232: Flexible curried procedures232: Flexible curried proceduresby Wolfgang Corcoran-Mathe, based on a design by Jason Hemann and Daniel P. Friedman.
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Scheme lacks a flexible way to create and apply curried procedures.
This SRFI describes curried, a variant of
lambda that creates true curried procedures which also
behave just like ordinary Scheme procedures. They can be applied to
their arguments one by one, all at once, or anywhere in between,
without any novel syntax. curried also supports
nullary and variadic procedures, and procedures created with it have
predictable behavior when applied to surplus arguments.
Currying is a well-known tactic in functional programming, and
many languages (e.g. Standard ML and Haskell) provide convenient ways to
define curried procedures. True
currying, though, is somewhat uncommon in Scheme. Beyond the fundamental
building block of first-class procedures, Scheme provides little support
for defining curried procedures, and the clumsiness of applying such
procedures may deter programmers from using them at all. Some attempts
have been made at improving the situation. The
“higher-order lambda” notation of
SRFI 219
extends the syntax of define with a simple
shorthand for writing curried procedures. The complexity of applying
curried procedures remains, however:
(define ((add-c x) y)
(+ x y))
(add-c 2 3) ⇒ ; error: too many arguments
((add-c 2) 3) ⇒ 5
Perhaps because of this clumsiness, Scheme has another form which
provides something a little like currying. The cut
operator, introduced by
SRFI 26, is not a
currying operator; instead, it allows procedures to be easily
specialized on given parameters. However, this is not too difficult
in ordinary Scheme (cf. Al* Petrofsky's
comment
to the SRFI 26 mailing list), and, as such, cut is not
frequently seen in Scheme programs. cut also introduces a
novel notation using the tokens <> and
<...>, which may pose an obstacle for new users.
The curried form described in this SRFI allows the
creation of true curried procedures which can also be treated as
normal, uncurried Scheme procedures. For example, a
curried procedure of two arguments may be applied to one
argument to return a new procedure of one argument, or directly to two
arguments to return its value. curried also provides
consistent semantics for nullary and variadic procedures, and
predictable behavior for cases in which a procedure is
applied to more arguments than it accepts. All this is accomplished
without the introduction of any novel forms; syntactically,
curried is very similar to lambda, which should
make it easy to adopt in existing Scheme programs.
This SRFI is not meant to replace SRFIs 26 or 219. In some
cases, these existing techniques may be preferable; the
SRFI 219 notation does much less "under the table" than does
curried, and cut provides greater flexibility
in argument specialization. The curried form provides
another option, one that will hopefully also prove useful and
convenient.
(curried ⟨formals⟩ ⟨body⟩)
Syntax: ⟨formals⟩ and ⟨body⟩ are as described in section 4.1.4 of the R⁷RS.
Semantics: If ⟨formals⟩ is an empty list, then a
curried expression evaluates to ⟨body⟩. Otherwise, it
evaluates to a procedure with the same environment semantics as
lambda (R⁷RS 4.1.4). The application of a procedure
created by curried has the following semantics:
If ⟨formals⟩ is of the form (v₁ v₂ … vₙ), then
If the procedure is applied to k arguments, where
k < n, then “partial application”
is performed: the result is a procedure with curried
semantics with an extended environment in which the formal
parameters v₁, v₂, …, vₖ are
bound to the supplied arguments in order; this procedure can
accept further arguments as described in this section.
Note that, if k = 0, the result is a procedure which is extensionally the same as the original procedure.
If the procedure is applied to k arguments, where k > n, then ⟨body⟩ is evaluated with the formal parameters v₁, v₂, …, vₙ bound to the first n arguments, and the result is applied to the remaining k − n arguments. It is an error if the value of ⟨body⟩ is not a procedure.
If the procedure is applied to exactly n arguments,
then evaluation proceeds as with lambda.
If ⟨formals⟩ is of the form (v₁ … vₙ . vₙ₊₁), then application proceeds as in the proper list case, except that, if n or more arguments are supplied, then the formal parameters v₁, …, vₙ are bound to the first n arguments and vₙ₊₁ is bound to a list of any remaining arguments, in order.
If ⟨formals⟩ is of the form ⟨variable⟩ (a single identifier),
then application proceeds as if the procedure were created with
lambda.
(define-curried (⟨variable⟩ ⟨formals⟩) ⟨body⟩)
Equivalent to
(define⟨variable⟩(curried⟨formals⟩ ⟨body⟩))
Here, curried is used to create procedures which can
be concisely parameterized:
(define-curried (add* x y) (+ x y))
(map (add* 2) '(1 2 3))
⇒ (3 4 5)
(define-curried (fold* proc base lis)
(fold proc base lis))
(let ((sum (fold* + 0))
(product (fold* * 1))
(lis '(1 2 3 4 5)))
(list (sum lis) (product lis)))
⇒ (15 120)
“Polyvariadic” refers to a procedure denoted by an
expression of the form
(lambda (v₁ … vₙ . vₙ₊₁) ⟨body⟩).
A polyvariadic procedure created with curried can be
partially applied until a procedure of at least one argument
remains:
(define foo (curried (a b . rest) (list a b rest)))
((foo 1) 2 3 4) ⇒ (1 2 (3 4))
((foo 'a) 'b) ⇒ (a b ())
When a curried procedure is applied to more arguments
than it accepts, the remaining arguments are passed to the procedure
body:
((curried (a)
(curried (b)
(curried (c) (+ a b c)))) 1 2 3) ⇒ 6
Since nullary curried expressions evaluate to their
bodies, application “passes through” any number of
such expressions:
((curried () (curried (x y) (+ x y))) 2 3)
⇒ 5
(((curried (x)
(curried ()
(curried (y z)
(list x (* y z))))) 4 5) 6)
⇒ (4 30)
The sample implementation is portable to R⁶RS and R⁷RS
syntax-rules implementations and can be found
in the repository for this SRFI.
curried is based heavily on the lambda*
form created by Jason Hemann and Daniel P. Friedman and described
elegantly in “λ*: Beyond Currying”. Their work is
gratefully acknowledged.
Thanks to the SRFI editor and to those who provided feedback on
this SRFI via the mailing list or the #scheme IRC
channel.
Jason Hemann & Daniel P. Friedman, “λ*: Beyond Currying” (2013). Available on the Web.
© 2022 Wolfgang Corcoran-Mathe.
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