\documentstyle[12pt]{article}

\newcommand{\lisp}{\tt}
\newcommand{\iter}{{\lisp iterate}}
\newcommand{\Iter}{{\lisp Iterate}}
\newcommand{\Do}{{\lisp do}}
\newcommand{\Doing}{{\lisp doing}}
\newcommand{\looP}{{\lisp loop}}
\newcommand{\LooP}{{\lisp Loop}}
\newcommand{\setf}{{\lisp setf}}
\newcommand{\iman}{{\em The Iterate Manual\/}}

\begin{document}


\title{Don't Loop, Iterate}
\author{Jonathan Amsterdam}

\maketitle

\section{Introduction}

Above all the wonders of Lisp's pantheon stand its metalinguistic
tools; by their grace have Lisp's acolytes been liberated from the
rigid asceticism of lesser faiths.  Thanks to Macro and kin, the
jolly, complacent Lisp hacker can gaze through a fragrant cloud of
setfs and defstructs at the emaciated unfortunates below, scraping out
their meager code in inflexible notation, and sneer superciliously.
It's a good feeling.

But all's not joy in Consville.  For---I beg your pardon, but---there
really is no good way to {\em iterate\/} in Lisp.  Now, some are happy
to map their way about, whether for real with {\lisp mapcar} and
friends, or with the make-believe of Series; others are so satisfied
with \Do\ it's a wonder they're not C hackers.\footnote{Hey, don't get
mad---I'll be much more polite later, when the real paper starts.}
Still others have gotten by with \looP, but are getting tired of
looking up the syntax in the manual over and over again.  And in the
elegant schemes of some, only tail recursion and lambdas figure.  But
that still leaves a sizeable majority of folk---well, me, at
least---who would simply like to {\em iterate,} thank you, but in a
way that provides nice abstractions, is extensible, and looks like
honest-to-God Lisp.

In what follows I describe a macro package, called \iter, that
provides the power and convenient abstractions of \looP\ but in a more
syntactically palatable way.  \iter\ also has many features that
\looP\ lacks, like generators and better support for nested loops.
\iter\ generates inline code, so it's more efficient than using the
higher-order function approach.  And \iter\ is also extensible---it's
easy to add new clauses to its vocabulary in order to express new
patterns of iteration in a convenient way.

\iter\ is fully documented in AI Lab Memo No. 1236, \iman.

\section{More about \Iter}

A Common Lisp programmer who wonders what's lacking with present-day
iteration features would do well to consider \setf.  Of course,
\setf\ doesn't iterate, but it has some other nice properties.  It's
easy to use, for one thing.  It's extensible---you can define new
\setf\ methods very easily, so that \setf\ will work with new forms.
\setf\ is also efficient, turning into code that's as good as anyone
could write by hand.  Arguably, \setf\ provides a nice abstraction:
it allows you to view value-returning forms, like {\lisp (car ...)} or
{\lisp (get ...)} as locations that can be stored into.  Finally and
most obviously, \setf\ {\em looks\/} like Lisp; it's got a syntax
right out of {\lisp setq}.

\iter\ attempts to provide all of these properties.  Here is a simple
use of \iter\ that returns all the elements of {\lisp num-list}
that are greater than three:

\pagebreak[4]

\begin{verbatim}
(iterate (for el in num-list)
         (when (> el 3)
           (collect el)))
\end{verbatim}

An \iter\ form consists of the symbol \iter\ followed by some Lisp
forms.  Any legal Lisp form is allowed, as well as certain forms that
\iter\ treats specially, called {\em clauses}.  {\lisp for...in} and
{\lisp collect} are the two clauses in the above example.  An \iter\
clause can appear anywhere a Lisp form can appear; \iter\ walks its
body, looking inside every form, processing \iter\ clauses when it
finds them.  It even expands macros, so you can write macros that
contain \iter\ clauses.  Almost all clauses use the syntax of function
keyword-argument lists: alternating keywords and arguments.  \iter\
keywords don't require a preceding colon, but you can use one if you
like.

\iter\ provides many convenient iteration abstractions, most of
them familiar to \looP\ users.  Iteration-driving clauses (those
beginning with {\lisp for}) can iterate over numbers, lists, arrays,
hashtables, packages and files.  There are clauses for collecting
values into a list, summing and counting, maximizing, finding maximal
elements, and various other things.  Here are a few examples, for
extra flavor.

\pagebreak[2]
To sum a list of numbers:
\begin{verbatim}
(iterate (for i in list)
         (sum i))
\end{verbatim}

\pagebreak[2]
To find the length of the shortest element in a list:
\begin{verbatim}
(iterate (for el in list)
         (minimize (length el)))
\end{verbatim}

\pagebreak[2]
To find the shortest element in a list:
\begin{verbatim}
(iterate (for el in list)
         (finding el minimizing (length el)))
\end{verbatim}

\pagebreak[2]
To return {\lisp t} only if every other element of a list is odd:
\begin{verbatim}
(iterate (for els on list by #'cddr)
         (always (oddp (car els))))
\end{verbatim}

\pagebreak[2]
To split an association list into two separate lists (this example
uses \iter's ability to do destructuring):
\begin{verbatim}
(iterate (for (key . item) in alist)
         (collect key into keys)
         (collect item into items)
         (finally (return (values keys items))))
\end{verbatim}


\section{Comparisons With Other Iteration Methods}

As with any aspect of coding, how to iterate is a matter of taste.  I
do not wish to dictate taste or even to suggest that \iter\ is a
``better'' way to iterate than other methods.  I would, however, like
to examine the options, and explain why I prefer \iter\ to its
competitors.

\subsection{\Do, {\lisp dotimes} and {\lisp dolist}}

The \Do\ form has long been a Lisp iteration staple.  It provides for
binding of iteration variables, an end-test, and a body of arbitrary
code.  It can be a bit cumbersome for simple applications, but
the most common special cases---iterating over the integers
from zero and over the members of a list---appear more conveniently as
{\lisp dotimes} and {\lisp dolist}.

\Do's major problem is that it provides no abstraction.  For example,
collection is typically handled by binding a variable to {\lisp nil},
pushing elements onto the variable, and {\lisp nreverse}ing the
result before returning it.  Such a common iteration pattern should be
easier to write.  (It is, using {\lisp mapcar}---but see below.)

Another problem with \Do, for me at least, is that it's hard to read.
The crucial end-test is buried between the bindings and the body,
marked off only by an extra set of parens (and some indentation).  It
is also unclear, until after a moment of recollection, whether the
end-test has the sense of a ``while'' or an ``until.''

Despite its flaws, \Do\ is superior to the iteration facilities of
every other major programming language except CLU.  Perhaps that is
the reason many Lisp programmers don't mind using it.


\subsection{Tail Recursion}
% (check hyphenation)

Tail-recursive implementations of loops, like those found in Scheme
code \cite{SchemeBook}, are parsimonious and illuminating.  They have
the advantage of looking like recursion, hence unifying the notation
for two different types of processes.  For example, if only
tail-recursion is used, a loop that operates on list elements from
front to back looks very much like a recursion that operates on them
from back to front.

However, using tail-recursion exclusively can lead to cumbersome code
and a proliferation of functions, especially when one would like to
embed a loop inside a function.  Tail-recursion also provides no
abstraction for iteration; in Scheme, that is typically done with
higher-order functions.


\subsection{Higher-order Functions}

Lisp's age-old mapping functions, recently revamped for Common Lisp
\cite{CLM}, are another favorite for iteration.  They provide a
pleasing abstraction, and it's easy to write new higher-order
functions to express common iteration patterns.  Common Lisp already
comes with many such useful functions, for removing, searching, and
performing reductions on lists.  Another Common Lisp advantage is that
these functions work on any sequence---vectors as well as lists.

One problem with higher-order functions is that they are inefficient,
requiring multiple calls on their argument function.  While the the
built-ins, like {\lisp map} and {\lisp mapcar}, can be open-coded,
that cannot be so easily done for user-written functions.  Also, using
higher-order functions often results in the creation of intermediate
sequences that could be avoided if the iteration were written out
explicitly. 

The second problem with higher-order functions is very much a matter
of personal taste.  While higher-order functions are theoretically
elegant, they are often cumbersome to read and write.  The unpleasant
sharp-quote required by Common Lisp is particularly annoying here, and
even in Scheme, I find the presence of a lambda with its argument list
visually distracting.

Another problem is that it's difficult to express iteration of
sequences of integers without creating such sequences explicitly as
lists or arrays.  One could resort to tail-recursion or {\lisp
dotimes}---but then it becomes very messy to  express double
iterations where one driver is over integers.  Multiple iteration is
easy in \iter, as illustrated by the following example, which creates
an alist of list elements and their positions: 

{\samepage
\begin{verbatim}
(iterate (for el in list)
         (for i from 0)
         (collect (cons el i)))
\end{verbatim}
}


\subsection{Streams and Generators}

For really heavy-duty iteration jobs, nothing less than a
coroutine-like mechanism will do.  Such mechanisms hide the state of
the iteration behind a convenient abstraction.  A {\em generator\/} is
a procedure that returns the next element in the iteration each time
it is called.  A {\em stream\/} (in the terminology of
\cite{SchemeBook}) is a data structure which represents the iteration,
but which computes the next element only on demand.  Generators and
streams support a similar style of programming.  Here, for example, is
how you might enumerate the leaves of a tree (represented as a Lisp
list with atoms at the leaves) using streams:
\begin{verbatim}
(defun tree-leaves (tree)
  (if (atom tree)
      (stream-cons tree empty-stream)
      (stream-append (tree-leaves (car tree)) 
                     (tree-leaves (cdr tree)))))
\end{verbatim}

Although {\lisp tree-leaves} looks like an ordinary recursion, 
it will only do enough work to find the first leaf before returning.
The stream it returns can be accessed with {\lisp stream-car}, which
will yield the (already computed) first leaf of the tree, or with
{\lisp stream-cdr}, which will initiate computation of the next leaf.

Such a computation would be cumbersome to write using \iter, or any of
the other standard iteration constructs; in fact, it is not even
technically speaking an iteration, if we confine that term to
processes that take constant space and linear time.  Streams, then,
are definitely more powerful than standard iteration machinery.

Unfortunately, streams are very expensive, since they must somehow
save the state of the computation.  Generators are typically cheaper,
but are less powerful and still require at least a function call.  
So these powerful tools should be used only when necessary, and that
is not very often; most of the time, ordinary iteration suffices.

There is one aspect of generators that \iter\ can capture, and that is
the ability to produce elements on demand.  Say we wish to
create an alist that pairs the non-null elements of a list with the
positive integers.  We saw above that it is easy to iterate over
a list and a series of numbers simultaneously, but here we would like
to do something a little different: we want to iterate over the list
of elements, but only draw a number when we need one (namely, when a list
element is non-null). The solution employs the \iter\ {\lisp generate}
keyword in place of {\lisp for} and the special clause
{\lisp next}: 

{\samepage
\begin{verbatim}
(iterate (for el in list)
         (generate i from 1)
         (if el
             (collect (cons el (next i)))))
\end{verbatim}
}

Using {\lisp next} with any driver variable changes how that driver
works.  Instead of supplying its values one at a time on each iteration,
the driver computes a value only when a {\lisp next} clause is
executed.  This ability to obtain values on demand greatly increases
\iter's power.  Here, {\lisp el} is set to the next element of the
list on each iteration, as usual; but {\lisp i} is set only when 
{\lisp (next i)} is executed.

\subsection{Series}

Richard C. Waters has developed a very elegant notation called Series
which allows iteration to be expressed as sequence-mapping somewhat in
the style of APL, but which compiles to efficient looping code
\cite{Series}.

My reasons for not using Series are, again, matters of taste.  Like many
elegant notations, Series can be somewhat cryptic.  Understanding what a
Series expression does can require some effort until one has mastered
the idiom.  And if you wish to share your code with others, they will
have to learn Series as well.  \iter\ suffers from this problem to some
extent, but since the iteration metaphor it proposes is much more
familiar to most programmers than that of Series, it is considerably
easier to learn and read.

\subsection{{\lisp Prog} and {\lisp Go}}

Oh, don't be silly.


\subsection{\LooP}

\looP\ is the iteration construct most similar to \iter\ in
appearance.  \looP\ is a macro written originally for MacLisp and in
widespread use \cite{Loop}.  It has been adopted as part of Common
Lisp.  \looP\ provides high-level iteration with abstractions for
collecting, summing, maximizing and so on.  Recall our first \iter\
example:

\pagebreak[2]
\begin{verbatim}
(iterate (for el in num-list)
         (when (> el 3)
           (collect el)))
\end{verbatim}

\pagebreak[2]
Expressed with \looP, it would read
n
\begin{verbatim}
(loop for el in list
      when (> el 3)
        collect el)
\end{verbatim}

The similarity between the two macros should immediately be apparent.
Most of \iter's clauses were borrowed from \looP.  But compared to
\iter, \looP\ has a paucity of parens.  Though touted as more readable
than heavily-parenthesized code, \looP's Pascalish syntax creates
several problems.  First, many dyed-in-the-wool Lisp hackers simply
find it ugly.  Second, it requires learning the syntax of a whole new
sublanguage.  Third, the absence of parens makes it hard to parse,
both by machine and, more importantly, by human.  Fourth, one often
has to consult the manual to recall lesser-used aspects of the strange
syntax.  Fifth, there is no good interface with the rest of Lisp, so
\looP\ clauses cannot appear inside Lisp forms and macros cannot
expand to pieces of \looP.  And Sixth, pretty-printers and indenters
that don't know about \looP\ will invariably display it wrongly.  This
is particularly a problem with program-editor indenters.  A reasonably
clever indenter, like that of Gnu Emacs, can indent nearly any normal
Lisp form correctly, and can be easily customized for most new forms.
But it can't at present handle \looP.  The syntax of \iter\ was
designed to keep parens to a minimum, but conform close enough to Lisp
so as not to confuse code-display tools.  Gnu Emacs indents \iter\
reasonably with no modifications.

Indenting is a mere annoyance; \looP's lack of extensibility is a more
serious problem.  The original \looP\ was completely extensible, but
the Symbolics version only provides for the definition of new
iteration-driving clauses, and the Common Lisp specification does not
have any extension mechanism.  But extensibility is a boon.  Consider
first the problem of adding the elements of a list together, which can
be accomplished with \iter\ by

\pagebreak[3]
\begin{verbatim}
(iterate (for el in list)
         (sum el))
\end{verbatim}

and in \looP\ with

\begin{verbatim}
(loop for el in list
      sum el)
\end{verbatim}

But now, say that you wished to compute the sum of the square roots of the
elements.  You could of course write, in either \looP\ or \iter,

\begin{verbatim}
(iterate (for el in list)
         (sum (sqrt el)))
\end{verbatim}

But perhaps you find yourself writing such loops often enough to make
it worthwhile to create a new abstraction.  There is nothing you can
do in \looP, but in \iter\ you could simply write a macro:

\begin{verbatim}
(defmacro (sum-of-sqrts expr &optional into-var)
  `(sum (sqrt ,expr) into ,into-var))
\end{verbatim}

{\lisp sum-of-sqrts} is a perfectly ordinary Lisp macro. Since
\iter\ expands all macros and processes the results, {\lisp
(sum-of-sqrts el)} will behave exactly as if we'd written {\lisp
(sum (sqrt el))}.

There's also a way to define macros that use \iter's clause syntax.
It's fully documented in \iman.

%Just to beat a dead horse, I'd like to point out that there's no way
%to define {\lisp for...maximizing} in 
%\looP.\footnote{In fact, it was in part the frustration of knowing
%that \looP\ could generate code to maximize a value, but could not be
%easily altered to supply the element associated with that maximum,
%that prompted me to write \iter.}

\section{Implementation}

A Common Lisp implementation of \iter\ has existed for well over a year.
It runs under Lucid for HP 300's, Sun 3's and SPARCstations, and on
Symbolics Lisp machines.  See \iman\ for details.

\section{Conclusion}

Iteration is a matter of taste.  I find \iter\ more palatable than
other iteration constructs: it's more readable, more efficient than
most, provides nice abstractions, and can be extended.  

If you're new to Lisp iteration, start with \iter---look before you
\looP.  If you're already using \looP\ and like the power that it
offers, but have had enough of its syntax and inflexibility, then my
advice to you is, don't \looP---\iter.

\pagebreak

\begin{flushleft}
\bf Acknowledgements
\end{flushleft}

Thanks to David Clemens for many helpful suggestions and for the
egregious pun near the end.  Conversations with Richard Waters prompted
me to add many improvements to \iter.  Alan Bawden, Sundar Narasimhan,
and Jerry Roylance also provided useful comments.  David Clemens and
Oren Etzioni shared with me the joys of beta-testing.

\begin{thebibliography}{9}

\bibitem{SchemeBook} Abelson, Harold and Gerald Jay Sussman. {\em
Structure and Interpretation of Computer Programs.} Cambridge, MA: The
MIT Press, 1985.


\bibitem{Loop} ``The loop Iteration Macro.'' In {\em Symbolics Common
Lisp---Language Concepts}, manual 2A of the Symbolics documentation,
pp. 541--567.

\bibitem{CLM} Steele, Guy L.  {\em Common Lisp: The Language}.
Bedford, MA: Digital Press, 1984.

\bibitem{Series} Waters, Richard C.  {\em Optimization of Series
Expressions: Part I: User's Manual for the Series Macro Package}.  MIT
AI Lab Memo No. 1082.

\end{thebibliography}

\end{document}



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