sequences of operations, especially on datastructures which contain other information. One especially useful procedure for functional programming which operates on lists is map, which applies its first argument's procedure to each element in its series.

For example, string-length gives the number of characters which exist in a given string:

REPL> (string-length "cat")
; => 3
REPL> (string-length "gorilla")
; => 7

So, using map, we could easily construct a list representing the length of each of its strings:

REPL> (map string-length '("cat" "dog" "gorilla" "salamander"))
; => (3 3 7 10)

We could also supply a procedure we define:

REPL> (define (symbol-length sym)
        (string-length (symbol->string sym)))
REPL> (map symbol-length '(basil oregano parsley thyme))
; => (5 7 7 5)

In fact, there is no requirement that we name the procedure… we can use lambda to construct an anonymous procedure which we pass to map directly:

REPL> (map (lambda (str)
             (string-append "I just love "
                            (string-upcase str)
                            "!!!"))
           '("strawberries" "bananas" "grapes"))
; => ("I just love STRAWBERRIES!!!"
;     "I just love BANANAS!!!"
;     "I just love GRAPES!!!")

map performs some extra work by building up a list of results every time. But what if we wanted to simply display our love of some food to the screen using display and did not care about operating on the data any further? We could use for-each, which has the same structure as map but does not build a result:

REPL> (for-each (lambda (str)
                  (display
                   (string-append "I just love "
                                  (string-upcase str)
                                  "!!!\n")))
                '("ice cream" "fudge" "cookies"))
; prints:
;   I just love ICE CREAM!!!
;   I just love FUDGE!!!
;   I just love COOKIES!!!

Scheme has the surprising property that iteration is actually defined in terms of recursion!

Here is what we mean. We could define our own version of for-each:

(define (for-each proc lst)
  (if (eq? lst '())  ; End of the list?
      'done          ; We're done, so simply return "done"
      (let ((item (car lst)))  ; Otherwise... let's fetch this item
        (proc item)            ; Call the procedure with this item
        (for-each proc (cdr lst)))))  ; Iterate with the remaining work

This calls proc successively with each item from lst until it runs out of items. If you have experience with other programming languages, your expectation would probably be that this design could accidentally "blow the stack". However, Scheme is smart: it sees that there is no more work left to be done within the current version of for-each once we reach the last line… in other words, where for-each calls itself is in the "tail position". Because of this, Scheme is able to skip allocating a new frame on the stack and "jump" back to the beginning of for-each again with the new variables allocated. This is called tail call elimination and all iteration facilities are actually defined this way in terms of recursion in Scheme.

It is also possible to build tree-recursive procedures. The following (somewhat advanced, if you don't follow this it's ok) procedure builds a binary tree:

(define (build-tree depth)
  (if (= depth 0)
      '(0)
      (list depth
            (build-tree (- depth 1))
            (build-tree (- depth 1)))))

REPL> (build-tree 3)
; => (3 (2 (1 (0)
;             (0))
;          (1 (0)
;             (0)))
;       (2 (1 (0)
;             (0))
;          (1 (0)
;             (0))))

Or, better visualized:

       3
      / \
     /   \
    2     2
   / \   / \
  1  1   1  1
 /\  /\ /\  /\
0 0 0 0 0 0 0 0

However, unlike for-each, build-tree does not call itself in the tail position. There is no way to simply "jump" to the beginning of the procedure without allocating work to be done on the stack with the way this code is written: more work needs to be done, as list sits waiting for its results. As such, unlike for-each, build-tree is recursive but not iterative.

Finally, come conveniences. Here are two variants on let, both useful for recursive and iterative procedures. The first is letrec which allows for procedures to call and refer to themselves or others defined by the letrec, regardless of definition ordering:

REPL> (letrec ((alice
                (lambda (first?)
                  (report-status "Alice" first?)
                  (if first? (bob #f))))
               (bob
                (lambda (first?)
                  (report-status "Bob" first?)
                  (if first? (alice #f))))
               (report-status
                (lambda (name first?)
                  (display
                   (string-append name " is "
                                  (if first?
                                      "first"
                                      "second")
                                  "!\n")))))
        (alice #t)
        (display "-----\n")
        (bob #t))
; prints:
;   Alice is first!
;   Bob is second!
;   -----
;   Bob is first!
;   Alice is second!

The second useful abstraction is the named let variant of let, where a looping name identifier appears as the first argument:

REPL> (let loop ((words '("carrot" "potato" "pea" "celery"))
                 (num-words 0)
                 (num-chars 0))
        (if (eq? words '())
            (format #f "We found ~a words and ~a chars!"
                    num-words num-chars)
            (loop (cdr words)
                  (+ num-words 1)
                  (+ num-chars (string-length (car words))))))
; => "We found 4 words and 21 chars!"

What a named let does is define the named procedure (here named loop) and immediately invokes it with the initial bindings of the let. The procedure is available within the body of the let for convenient recursive (perhaps iterative) calls.

This section is included for completeness. Notably, Goblins provides a different approach to much of this here which we will discuss towards the end.

Scheme ships with a way to reassign the current value of a variable using set!:

REPL> (define chest 'sword)
REPL> chest
; => sword
REPL> (set! chest 'gold)
REPL> chest
; => gold

This can even be combined with the techniques shown in Closures. For instance, here's an example of an object that counts down from an initial number n until it reaches its zero, and then always returns zero afterwards.

REPL> (define (make-countdown n)
        (lambda ()
          (define last-n n)
          (if (zero? n)
              0
              (begin
                (set! n (- n 1))
                last-n))))
REPL> (define cdown (make-countdown 3))
REPL> (cdown)
; => 3
REPL> (cdown)
; => 2
REPL> (cdown)
; => 1
REPL> (cdown)
; => 0
REPL> (cdown)
; => 0

There are several interesting things about this example:

• We have introduced time and change into our computations. Before the introduction of side effects such as assignment, calling a procedure with the same arguments will always produce the same result. But in the above example, cdown changes its response over time (even without being passed any arguments on invocation).
• Since we want to show the initial number the first time the procedure is called, we have to capture last-n before using set! to change n. If we accidentally reverse this order, we will introduce a bug where cdown would have started with 2 instead of 3 in the example above.
• Here we also see an interesting new piece of syntax: begin. begin executes several expressions in sequence, returning the value of the last expression.

This last one is interesting. Prior to introducing effects (such as the assignment shown above, displaying to the screen, logging to a file or database, etc), there is never any reason for begin. To understand this, recall the substitution method demonstrated at the beginning of this tutorial:

(* (- 8 (/ 30 5)) 21)   ; beginning expression
(* (- 8 6) 21)          ; simplify: (/ 30 5) => 6
(* 2 21)                ; simplify: (- 8 6)  => 2
42                      ; simplify: (* 2 21) => 42

Before effects, every procedure invoked is to compute a new part of the program. But since each branch of if only evaluates one expression, we must provide a way to sequence the alternate clause so that we can both set! and then return a value.¹⁵ In other words, a purely functional program is really built to take a series of inputs and precisely compute a value, the same value, every time. This is a clean set of substitutions all the way up and down the evaluation. (In other worlds, before introducing time, we will have programs which are fully deterministic.)

However, by introducing mutation and side effects, we have introduced a powerful, but dangerous, new construct into our program: time. Our programs are no longer purely functional, time has made them imperative: do this, then do that. Time is change, and change requires sequences of events, not mere substitutions. And time means that the same programs and procedures run with the same inputs will not always produce the same outputs. We have traded a timeless world for one that changes.

Despite the caution, change can be desirable. We live in a world with time and change, and so too often do our programs. Scheme has a (somewhat inconsistent) naming convention for observing time and change: the addition of a ! suffix, as we have seen with set!. The ! can be seen as a kind of warning, as if the user is shouting about the possibility of mutation. (However, the runtime of Scheme provides no guarantee that the presence or absence of this suffix says anything about mutation whatsoever.)

However, set! is not the only form of change and mutation available in standard (and nonstandard) Scheme.¹⁶ Another example is mutable vectors and vector-set!:

REPL> (define vec (vector 'a 'b 'c))
REPL> vec
; => #(a b c)
REPL> (vector-ref vec 1)
; => b
REPL> (vector-set! vec 1 'boop)
REPL> (vector-ref vec 1)
; => boop
REPL> vec
; => #(a boop c)

Both of these examples resemble mutation. However, we have already seen a different form of side effects in this tutorial, namely display, which writes to the screen. In fact, display itself builds on the idea of ports, which are mechanisms in Scheme for reading and writing from and to input and output devices.

All of these carry the same challenges of set!. Put simply, the introduction of ambient time makes our programs less timeless. However, if turns out that we cannot remove all time and change from our computers, as illustrated in this nested set of quotes:

As Simon Peyton Jones, a well-known functional programmer, likes to say, "All you can do without side effects is push a button and watch the box get hot for a while." (Which isn't technically true, since even the box getting hot is a side effect.)

— From Land of Lisp by Conrad Barski, M.D.

As Simon and Conrad point out, the challenge with functional programming is that even though side effects can be dangerous, they are in a sense all the user really cares about. At some point, in order for a computer to be useful, input must be read from the user and output must be given back, and these are inherently side-effectful. Even using the radiant heat of a busy computer to warm your house is a side effect. At some point, we must both enter and leave the realm of pure mathlandia.¹⁷

Let's say we'd like some new syntax. For instance, maybe we want to run multiple pieces of code in sequence when a condition is met. We could write:

(if (our-test)
    (begin
      (do-thing-1)
      (do-thing-2)))

But this is kind of ugly. What if we created some new syntax specifically for this purpose?

(when (our-test)
  (do-thing-1)
  (do-thing-2))

when cannot be built as a function because we do not want to execute (do-thing-1) or (do-thing-2) unless (our-test) passes. We need new syntax.

Could we build the new syntax ourselves? Remembering that we can "write Lisp in Lisp", the answer seems to be yes:

REPL> (define (when test . body)
        `(if ,test
             ,(cons 'begin body)))
REPL> (when '(our-test)
        '(do-thing-1)
        '(do-thing-2))
; => (if (our-test)
;        (begin
;          (do-thing-1)
;          (do-thing-2)))

This does build out the appropriate syntax! And it does demonstrate that our claim that Lisp can "write code which writes code" is indeed true.¹⁸

However, there are two obvious problems with this first attempt:

• We had to quote each argument passed to when. This is annoying to do.
• when does not actually run its code, it just returns the quoted structure that the code should expand to.

However, with just one tweak our procedure can be turned into a "macro": a special kind of procedure used by the compiler to expand code. Here is all we need to do:

(define-macro (when test . body)
  `(if ,test
       ,(cons 'begin body)))

All we needed to do was rename define to define-macro! Now Scheme knows it should use this for code expansion. This allows us to define new kinds of syntax forms.

define-macro shows very clearly what macros in Lisp and Scheme do: they operate on structure. Manually building up a list structure like this is how macros in Common Lisp work. However, this is not the general way to write macros in Scheme. Scheme macros look very similar though:

(define-syntax-rule (when test body ...)
  (if test
      (begin body ...)))

define-syntax-rule uses pattern matching to implement macros. The first argument to define-syntax-rule describes the pattern which the user will enter, and the second describes the template which will be expanded.¹⁹ We can also notice that body ... appears in both the pattern and the template; the ... ellipsis in the pattern represents that multiple expressions will be captured from the user's input and the ... in the template indicates where the repeating should occur. We can see that we do not need to manually quote things using this mechanism; Scheme cleverly takes care of it for us.

Ultimately, the Scheme version of syntax definitions is less obvious as to how it works under the hood than the define-macro version is. However, there is an issue that arrives in syntax transformation systems called hygiene: that a syntax form / macro not introduce unexpected temporary identifiers into the body of the form it expands into. We will not get into the debate in this primer, but both Common Lisp and Scheme's macros have significant tradeoffs, with Scheme being much more likely to be properly "hygienic", easier to write for simple syntax forms, but harder to write for more complicated ones, and less obvious as to how they work under the hood. For this reason, even though you will likely never use the define-macro approach in Scheme, it is a useful way to understand the idea behind "code that writes code".

Now that we know how to produce new syntax the Scheme way, let's see if we can make our life more convenient than before. Let's revisit our use of for-each from earlier:

REPL> (for-each (lambda (str)
                  (display
                   (string-append "I just love "
                                  (string-upcase str)
                                  "!!!\n")))
                '("strawberries" "bananas" "grapes"))
; prints:
;   I just love STRAWBERRIES!!!
;   I just love BANANAS!!!
;   I just love GRAPES!!!

This works, but it is also unnecessarily tedious. That lambda is an unnecessary piece of detail! A small new syntax definition lets us clean things up:

(define-syntax-rule (for (item lst) body ...)
  (for-each (lambda (item)
              body ...)
            lst))

Let's give it a try:

REPL> (for (str '("strawberries" "bananas" "grapes"))
        (display
         (string-append "I just love "
                        (string-upcase str)
                        "!!!\n")))
; prints:
;   I just love STRAWBERRIES!!!
;   I just love BANANAS!!!
;   I just love GRAPES!!!

It works! This is much easier to read.²⁰

We need not stop here. The methods feature in Spritely Goblins is an example of a macro. Here is a simplified version:

(define-syntax-rule (methods ((method-id method-args ...)
                              body ...) ...)
  (lambda (method . args)
    (letrec ((method-id
              (lambda (method-args ...)
                body ...)) ...)
      (cond
       ((eq? method (quote method-id))
        (apply method-id args)) ...
       (else
        (error "No such method:" method))))))

We can both see here simultaneously how expressive Scheme style pattern matching examples are, but also how with multiple layers of ellipses (the ...), it can be a bit challenging to see how the code expander is figuring out how to unpack things.

But let's not worry about that for now, and instead show an example of usage:

REPL> (define (make-enemy name hp)
        (methods
         ((get-name)
          name)
         ((damage-me weapon hp-lost)
          (cond
           ((dead?)
            (format #t "Poor ~a is already dead!\n" name))
           (else
            (set! hp (- hp hp-lost))
            (format #t "You attack ~a, doing ~a damage!\n"
                    name hp-lost))))
         ((dead?)
          (<= hp 0))))
REPL> (define hobgob
        (make-enemy "Hobgoblin" 25))
REPL> (hobgob 'get-name)
; => "Hobgoblin"
REPL> (hobgob 'dead?)
; => #f
REPL> (hobgob 'damage-me "club" 10)
; prints: You attack Hobgoblin, doing 10 damage!
REPL> (hobgob 'damage-me "sword" 20)
; prints: You attack Hobgoblin, doing 20 damage!
REPL> (hobgob 'damage-me "pickle" 2)
; prints: Poor Hobgoblin is already dead!
REPL> (hobgob 'dead?)
; => #t

We can go further. We can extend Scheme to include logic programming, we can add pattern matching, etc etc etc. Indeed, we will use a pattern matching system included in Guile's standard library in the next subsection.

Because of the syntactic extensibility of Lisp/Scheme, advanced programming language features can be implemented as libraries rather than as entirely separate sub-languages. Multiple problem domains can be combined into one system. For this reason, we say that languages in the Lisp language family support composable domain specific languages.

It is also liberating. In other programming languages, users must pray at the altar of the programming language implementers for features to show up in the next official language release, features which would be only a few small and simple lines of code in the hand of a Lisp/Scheme user.

This is true power. But there is more. In the next section we will unlock Scheme itself, allowing us to configure and experiment with its underlying mechanisms, in a surprisingly compact amount of code.

Here is a working implementation of Scheme written in Scheme:

(use-modules (ice-9 match))

(define (env-lookup env name)
  (match (assoc name env)
    ((_key . val)
     val)
    (_
     (error "Variable unbound:" name))))

(define (extend-env env names vals)
  (if (eq? names '())
      env
      (cons (cons (car names) (car vals))
            (extend-env env (cdr names) (cdr vals)))))

(define (evaluate expr env)
  (match expr
    ;; Support builtin types
    ((or #t #f (? number?))
     expr)
    ;; Quoting
    (('quote quoted-expr)
     quoted-expr)
    ;; Variable lookup
    ((? symbol? name)
     (env-lookup env name))
    ;; Conditionals
    (('if test consequent alternate)
     (if (evaluate test env)
         (evaluate consequent env)
         (evaluate alternate env)))
    ;; Lambdas (Procedures)
    (('lambda (args ...) body)
     (lambda (. vals)
       (evaluate body (extend-env env args vals))))
    ;; Procedure Invocation (Application)
    ((proc-expr arg-exprs ...)
     (apply (evaluate proc-expr env)
            (map (lambda (arg-expr)
                   (evaluate arg-expr env))
                 arg-exprs)))))

Without comments, blank lines, and the pattern matching import at the top (not necessary, but convenient), this is a mere 30 lines of code. This evaluator, while bare bones, is complete enough to be able to compute anything we can imagine. (You could even write another similar Scheme evaluator on top of this one!)²¹

Our evaluator takes two arguments, a Scheme expression expr and an environment env. Scheme's lispy structure is of great benefit here, since as we have learned we can easily quote entire sections of code. (Indeed, that is exactly what we are going to do.) The env of the second argument is an association list mapping symbols for names and their associated procedures.

Seeing is believing. Let's do some simple arithmetic, passing in some procedures to the default environment which can do some math:

(define math-env
  `((+ . ,+)
    (- . ,-)
    (* . ,*)
    (/ . ,/)))

As we can see, the first "substitution method" example we wrote works just fine using this environment:

REPL> (evaluate '(* (- 8 (/ 30 5)) 21)
                math-env)
; => 42

What do you know, that's the same answer we got in our own program!

We can also make a lambda and apply it. Let's make one that can square a number:

REPL> (evaluate '((lambda (x)
                    (* x x))
                  4)
                math-env)
; => 16

Nice, works perfectly.

Let's do something more advanced. Supplying only two operators, + and =, we are able to compute the Fibonacci sequence:

REPL> (define fib-program
        '((lambda (prog arg)   ; boot
            (prog prog arg))
          (lambda (fib n)      ; main program
            (if (= n 0)
                0
                (if (= n 1)
                    1
                    (+ (fib fib (+ n -1))
                       (fib fib (+ n -2))))))
          10))                 ; argument
REPL> (define fib-env
        `((+ . ,+)
          (= . ,=)))
REPL> (evaluate fib-program fib-env)
; => 55

This seems like magic. But it works! The evaluator really is performing the underlying computation, using merely addition (on both positive and negative numbers) and numeric equality check procedures, which we have provided.

The main program needs to be able to call itself, so the first procedure (labeled boot) takes a program and an argument and invokes the procedure with itself and that argument.²² The second procedure (labeled main program) takes itself as the argument fib (supplied by our boot procedure) as well as an argument of n (also supplied by the boot procedure)… and it works! Our evaluator recursively builds up the Fibonacci sequence.

Our evaluator can also be easily understood. Let us break it down section by section.

(define (env-lookup env name)
  (match (assoc name env)
    ((_key . val)
     val)
    (_
     (error "Variable unbound:" name))))

This one is easy. We are defining environments as association lists, so all env-lookup does is search for a matching name in the list. Newer additions will be found first, meaning that the same name defined in a deeper scope will shadow the parent scope. This can be seen by usage:

REPL> (env-lookup '((foo . newer-foo)
                    (bar . bar)
                    (foo . older-foo))
                  'foo)
; => 'newer-foo

The next one is a utility:

(define (extend-env env names vals)
  (if (eq? names '())
      env
      (cons (cons (car names) (car vals))
            (extend-env env (cdr names) (cdr vals)))))

extend-env takes an environment and a list of names and a parallel list of values. This is a convenience which we will use in procedure definitions later. Once again, easily understood by usage:

REPL> (extend-env '((foo . foo-val))
                  '(bar quux)
                  '(bar-val quux-val))
; => ((bar . bar-val)
;     (quux . quux-val)
;     (foo . foo-val))

And now we are onto the evaluator. The shell of evaluate looks like so:

(define (evaluate expr env)
  (match expr
    (<MATCH-PATTERN>
     <MATCH-BODY> ...) ...))

evaluate takes two arguments:

• expr: the expression to evaluate
• env: the environment in which we will evaluate the expression

For the body of evaluate, we are dispatching our behavior depending on which patterns match expr. We are using match from Guile's pattern matching syntax (which came from our module import at the top). The short of it is though that if a <MATCH-PATTERN> matches, we will then stop searching for matches and evaluate <MATCH-BODY> (possibly with bindings set up from the <MATCH-PATTERN>).

So, now all we need to do is look at each pattern we support. The first is easy:

;; Support builtin types
((or #t #f (? number?))
 expr)

The or says we can match any one of its contained patterns. The first two are literally the true and false values from Scheme itself. The parentheses starting with a ? symbol indicates that we will try matching against a predicate, in this case number?. If any of these match, we simply return the very same expr we are matching against… borrowing booleans and numbers straight from the underlying Scheme implementation.

In other words, the above powers:

REPL> (evaluate #t '())
; => #t
REPL> (evaluate #f '())
; => #f
REPL> (evaluate 33 '())
; => 33
REPL> (evaluate -2/3 '())
; => -2/3

That was easy! The next one is also easy:

;; Quoting
(('quote quoted-expr)
 quoted-expr)

Recall that 'foo is just shorthand for (quote foo), and likewise '(1 2 3) is shorthand for (quote (1 2 3)). In this pattern, we look for anything matching a list starting with the 'quote symbol and a second element which is the expression to be quoted.

In other words, the above powers:

REPL> (evaluate ''foo '())
; => foo
REPL> (evaluate ''(1 2 3) '())
; => (1 2 3)
REPL> (evaluate (quote (quote (1 2 3))) '())
; => (1 2 3)

Those last two are the same. Note that we quote twice: once for quoting the entire program to be run, and once within the quoted program to say we want to quote an expression.

So far so good. The next one is still quite easy:

;; Variable lookup
((? symbol? name)
 (env-lookup env name))

The (? symbol? name) part binds name to the matching component. (In this case, name will be bound to the same value as the expr matched against, but this improves readability a little.)

As for the body… why, this is quite simple! We have already reviewed how env-lookup works. In other words if we see a symbol (not a quoted one of course, that has already been handled), we look up its corresponding value in the environment.

In other words, the above powers:

REPL> (evaluate 'x '((x . 33)))
; => 33

However, it will also empower variable lookups we define through lambda applications:

REPL> (evaluate '((lambda (x) x) 33) '())
; => 33

Of course, we have not yet gotten to lambda! But we are nearly there.

The next one, conditionals, also turns out to be fairly easy:

;; Conditionals
(('if test consequent alternate)
 (if (evaluate test env)
     (evaluate consequent env)
     (evaluate alternate env)))

In other words, a list starting with the symbol 'if will be matched, with the three sub-expressions following 'if bound to the variables test, consequent, and alternate in the match body. We use the underlying Scheme if, and first evaluate test against the current environment env (notice the recursion!), and the host Scheme's if helps us whether to evaluate the consequent or alternate inside of env, again using evaluate recursively.

Okay, now it's time to build procedures. This one is a little bit more complicated, but ultimately not too complicated either:

;; Lambdas (Procedures)
(('lambda (args ...) body)
 (lambda (. vals)
   (evaluate body (extend-env env args vals))))

The pattern here looks for a list starting with 'lambda, with the second list member being the set of arguments, with the body being captured as, well, body. We then return a procedure which is ready to be evaluated with the same number of arguments.²³ The inner body of the procedure we return recursively calls evaluate against the body expression of the lambda we are matching against, but with a newly extended environment, binding together the names within args and the vals from the procedure invocation.²⁴

In other words, the above powers:

REPL> ((evaluate '(lambda (x y) x) '())
       'first 'second)
; => first
REPL> ((evaluate '(lambda (x y) y) '())
       'first 'second)
; => second

There is only one more piece left… application!

;; Procedure Invocation (Application)
((proc-expr arg-exprs ...)
 (apply (evaluate proc-expr env)
        (map (lambda (arg-expr)
               (evaluate arg-expr env))
             arg-exprs)))

This is the general-purpose procedure application piece of the puzzle! At this point, the pattern will match any list with one or more items, determining that this must mean a procedure applied to arguments. We evaluate the proc-expr, representing the procedure to be evaluated, within the current arguments, calling evaluate recursively with the current environment, env. We also gather all the arg-expr argument expressions passed to this procedure by calling evaluate recursively on each with the current environment, env²⁵.

In other words, the above powers:

REPL> (evaluate '(* (- 8 (/ 30 5)) 21)
                math-env)
; => 42

And with all pieces combined, we have enough power not only to compute the Fibonacci sequence, but any computable problem imaginable!

To be fair, this does borrow a portion of Scheme's underlying power, but not as much as it may appear… certainly less than many languages implemented on top of other languages do (certainly, certainly far less than Clojure borrows from Java, for instance, or nearly any popular language borrows from C's standard library).²⁶ And it is not so complete as to implement any of the Scheme standards. But without too much extra work, we could get there, and it is enough for demonstration. But we also get to choose how much power we give the language, by modifying the initial environment the code evaluates in.

It is also a capability-secure language! Aside from going into an infinite loop and consuming too many resources in terms of memory or CPU power, there is nothing particularly dangerous this language can do. However, we can decide how much power we would like to give it. If we choose, we can provide an environment with mutable cells, or one with access to the filesystem. The choice is ours.

And with tiny tweaks, our evaluator can operate in different and marvelous ways. We can add new syntax. We can add new syntax to add new syntax (macros)! We can change evaluation order, we can add static type analysis, we can do many things.

We promised that you would have learned Scheme from this tutorial. If you have reached this point, you have reached much more: you are no longer just a user of Scheme, but a builder of Scheme. The power is yours!²⁷