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Lazy sequences (or lseqs, pronounced "ell-seeks") are a generalization of lists. In particular, an lseq is either a proper list or a dotted list whose last cdr is a SRFI 121 generator. A generator is a procedure that can be invoked with no arguments in order to lazily supply additional elements of the lseq. When a generator has no more elements to return, it returns an end-of-file object. Consequently, lazy sequences cannot reliably contain end-of-file objects.
This proposal provides a set of procedures suitable for operating on lazy sequences based on SRFI 1.
Lazy sequences are more heavyweight than generators, on which they are based, but they are more lightweight than SRFI 41 streams. However, streams are even, as explained in the SRFI 41 rationale; that is, the initial state of a stream does not have any elements that have already been realized. By contrast, lazy sequences are odd, meaning that at least one element is realized at all times unless the lseq is empty. Therefore, when constructing an lseq in an iterative lazy algorithm, only the cdr side of the lazy pair is lazily evaluated; the car side is evaluated immediately, even if it is never used.
In most cases this doesn't matter, because calculating one additional item is a negligible overhead. However, when you create a self-referential lazy structure, in which the earlier elements of a sequence are used to calculate the latter elements of itself, a bit of caution is needed; code that is valid for circular streams may not terminate if it is mechanically converted to use lazy sequences. This eagerness is also visible when side effects are involved; for example, a lazy character sequence reading from a port may read one character ahead.
This proposal is less comprehensive than SRFI 1, because it omits
many procedures that process every element of their list arguments (at least,
when used in the absence of call/cc). Lseqs are meant
to be used with ordinary Scheme functions, which are strict; consequently,
neither left nor right folds are able to work on infinite sequences,
so the only difference between folding over an lseq and realizing it as
a list and using list fold is space efficiency (for which reason
lseq-fold is provided). The
linear-update procedures of SRFI 1 are also left out, as lazy sequences
are not intended to be mutated.
Here is a short list of the procedures provided by this SRFI.
generator->lseq
lseq? lseq=
lseq-car lseq-cdr lseq-first lseq-second lseq-third lseq-fourth lseq-fifth lseq-sixth lseq-seventh lseq-eighth lseq-ninth lseq-tenth lseq-rest lseq-ref lseq-take lseq-drop lseq-split-at
lseq-realize lseq->generator lseq-length lseq-append lseq-concatenate lseq-zip lseq-unzip1 lseq-unzip2 lseq-unzip3 lseq-unzip4 lseq-unzip5
lseq-map lseq-fold lseq-reduce lseq-for-each lseq-pair-for-each
lseq-member lseq-memq lseq-memv lseq-find lseq-find-rest lseq-any lseq-every lseq-index lseq-take-while lseq-drop-while lseq-span lseq-break
lseq-assoc lseq-assq lseq-assv
lseq-comparator make-lseq-comparator
Except as noted, if any of these procedures accepts multiple lseq arguments, then at least one of them must be finite: that is, it must either be a proper list, or contain a generator that eventually returns an end-of-file object.
The templates given below obey the following conventions for procedure formals:
| lseq | A lazy sequence |
|---|---|
| x, y, a, b | Any value |
| object, value | Any value |
| n, i | A natural number (an integer >= 0) |
| proc | A procedure |
| pred | A procedure whose return value is treated as a boolean |
| generator | A procedure with no arguments that returns a sequence of values |
| = | A boolean procedure taking two arguments |
To interpret the examples, pretend that they are executed on a Scheme that prints lazy sequences with the syntax of lists.
Every list constructor procedure is also an lseq constructor procedure.
The procedure generator->lseq constructs an lseq based on the
values of a generator. In order to prepend a realized value to a generator,
simply use cons; to prepend more than one value, use SRFI 1's
cons*.
generator->lseq generator -> lseq
Returns an lseq whose elements are the values generated by generator. The exact behavior is as follows:
(generator->lseq (make-repeating-generator 'c)) => (c c c ...)
lseq? x -> boolean
Returns #t iff x is an lseq, otherwise #f.
This procedure may return #t if x is an improper list
whose last car is a procedure that requires arguments, since there is no
portable way to examine a procedure to determine how many arguments it requires.
lseq= elt= lseq1 ... -> boolean
Determines lseq equality, given an element-equality procedure.
The lseq A equals the lseq B
if they are of the same length,
and their corresponding elements are equal,
as determined by elt=.
If the element-comparison procedure's first argument is
from lseqi,
then its second argument is from lseqi+1,
i.e. it is always called as
(elt= a b)
for a an element of lseq A,
and b an element of lseq B.
In the n-ary case,
every lseqi is compared to
lseqi+1
(as opposed, for example, to comparing
lseq1 to lseqi,
for i>1).
If there are no lseq arguments at all,
lseq= simply returns true.
The dynamic order in which the elt= procedure is
applied to pairs of elements is not specified.
For example, if lseq= is applied
to three lseqs, A, B, and C,
it may first completely compare A to B,
then compare B to C,
or it may compare the first elements of A and B,
then the first elements of B and C,
then the second elements of A and B, and so forth.
The equality procedure must be consistent with eq?.
That is, it must be the case that
(eq? x y) => (elt= x y).
Note that this implies that two lseqs which are eq?
are always lseq=, as well; implementations may exploit this
fact to "short-cut" the element-by-element comparisons.
(lseq= eq?) => #t ; Trivial cases (lseq= eq? '(a)) => #t
lseq-car lseq -> object
lseq-first lseq -> object
These procedures are synonymous.
They return the first element of lseq. They
are included for completeness, as they are the same as car.
It is an error to apply them to an empty lseq.
lseq-cdr lseq -> lseq
lseq-rest lseq -> lseq
These procedures are synonymous. They return an lseq with the contents of lseq except for the first element. The exact behavior is as follows:
Implementations that inline cdr are advised to inline lseq-cdr if
possible.
lseq-second lseq -> object
lseq-third lseq -> object
lseq-fourth lseq -> object
lseq-fifth lseq -> object
lseq-sixth lseq -> object
lseq-seventh lseq -> object
lseq-eighth lseq -> object
lseq-ninth lseq -> object
lseq-tenth lseq -> object
Returns the second, third, ..., or tenth element of lseq.
(lseq-third '(a b c d e)) => c
lseq-ref lseq i -> value
Returns the ith element of lseq.
(This is the same as
(lseq-first (lseq-drop lseq i)).)
It is an error if i >= n,
where n is the length of lseq.
(lseq-ref '(a b c d) 2) => c
lseq-take lseq i -> list
lseq-drop lseq i -> lseq
lseq-take returns the first i elements of lseq as a proper list.
lseq-drop returns all but the first i elements of lseq.
(lseq-take '(a b c d e) 2) => (a b) (lseq-drop '(a b c d e) 2) => (c d e)
lseq-drop is exactly equivalent to performing i lseq-rest operations on lseq.
lseq-split-at lseq i -> [list lseq]
Splits the lseq lseq at index i, returning two values, a list of the first i elements, and an lseq of the remaining elements. It is equivalent to
(values (lseq-take lseq i) (lseq-drop lseq i))
lseq-realize lseq -> list
Repeatedly applies lseq-cdr to lseq
until its generator (if there is one) has been exhausted,
and returns lseq, which is now
guaranteed to be a proper list. This
procedure can be called on an arbitrary lseq before passing
it to a procedure which only accepts lists. However, if the
generator never returns an end-of-file
object, lseq-realize will never return.
lseq->generator lseq -> generator
Returns a generator which when invoked will return all the elements of lseq, including any that have not yet been realized.
lseq-length lseq -> integer
Returns the length of its argument, which is the non-negative integer n such that lseq-rest
applied n times to the lseq produces an empty lseq.
lseq-append lseq …Returns an lseq that lazily contains all the elements of the lseqs.
lseq-concatenate lseqThe lseq argument is an lseq of lseqs. Returns an lseq that whose elements are all the elements of the first lseq, then all the elements of the second one, then the third, etc.
It is similar to (apply lseq-append (lseq-realize lseq)), except
that lseq-concatenate can work even if lseq contains an infinite
number of lseqs.
lseq-zip lseq1 lseq2 ... -> lseq
If lseq-zip is passed n lseqs, it returns an lseq as long as the shortest
of these lseqs, each element of which is an n-element list comprised
of the corresponding elements from the arguments. The lseqs can be infinite, as long as they are not all infinite.
(lseq-zip '(one two three)
(generator->lseq (make-iota-generator +inf.0 1 1))
(generator->lseq (make-repeating-generator) '(odd even))))
=> ((one 1 odd) (two 2 even) (three 3 odd))
(lseq-zip '(1 2 3)) => ((1) (2) (3))
lseq-unzip1 list-of-lseqs -> lseq
lseq-unzip2 list-of-lseqs -> [lseq lseq]
lseq-unzip3 list-of-lseqs -> [lseq lseq lseq]
lseq-unzip4 list-of-lseqs -> [lseq lseq lseq lseq]
lseq-unzip5 lslist-of-lseqs -> [lseq lseq lseq lseq lseq]
lseq-unzip1 takes a list of lseqs,
where every lseq must contain at least one element,
and returns an lseq containing the initial element of each such lseq.
That is, it returns (lseq-map lseq-first lseqs).
lseq-unzip2 takes a list of lseqs, where every lseq must contain at least
two elements, and returns two values: an lseq of the first elements,
and an lseq of the second elements. lseq-unzip3 does the same for the first
three elements of the lseqs, and so forth.
(lseq-unzip2 (list '(1 one) '(2 two) '(3 three))) =>
(1 2 3)
(one two three)
lseq-fold kons knil lseq1 lseq2 ... -> value
The fundamental lseq iterator.
First, consider the single lseq-parameter case. If lseq1 = (e1 e2 ... en), then this procedure returns
(kons en ... (kons e2 (kons e1 knil)) ... )
That is, it obeys the (tail) recursion
(lseq-fold kons knil lis) = (lseq-fold kons (kons (lseq-first lis) knil) (lseq-last lis)) (lseq-fold kons knil '()) = knilExamples:
(lseq-fold + 0 lseq) ; Add up the elements of lseq.
(lseq-fold cons '() lseq) ; Reverse lseq.
;; How many symbols in lseq?
(lseq-fold (lambda (x count) (if (symbol? x) (+ count 1) count))
0
lseq)
;; Length of the longest string in lseq:
(lseq-fold (lambda (s max-len) (max max-len (string-length s)))
0
lseq)
If n lseq arguments are provided, then the kons function must take n+1 parameters: one element from each lseq, and the "seed" or fold state, which is initially knil. The fold operation terminates when the shortest lseq runs out of values:
(lseq-fold lseq* '() '(a b c) '(1 2 3 4 5)) => (c 3 b 2 a 1)
lseq-reduce f ridentity lseq -> value
lseq-reduce is a variant of lseq-fold.
ridentity should be a "right identity" of the procedure f — that is, for any value x acceptable to f,
(f x ridentity) = x
lseq-reduce has the following definition:
(lseq-fold f (lseq-first lseq) (lseq-last lseq)).
(lseq-fold f ridentity lseq).
Note that ridentity is used only in an empty-lseq case.
You typically use lseq-reduce when applying f is expensive and you'd
like to avoid the extra application incurred when lseq-fold applies
f to the head of lseq and the identity value,
redundantly producing the same value passed in to f.
For example, if f involves searching a file directory or
performing a database query, this can be significant.
In general, however, lseq-fold is useful
in many contexts where lseq-reduce is not
(consider the examples given in the lseq-fold definition — only one of the
folds uses a function with a right identity.
The other four may not be performed with lseq-reduce).
;; take the max of an lseq of non-negative integers. (lseq-reduce max 0 nums)
lseq-map proc lseq1 lseq2 ... -> lseq
proc is a procedure taking as many arguments
as there are lseq arguments and returning a single value.
lseq-map applies proc element-wise to the elements
of the lseqs and returns an lseq of the results,
in order.
The dynamic order in which proc
is applied to the elements of the lseqs is unspecified.
(lseq-map lseq-second '((a b) (d e) (g h))) => (b e h)
(lseq-map (lambda (n) (expt n n))
(make-iota-generator +inf.0 1 1)
=> (1 4 27 256 3125 ...)
(lseq-map + '(1 2 3) '(4 5 6)) => (5 7 9)
(let ((count 0))
(lseq-map (lambda (ignored)
(set! count (+ count 1))
count)
'(a b))) => (1 2) or (2 1)
lseq-for-each proc lseq1 lseq2 ... -> unspecified
lseq-pair-for-each proc lseq1 lseq2 ... -> unspecified
The arguments to lseq-for-each are like the arguments to
lseq-map, but
lseq-for-each calls proc for its side effects rather
than for its values.
Unlike lseq-map, lseq-for-each is guaranteed to call
proc on the elements of the lseqs in order from the first
element(s) to the last,
and the value returned by lseq-for-each is unspecified.
The procedure lseq-pair-for-each is the same as
lseq-for-each, except that it
calls proc on each pair rather than
each element.
(let ((v (make-vector 5)))
(lseq-for-each (lambda (lseq-)
(vector-set! v i (* i i)))
'(0 1 2 3 4))
v) => #(0 1 4 9 16)
The following procedures all search lseqs for the leftmost element satisfying some criteria.
lseq-find pred lseq -> value
Return the first element of lseq that satisfies predicate pred; false if no element does.
(lseq-find even? '(3 1 4 1 5 9)) => 4
Note that lseq-find has an ambiguity in its lookup semantics — if lseq-find
returns #f, you cannot tell (in general) if it found a #f element
that satisfied pred, or if it did not find any element at all. In
many situations, this ambiguity cannot arise — either the lseq being
searched is known not to contain any #f elements, or the lseq is
guaranteed to have an element satisfying pred. However, in cases
where this ambiguity can arise, you should use lseq-find-tail instead of
lseq-find — lseq-find-tail has no such ambiguity:
(cond ((lseq-find-tail pred lseq) => (lambda (lseq) ...))
(else ...)) ; Search failed.
lseq-find-tail pred lseq -> lseq or false
Return the longest tail of lseq whose first element satisfies pred. If no element does, return false.
lseq-find-tail can be viewed as a general-predicate variant of the lseq-member
function.
Examples:
(lseq-find-tail even? '(3 1 37 -8 -5 0 0)) => (-8 -5 0 0) (lseq-find-tail even? '(3 1 37 -5)) => #f ;; imember x lseq: (lseq-find-tail (lambda (elt) (equal? x elt)) lseq)
lseq-find-tail is essentially lseq-drop-while,
where the sense of the predicate is inverted:
lseq-find-tail searches until it finds an element satisfying
the predicate; lseq-drop-while searches until it finds an
element that doesn't satisfy the predicate.
lseq-take-while pred lseq -> lseq
Returns the longest initial prefix of lseq whose elements all satisfy the predicate pred.
(lseq-take-while even? '(2 18 3 10 22 9)) => (2 18)
lseq-drop-while pred lseq -> lseq
Drops the longest initial prefix of lseq whose elements all satisfy the predicate pred, and returns the rest of the lseq.
(lseq-drop-while even? '(2 18 3 10 22 9)) => (3 10 22 9)
lseq-span pred lseq -> [lseq lseq]
lseq-break pred lseq -> [lseq lseq]
lseq-span splits the lseq into the longest initial prefix whose
elements all satisfy pred, and the remaining tail.
lseq-break inverts the sense of the predicate:
the tail commences with the first element of the input lseq
that satisfies the predicate.
In other words:
lseq-span finds the initial span of elements
satisfying pred,
and lseq-break breaks the lseq at the first element
satisfying pred.
lseq-span is equivalent to
(values (lseq-take-while pred lseq)
(lseq-drop-while pred lseq))
(lseq-span even? '(2 18 3 10 22 9)) => (2 18) (3 10 22 9) (lseq-break even? '(3 1 4 1 5 9)) => (3 1) (4 1 5 9)
lseq-any pred lseq1 lseq2 ... -> value
Applies the predicate across the lseqs, returning true if the predicate returns true on any application.
If there are n lseq arguments lseq1 ... lseqn, then pred must be a procedure taking n arguments and returning a boolean result.
lseq-any applies pred to the first elements of the lseqi parameters.
If this application returns a true value, lseq-any immediately returns
that value. Otherwise, it iterates, applying pred to the second
elements of the lseqi parameters, then the third, and so forth.
The iteration stops when a true value is produced or one of the lseqs runs
out of values; in
the latter case, lseq-any returns #f.
The application of pred to the last element of the
lseqs is a tail call.
Note the difference between lseq-find and lseq-any — lseq-find returns the element
that satisfied the predicate; lseq-any returns the true value that the
predicate produced.
Like lseq-every, lseq-any's name does not end with a question mark — this is to
indicate that it does not return a simple boolean (#t or #f), but a
general value.
(lseq-any integer? '(a 3 b 2.7)) => #t
(lseq-any integer? '(a 3.1 b 2.7)) => #f
(lseq-any < '(3 1 4 1 5)
'(2 7 1 8 2)) => #t
lseq-every pred lseq1 lseq2 ... -> value
Applies the predicate across the lseqs, returning true if the predicate returns true on every application.
If there are n lseq arguments lseq1 ... lseqn, then pred must be a procedure taking n arguments and returning a boolean result.
lseq-every applies pred to the first elements of the lseqi parameters.
If this application returns false, lseq-every immediately returns false.
Otherwise, it iterates, applying pred to the second elements of the
lseqi parameters, then the third, and so forth. The iteration stops
when a false value is produced or one of the lseqs runs out of values.
In the latter case, lseq-every returns
the true value produced by its final application of pred.
The application of pred to the last element of the lseqs
is a tail call.
If one of the lseqi has no elements, lseq-every simply returns #t.
Like lseq-any, lseq-every's name does not end with a question mark — this is to
indicate that it does not return a simple boolean (#t or #f), but a
general value.
lseq-index pred lseq1 lseq2 ... -> integer or false
Return the index of the leftmost element that satisfies pred.
If there are n lseq arguments lseq1 ... lseqn, then pred must be a function taking n arguments and returning a boolean result.
lseq-index applies pred to the first elements of the lseqi parameters.
If this application returns true, lseq-index immediately returns zero.
Otherwise, it iterates, applying pred to the second elements of the
lseqi parameters, then the third, and so forth. When it finds a tuple of
lseq elements that cause pred to return true, it stops and returns the
zero-based index of that position in the lseqs.
The iteration stops when one of the lseqs runs out of values; in this
case, lseq-index returns #f.
(lseq-index even? '(3 1 4 1 5 9)) => 2 (lseq-index < '(3 1 4 1 5 9 2 5 6) '(2 7 1 8 2)) => 1 (lseq-index = '(3 1 4 1 5 9 2 5 6) '(2 7 1 8 2)) => #f
lseq-member x lseq [=] -> lseq
lseq-memq x lseq -> lseq
lseq-memv x lseq -> lseq
These procedures return the longest tail of lseq whose first element is
x, where the tails of lseq are the
non-empty lseqs returned by
(lseq-drop lseq i)
for i less than the length of lseq.
If x does
not occur in lseq, then #f is returned.
lseq-memq uses eq? to compare x
with the elements of lseq,
while lseq-memv uses eqv?, and
lseq-member uses =, which defaults to equal?.
(lseq-memq 'a '(a b c)) => (a b c)
(lseq-memq 'b '(a b c)) => (b c)
(lseq-memq 'a '(b c d)) => #f
(lseq-memq (lseq 'a) '(b (a) c)) => #f
(lseq-member (lseq 'a)
'(b (a) c)) => ((a) c)
(lseq-memq 101 '(100 101 102)) => *unspecified*
(lseq-memv 101 '(100 101 102)) => (101 102)
The comparison procedure is used to compare the elements ei of lseq to the key x in this way:
(= x ei) ; lseq is (E1 ... En)
That is, the first argument is always x, and the second argument is
one of the lseq elements. Thus one can reliably find the first element
of lseq that is greater than five with
(lseq-member 5 lseq <)
Note that fully general lseq searching may be performed with
the lseq-find-tail and lseq-find procedures, e.g.
(lseq-find-tail even? lseq) ; Find the first elt with an even key.
An "lazy association list" (or "lazy alist") is an lseq of pairs. The car of each pair contains a key value, and the cdr contains the associated data value. They can be used to construct simple look-up tables in Scheme. Note that lazy alists are probably inappropriate for performance-critical use on large data; in these cases, immutable maps or some other alternative should be employed.
lseq-assoc key lseq-alist [=] -> lseq or #f
lseq-assq key lseq-alist -> lseq or #f
lseq-assv key lseq-alist -> lseq or #f
These procedures
find the first pair in lseq-alist whose car field is key,
and returns that pair.
If no pair in lseq-alist has key as its car,
then #f is returned.
lseq-assq uses eq? to compare key
with the car fields of the ipairs in lseq-alist,
while lseq-assv uses eqv?
and lseq-assoc uses =, which defaults to equal?.
(define e '((a 1) (b 2) (c 3))) (lseq-assq 'a e) => (a 1) (lseq-assq 'b e) => (b 2) (lseq-assq 'd e) => #f (lseq-assq (lseq 'a) '(((a)) ((b)) ((c)))) => #f (lseq-assoc (lseq 'a) '(((a)) ((b)) ((c)))) => ((a)) (lseq-assq 5 '((2 3) (5 7) (11 13))) => *unspecified* (lseq-assv 5 '((2 3) (5 7) (11 13))) => (5 7)
The comparison procedure is used to compare the elements ei of lseq to the key parameter in this way:
(= key (lseq-first ei)) ; lseq is (E1 ... En)
(lseq-assoc 5 lseq-alist <)
Note that fully general lazy alist searching may be performed with
the lseq-find-tail and lseq-find procedures, e.g.
;; Look up the first association in lazy alist with an even key: (lseq-find (lambda (a) (even? (lseq-first a))) lazy alist)
lseq-pair-comparator
The lseq-comparator object is a SRFI-114 comparator suitable for comparing lseqs.
Note that it is not a procedure.
It compares lseqs using default-comparator on their first elements. If they are not equal, that value is returned. If they are equal, lseq-comparator is used on the lseq-rest of the lseqs and that value is returned.
make-lseq-comparator comparator -> comparator
The make-lseq-comparator procedure returns a comparator suitable for comparing lseqs
using element-comparator to compare the elements.
The files in the implementation are as follows:
FIXME
Without the work of Olin Shivers on SRFI 1, this SRFI would not exist. Everyone acknowledged there is transitively acknowledged here. This is not to imply that either Olin or anyone else necessarily endorses the final results, of course.