APL THINKING: EXAMPLES
Murray Eisenberg
Mathematics h Statistics Dept.
University of Massachusetts
Amherst, MA 01003 IJSA
(413) 545 - 2859
Abstract
In an effort to understand ‘APL thinking”,
we examine a few selected examples of using
APL to solve specific problems, namely:
compute the median of a numerical vector;
simulate the Replicate function; string
search: carry forward work-to-be-done in
excess of capacity: rotate concentric
rectangular rings in a matrix: find column
indices of pivots in an echelon matrix.
These examples are drawn from our teaching
experience as well as from APL literature.
We are particularly interested in studyinq
thinking processes underlying alternative
solutions to such problems -- i.e., our
goal is to ‘get inside the head” of the APL
programmer, Analyses include rWXnStrUCtin9
thoughts, comparing alternative approaches,
and, in general, scrutinizing supposed
characteristics of APL thinking.
Introduction
Is there such a thing as “APL thinking”? If
so, what is it? When a programmer claims to
solve a problem in an ‘APL way’, We must
ask what is meant by that. And, when
someone implies that the APL way is better,
or that APL is a better language for
thinking about problems, we must ask how
APL helps (or hinders) problem-solving.
Specifically, just what is unique of
special or different about thinking with
APL? Such questions motivate us to study
how APL affects cognitive processes. This
should enable us to teach APL more effectively
and to promulgate its advantages.
In an effort to understand “APL thinking”,
we began by focussing on views within the
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Howard A. Peelle
School of Education
[Jniversity of Massachusetts
Amherst, MA 01003 IJSA
(413) 545 - 0135
APL community: we have scanned the APL
literature for definitions of or references
to APL thinkino: surveved ooinions on APL
thinking (in A&., Quote&ad,‘nec. 1985, and
by questionnaire at APL86 1 : interviewed
students, instructors, and professional
programmers: and analyzed APL learning bugs
and APL teaching bugs [1,21. Additionally,
we have gathered opinions in response to
our talks on the subject at the NY/SIGAPL
“APL as a Tool of Thought’ conference
(April 1986) and at the New England APL
IJser’s Group meeting (May 1986) and have
conducted a panel discussion at APL86.
We are still at an early stage of studying
APL thinking. So far, we have fpund that
there is a wide variety of interpretations
extant. To some eeople, “APL thinking”
suggests one-liners (expressions or defined
functions). To others it means array
(parallel) processing, thereby avoiding
branching, iteration, and recursion:
combining cases: working with large chunks:
or doing data-driven computing. Some
believe it lies in the notation -- syntax
and the symbols themselves -- or in concise
APL expressions, or in rich and powerful
primitives. some relate it to modularizing
code (epitomized by direct definition), or
a “qlass box” program, or the propensity to
generalize. A few attest that it involves
use of identities and proofs in formal
reasoning. It might also suggest tricky,
contorted programming techniques. Further,
it might even involve imagery, metaphors,
and visualization of mental representations
and transformations. Perhaps these are all
aspects of APL thinking.
We have also recognized a number of related
issues and challenges, including: the
matter of style, (e.g., use of idioms); how
to arrange studies (which seem to be low
priority in business and academia);
different APLs (e.g., APL2): distinguishing
between problem-solving and formula
translation; identifying problems which
make APL look good or bad; finding reliable
methods to get inside someone’s head and
reveal his or her thoughts; considering
what the programmer knows beforehand; how
to analyze protocols effectively; and
433
whether APL thinkinq differs among
cultures. Nevertheless, we are perserverinq
to focus on underlying cognitive processes.
Accordingly, here we are studying solutions
to selected problems which purportedly
involve APL thinking. Rather than our
supposing what APL thinking is in general,
these examples will have to speak for
themselves. The examples are drawn from our
experience teaching as well as from APL
literature; they are necessarily small in
scope but are intended to be provocative.
We are deliberately exposing thinking
processes -- au naturel.
Examples
1. The Median Problem
The problem is to write an AFL expression
or define a function to compute the median
of a vector of numbers. The median is, by
defin.ition, the middle value of the vector
when arranged in ascending (or descending)
order: if there is an even number of
numbers, however, the median is the average
of the two middle values.
At the onset, we note that this is not
really a ‘problem’ in the strict sense,
since the algorithm is included in the
problem statement itself. Nonetheless, in
the process of creating alternative
expressions in APL, there are opportunities
for varied ways of thinking.
A straightforward, case-by-case approach
directly translates the problem statement
into APL code:
Q Z+MEDIAN X;N
Cl1 Z+XC 4x1
[21 N+o. 5xpX
c31 -+(0=2(pX)/EVGN
II41 ODD: .z+z[rNl
:51 -’ c
C6 1 EVEN: 2+-O. S~Z[Nl+ZCNtIl
Q
BY contrast, the followinq “solution”,
which has become an old chestnut in the APL
community, is not so obvious:
0.sXtixcc4x)cIr-0.5 0.5xltpxJl
(from [31, p. 328)
APL novices often consider this as an
example of “bizarre” thinking. They tend to
make remarks such as “I wouldn’t have
thought of doing it that way”, or even,
“I’ll never be able to think like that!”
The question is: What thinking was involved
in doing it this way? (Would the creator
please identify himself or herself?) In the
absence of any known explanation of how the
above expression was originally conceived,
we can only speculate that it was based on
a deliberate attempt to combine two cases
graduate student, who explained that it ,‘was
an example of ‘APL thinking” because it
solved the problem in an unusual way --
434
into a single expression -- ‘the APL way”
-- which entailed forming arithmetically a
vector index of two middle values and, for
the sake of efficiency , indexing the
permutation vector directly.
Incidentally, as we compa c e these
solutions, we should note that we are not
considering which expression best conveys
the concept of median nor which is most
efficient in space and time,
Other variations, perhaps designed for
greater clarity, include the following.
One which breaks the code into steps:
Q Z+MEDIAN X;N
ClJ X+XC PX 1
c21 N+O. 5xltpX
C31 Z+O. 5xXI: LN3tXCfNl
V
(from 141, p.125)
One which uses subfunctions SORT, MEAN, and
ROUND :
V MIDDLE+MEDI AN DATA
El3 DATA+SURT DATA
c27 MIDDLE+MEAN DATACROUh’D (0 ftpDATA)+21
Q
V UP+SORT DATA
Cl3 UP+DATA[ ADATA
V
V CENTER+MEAN DATA
Cl3 CENTER+(tlDATA)ipDATA
Q
V OFFtROUND N
Cl1 OPF+l.NtO. 5
V
(from 151, p.274)
Both amalgamate the even and odd cases by
duplicating the middle value in the odd
case and then averaging the two values.
A different approach isolates the middle
value(s) by dropping the unneeded fr.ont and
back of the sorted vector;
Q ANS+MEDIAN DATA;A
CIJ DATA+DATAC ADATA
c23 A+L-0.5ttpDATAls2
c31 DATAeASt -A)SDATA
c41 ANS’-( t /DATA) ;( pDATA)
Q
(by a student)
Yet another approach involves reproducing
the entire vector:
MEAN (SORT DATA,DATA)CO ltpDATA1
This was of feted spontaneously by a