Lazy-ish iterators

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Colin <>




Generators The Way I Want Them Generated

(Technically not generators, but iterators.)

The GTWIWTG library is meant to be small, explorable, and understandable. The source code is meant to be legible and straightforward.

Every symbol exported from the GTWIWTG package has a useful docstring. Many docstrings include examples of use.

Table Of Contents


The gtwiwtg system is available from quicklisp and ultralisp.

(ql:quickload :gtwiwtg)
(use-package :gtwiwtg)

First, Some Action

Here are a few examples to show you what you can do. A more involved example apears at the end of the document, following the tutorial.

All The Primes

> (defun prime-p (n)
    "Naive test for primes."
       :for x :from 2 :upto (sqrt n)
       :when (zerop (mod n x)) :do (return nil)
       :finally (return t)))

> (defun all-primes ()
    "Creates a generator that produces an infinite series of primes."
     (filter! #'prime-p (range :from 2)))

> (take 10 (all-primes)) ;; (2 3 5 7 11 13 17 19 23 29)

Fun With Fibonacci

> (defun fibs ()
    "Creates an infinite series of Fibonacci numbers."
     (lambda (n-1 n-2) (+ n-1 n-2))
     1 0))

;; First ten Fibonacci numbers
> (take 10 (fibs)) ;; (1 2 3 5 8 13 21 34 55 89) 

;; Just the 40th Fibonacci number, indexed from 0
> (car (pick-out '(40) (fibs))) ;; 267914296

Cartesian Products

> (defun cartesian-product (list &rest lists) 
    "A generator for the Cartesian product of a some lists"
    (if (null lists)
        (map! 'list (seq list))
        (inflate! (lambda (elem)
                    (map! (lambda (tail) (cons elem tail))
                          (apply 'cartesian-product lists)))
                  (seq list))))

> (collect (cartesian-product '(1 2 3) '(a b c) '("foo" "bar")))
((1 A "foo") (1 A "bar") (1 B "foo") (1 B "bar") (1 C "foo") (1 C "bar")
 (2 A "foo") (2 A "bar") (2 B "foo") (2 B "bar") (2 C "foo") (2 C "bar")
 (3 A "foo") (3 A "bar") (3 B "foo") (3 B "bar") (3 C "foo") (3 C "bar"))


(defun powerset (list)
  "Generate the set of all subsets of a given set."
  (if (null list) (seq (list nil))
      (concat! (powerset (cdr list))
               (map! (lambda (sub) (cons (car list) sub))
                     (powerset (cdr list))))))

> (collect (powerset '()))

> (collect (powerset '(1)))
(NIL (1))

> (collect (powerset '(1 2)))
(NIL (2) (1) (1 2))

> (collect (powerset '(1 2 3)))
(NIL (3) (2) (2 3) (1) (1 3) (1 2) (1 2 3))

> (collect (powerset '(1 2 3 4)))
(NIL (4) (3) (3 4) (2) (2 4) (2 3) (2 3 4) (1) (1 4) (1 3) (1 3 4) (1 2)
 (1 2 4) (1 2 3) (1 2 3 4))

Note: The above is just for demonstration purposes and is much slower than a version you'd write without generators. Something like:

(defun powerset (xs) 
  (if (null xs) (list nil)
      (let ((subsets (powerset (cdr xs))))
        (append subsets
                (mapcar (lambda (subset) (cons (car xs) subset))

The "vanilla CL" version is MUCH faster, but may not be possible if you're generating powersets of very long lists. The trade off, as usual, is between speed and memory.

A Kind Of Grep

> (defun grepper (pattern file)
   (filter! (lambda (idx-line) (search pattern (second idx-line)))
            (zip! (range) (file-lines file))))

> (for (idx line) (grepper "defun" "examples.lisp")
     (format t "~4a: ~a~%" idx line))

12  : (defun prime-p (n)
19  : (defun all-primes ()
37  : (defun fibs ()
52  : (defun fill-and-insert (idx elem vec buffer)
69  : (defun thread-through (elem vec)
86  : (defun perms (vec)
104 : ;; (defun perms (vec)
115 : (defun grepper (pattern file)


GTWIWTG is a tiny library for creating and using generators.

If you have never heard of generators before, let me offer a definition, but not the definition.

For the purposes of this library, a generator is an object that can produce a series of values, one value at a time. Generators are sometimes convenient when you want to deal with series that are too long to fit into memory. They also help when you want to generate sequential data using recurrence relations, as in the Fibonacci example above.

Three Kinds Of Function

In GTWIWTG, there are three kinds of functions.

  1. functions that construct generators
  2. functions that combine generators
  3. functions and macros that consume generators.

The Breadwinning Constructors

The two most common generator constructors are:

  • (range &key (from 0) to (by 1) inclusive)
  • (seq sequence)

Here are some examples using range and seq to make generators.

  ;; all positive integers starting at 0
> (range)


 ;; positive integers from 0 to 9
> (range :to 10)


;; positive integers from 0 to 10 
> (range :to 10 :inclusive t) 


;; numbers between 4.0 and -15.7 incremented by -0.44
> (range :from 4 :to -15.7 :by -0.44) 


;; the characters in the string "hello"
> (seq "hello")


;; the symbols in the list
> (seq '(h e l l o))


;; the symbols in the vector 
> (seq #('h 'e 'l 'l 'o))


As you can see, generators are objects. Nothing is generated until you consume a generator. As a quick, but greatly impoverished, example, consider this:

;; get the first 4 numbers from the range starting at 20
> (take 4 (range :from 20))

(20 21 22 23)

Other Constructors

Here is a brief listing of the other generator constructors in GTWIWTG:

  • (times n) is shorthand for (range :to n)
  • (repeater &rest args) repeats its arguments in order, looping forever.
  • (noise &optional (arg 1.0)) an infinite sequence of random numbers
  • (from-thunk thunk) an infinite sequence of calls to (funcall thunk)
  • (from-thunk-until thunk &optional until clean-up) like from-thunk, but stops when (funcall until) is non nil. Runs the thunk clean-up when done.
  • (from-thunk-times thunk n) like from-thunk but stops after n times.
  • (from-recurrence fn n-1 &rest n-m) generate using a recurrence relation
  • (from-input-stream stream reader) turn a stream into a generator
  • (file-lines file) a file-backed generator. Produces lines from that file (strings)
  • (file-chars file) a file-backed generator. Produces characters from that file.
  • (file-bytes file) a file-backed generator. Produces bytes from that file.

You can see some of these in action in the examples section at the top of this document.

The Combination and Transformation Functions

You can create more intersting and more specific generators by using a few higher-order functions to combine and transform simple generators.

These transformations are desirable because they can be performed before any elements are produced.

That is, if you think of a generator as a computation that produces a series of values, then transformation functions allow you to incrementally "build up" a desired computation before it is run.

The three core transformation functions are:

  • (map! fn gen &rest gens) makes a new generator by mapping fn over other generators
  • (filter! pred gen) makes a new generator by discarding values that dont satisfy pred
  • (inflate! fn gen) The function fn should make new generators using the values produced by the generator gen. The inflate! function combines all those "intermediate" generators into a single generator.

Admittedly, the behavior of inflate! is difficult to grok by reading a description. Once you begin to use it, however, it becomes indispensible.

[NB: inflate! is really a kind of monadic bind operator in disguise.]

Here are some simple examples of their use:

;; map cons over two generators
> (map! #'cons (times 3) 
               (range :from 8))


;; consuming the above using collect
> (collect (map! #'cons (times 3) (range :from 8)))

((0 . 8) (1 . 9) (2 . 10))

;; Notice that map! stops generating after 3 steps even though 
;; (range :from 8) is an infinite generator. This is because (times 3)
;; only generates 3 values.

;; get just the even values from a generator: 
> (collect (filter! #'evenp (times 10)))

(0 2 4 6 8)

;; generate (times N) for each N in the range 1 to 4
> (for x (inflate! #'times (range :from 1 :to 4 :inclusive t))
    (when (zerop x) (terpri))
    (princ x) (princ #\Space))

0         ; (times 1)
0 1       ; (times 2)
0 1 2     ; (times 3)
0 1 2 3   ; (times 4)

The Other Combinations and Transformations

  • (zip! gen1 &rest gens) is shorthand for (map! #'list gen1 gen2 ...)
  • (indexed! gen) is shorthand for (zip! (range) gen)
  • (concat! gen &rest gens) concatenates generators
  • (skip! n gen) produces a generator by skipping the first n values in gen
  • (skip-while! pred gen) produces a generator by skipping elements of gen while pred is t
  • (merge! comp gen1 gen2 &rest gens) emulates the behavior of merge but for generators
  • (truncate! n gen) produces at most n of the values produced by gen
  • (inject! fn gen) shorthand for (map! (lambda (x) (funcall fn x) x) gen)
  • (intersperse! gen1 gen2 &rest gens) returns a generator that intermingles the values of its argument generators, in the order they appear in the argument list.

A Word Of Warning!

(Or, there's a reason those forms all end in !.)

You must be cautious when incrementally building up generators. The reason for caution is that generators cannot be "combined twice". If you are storing intermediate generators in a let binding, for example, you may be tempted to pass those bound variables into generator combination functions more than once. If you do, an error will be signalled.

The general rule is: if you pass a generator to more than one combining function (those whose names end in !), or if you pass the same generator to one such function at two argument positions, then an error will be raised and new the generator will not be built.

Internally, the library keeps track of whether or not generators have been combined with others. Don't quote me on it, but I think that the library will prevent you from making generators with surprising (i.e. erroneous) behavior.

Here is an example to show you the illegal behavior:

> (let ((ten-times (times 10)))
    (zip! ten-times ten-times))

; Evaluation aborted on #<SIMPLE-ERROR "~@<The assertion ~S failed~:[.~:; ~
                                           with ~:*~{~S = ~S~^, ~}.~]~:@>" {10046A61D3}>.

The gist is that we tried to zip a generator with itself. Such behavior is not allowed.

An ongoing goal is to make those errors nicer to look at so that you can more easily pin-point where you goofed.

The Fundamental Consumer

Finally! Once you have built up your generators using constructors and combinations, you want to actually use them for something. This is where consumers come in.

There is one fundamental consumer, a macro, called for. (Triumphant Horns Play)

Every other consumer in GTWIWTG uses for under the hood.

Here is how it looks when you use it:

> (for x (times 3)
    (print x))


> (for (x y) (zip! (seq "hello") (range))
    (format t "~a -- ~a~%" x y)
    (when (= 4 y) 
      (princ "world!")

h -- 0
e -- 1
l -- 2
l -- 3
o -- 4

> (let* ((ten-times (times 10))
         (doubled (map! (lambda (x) (* 2 x)) ten-times))
         (incremented (map! #'1+ doubled))
         (indexed (zip! (range) incremented)))
    (for (index number) indexed
      (princ index) 
      (princ " -- ") 
      (princ number) 

0 -- 1
1 -- 3
2 -- 5
3 -- 7
4 -- 9
5 -- 11
6 -- 13
7 -- 15
8 -- 17
9 -- 19

As you can see for has 3 basic parts: a binding form, a generator form, and a body.

The binding form is either a variable, like x above, or is a form suitable for use in the binding form of a DESTRUCTURING-BIND, like (x y) above.

On each iteration, the variables in the binding form are bound to successive values generated by the generator form. Notice that you do not need to inline your generator form, you can build it up and pass it in as in the third example above.

Finally, the body is evaluated for each iteration.

[Aside: for used to be called iter, but I didn't want to step on the toes of series and iterate users :P].

Generators are Consumed at Most Once

Even if you don't think you're "using up" the whole generator, a generator can only be passed to a single consumer. Once that consumer finishes, the generator is consumed. Here is an example:

>(let ((foo (seq "foobar")))
   (print (take 2 foo))
   (print (collect foo)))

(#\f #\o) 

Even though you only seemed to use the first two members of the generator foo, the take form will mark the generator as having been consumed in its entirety.

That is, even when the whole sequence was not actually generated, a consuming form leaves its generator in an unusable state. This approach has been taken in order to automatically close streams for stream-backed generators - i.e. it has been done in the spirit of letting you not have to think about how generators work.

You need only remember the rule: Generators Are Consumed At Most Once.

But Resumable Generators are Possible

An exception to the above comes in the form of resumable generators. To make a resumable generator call (make-resumable! <gen>) on a generator. Once you have passed a resumable generator to a consuming form you can still get some values out of it by passing it to resume!, which will create a brand new generator that picks up where the old one left off.


> (defvar *resumable-evens* 
           (make-resumable! (filter! 'evenp (range :from 1))))

> (take 10 *resumable-evens* )
(2 4 6 8 10 12 14 16 18 20)

> (setf *resumable-evens* (resume! *resumable-evens*))

> (take 10 *resumable-evens*)
(22 24 26 28 30 32 34 36 38 40)

The Accumulating Consumer

The next most common consuming form is fold, which lets you consume values produced by a generator while accumulating some data along the way.

Here is how you would do a classic summing operation:

> (fold (sum 0) (x (times 10)) 
        (+ sum x))

The syntax is (fold (acc init) (iter-var gen) update).

First, you declare and initialize an accumulator variable. In the above that is the form (sum 0), which declares a variable called sum initialized to 0.

Next comes your iteration variable and generator form. These have the same syntax as for. So in the above we bind a variable x to each successive value generated by (times 10).

Finally, you write a single update form whose value becomes bound to your accumulator variable. In the above example sum is set to (+ sum x).

The fold form returns the final value of the accumulator.

Here are some more folds:

;; some funky calculation 

> (fold (acc 0)
        ((x y) (zip! (times 10) (range :by -1)))
        (sqrt (+ acc (* x y))))
#C(0.444279 8.986663)

;; Example: building a data structure

> (fold (plist nil) 
        ((key val)
         (zip! (seq '(:name :occupation :hobbies))
               (seq '("buckaroo banzai" 
                      ("neuroscience" "particle physics" "piloting fighter jets")))))
         (cons key (cons val plist)))

 (:HOBBIES ("neuroscience" "particle physics" "piloting fighter jets")
  :OCCUPATION "rocker" :NAME "buckaroo banzai")

The Remaining Consumers

All of the remaining consumers are regular functions that have been built using for and fold. They are:

  • (collect gen) collects the values of gen into a list
  • (take n gen) collects the first n values of gen into a list
  • (pick-out indices gen) see example below
  • (size gen) consumes a generator, returning the number of values it produced
  • (maximum gen) returns the maximum among the values in gen (subject to change)
  • (minimum gen) see maximum
  • (average gen) returns the average of the values produced by gen
  • (argmax fn gen) returns a pair (val . x) where val is the value of gen for which (funcal fn val) is maximal. x is (funcall fn val)
  • (argmin fn gen) see argmax

The pick-out consumer is interesting enough to see a quick example of:

;; pick out characters and index 1 and index 4
> (pick-out '(1 4) (seq "generators"))
(#\e #\r)

;; you can do this in any order
> (pick-out '(4 1) (seq "generators"))
(#\r #\e)

;; you can even repeat indices
> (pick-out '(4 1 1 4 2) (seq "generators"))
(#\r #\e #\e #\r #\n)

Anaphoric Consumer Macros

If you would like to use for and fold macros with a little less visual noise (but sacrificing some of their flexibility), you can use the gtwiwtg.anaphora package. Here's an example:

> (use-package :gtwiwtg)          ;; gets you the core package
> (use-package :gtwiwtg.anaphora) ;; gets you the two extra anaphoric consumers

;; ordinary for
> (for x (times 3) (print x))


;; anaphoric for
> (afor (times 3) (print it))    ;; the variable IT is provided by AFOR

;; ordinary fold
> (fold (sum 0) (x (times 10)) (+ sum x))

;; anaphoric fold
> (afold 0 (times 10) (+ acc it))  ;; variables IT and ACC are provided by AFOLD

Making New Generators

Generators are subclasses of gtwiwtg::generator! that have at least two methods specialized on them:

  • (gtwiwtg::next gen) : advances the generator and gets its next value
  • (gtwiwtg::has-next-p gen) : checks whether or not the generator has a next value

Additionally, if your generator needs to perform cleanup after it is consumed, you can implement the :after method combination for the method

  • (gtwiwtg::stop gen) : is called by consumers to mark the generator as stopped.

None of the above are meant to be called by users of the library, which is why they are not exported symbols. But if you want to make your own generators you can.

A silly example:

> (defclass countdown (gtwiwtg::generator!)
    ((value :accessor countdown-value 
            :initarg :value 
            :initform 0)))

> (defmethod gtwiwtg::next ((g countdown))
    (decf (countdown-value g)))
> (defmethod gtwiwtg::has-next-p ((g countdown))
    (plusp (countdown-value g)))
;; you might also want a constructor 

> (defun countdown (n) (make-instance 'countdown :value n))

;; now you can use it:

> (for x (countdown 4) (print x))

You can see that next ASSUMES that there is a next value. This is one of the reasons you are not ment to call next manually. The for consumer automatically checks that there is a next value before trying to get it.

The Naughty Consumer

Now that the mysteries that make generators go have been explained in the previous section, you may be tempted to manually call next and has-next-p on your generators. If you must do this, you should use the with-generator macro:

> (with-generator (gen (seq "a1b2c3"))
     (when (gtwiwtg::has-next-p gen)
       (princ (gtwiwtg::next gen))

The with-generator form will ensure that the generator is properly closed. It could be useful with generators backed by input streams that need a custom logic, or perhaps in some case where you need to interleave operations between multiple generators. I'm not sure if you ever will need it, but the library provides it just in case.

The Permutations Example

One final example to show you what you can do. Here is a function that generates all of the permutations of a sequence passed to it, one at a time. It is a good example of the usefulness of inflate!.

(defun perms (vec)
  "Creates a generator that produces all of the permutations of the
   vector VEC, one at a time."
  (if (= 1 (length vec)) (seq (list vec))
      (let ((elem (elt vec 0))
            (subperms (perms (make-array (1- (length vec))
                                         :displaced-to vec         ; share vec's memory
                                         :displaced-index-offset 1
                                         :element-type (array-element-type vec)))))
        (inflate! (lambda (subperm) (thread-through elem subperm)) 

The basic flow is:

  1. single out the first element of the vector
  2. make a generator for permutations of the remainder of the vector
  3. return a generator that "adds back" the singled out element at each possible spot in each permutation.

The interesting bit about this is that we recursively compute permutation generators for the subvectors of vec in a classic divide-and-conquer way, and then use inflate! to combine those "generated sub-generators" into a single generator, which we return.

The above code is made significantly noisier by the use of displaced arrays. Displaced arrays let us share memory with the original vector.

For each "sub permutation", we create a new generator using a generator constructor called thread-through. This is the part where we "add back" the singled out element.

(defun thread-through (elem vec)
  "Creates a generator that produces a series of N vectors of length
   N, where N is one greater than the length of VEC.  The vectors
   produced by this generator have the same contents as VEC but have ELEM
   inserted at each possible spot, N spots in all. 

   Note: The generator reuses the memory that it returns on each step. If
   you intend to collect the values of the generator, you should copy
   them on each iteration."
  (let ((buffer (concatenate 'vector vec (list elem)))) ;; reusable buffer
    (map! (lambda (idx)
            (fill-and-insert idx elem vec buffer)
          (range :from 0 :to (length vec) :inclusive t))))

And this function uses a utility function called fill-and-insert that just fills a buffer, which I pulled out into its own function for clarity:

(defun fill-and-insert (idx elem vec buffer)
  "A utilty function that modifies BUFFER. 

The length of BUFFER is assumed to be one greater than the length of

This function fills the first IDX fields of BUFFER with the first IDX
fields of VEC. It fills the field of BUFFER at IDX with ELEM. And it fills
the remaining fields of BUFFER with the remaining fields of VEC.

  (loop :for i :below (length buffer)
     :when (= i idx) :do (setf (aref buffer idx) elem)
     :when (< i idx) :do (setf (aref buffer i)
                               (aref vec i))
     :when (> i idx) :do (setf (aref buffer i)
                               (aref vec (1- i))))  )

And here's a quick demo of its use:

(for perm (perms "abcd") 
  (print (concatenate 'string perm)))


We could have generated all 121645100408832000 permutations of "generators are cool", and, though it would have taken us an eternity (a little more than 1000 years on a single core of my machine), the memory consumption would stay at an even keel.

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