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\null
\vfill
\centerline{\titlefont A torture test for \logo ()*+,-.*}
\vskip 18pt
\centerline{by Donald E. Knuth}
\centerline{Stanford University}
\vskip 6pt
\centerline{({\sl Version 0, August 1984\/})}
\vfill
\centerline{\vbox{\hsize 4in
\noindent Programs that claim to be implementations of \MF84 are
supposed to be able to process the test routine contained in this
report, producing the outputs contained in this report.}}
\vskip 24pt
{\baselineskip 9pt
\eightrm\noindent
The preparation of this report was supported in part by the National Science
Foundation under grants IST-8201926 and MCS-8300984,
and by the System Development Foundation.

}\pageno=0\eject

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\section Introduction.
People often think that their programs are ``debugged'' when large applications
have been run successfully. But system programmers know that a typical large
application tends to use at most about 50 per cent of the instructions
in a typical compiler. Although the other half of the code---which tends
to be the ``harder half''---might be riddled with errors, the system seems
to be working quite impressively until an unusual case shows up on the
next day. And on the following day another error manifests itself, and so on;
months or years go by before certain parts of the compiler are even
activated, much less tested in combination with other portions of the system,
if user applications provide the only tests.

How then shall we go about testing a compiler? Ideally we would like to
have a formal proof of correctness, certified by a computer.
This would give us a lot of confidence,
although of course the formal verification program might itself be incorrect.
A more serious drawback of automatic verification is that the formal
specifications of the compiler are likely to be wrong, since they aren't
much easier to write than the compiler itself. Alternatively, we can
substitute an informal proof of correctness: The programmer writes his or
her code in a structured manner and checks that appropriate relations
remain invariant, etc. This helps greatly to reduce errors, but it cannot
be expected to remove them completely; the task of checking a large
system is sufficiently formidable that human beings cannot do it without
making at least a few slips here and there.

Thus, we have seen that test programs are unsatisfactory if they are simply
large user applications; yet some sort of test program is needed because
proofs of correctness aren't adequate either. People have proposed schemes
for constructing test data automatically from a program text, but such
approaches run the risk of circularity, since they cannot assume that a
given program has the right structure.

I have been having good luck with a somewhat different approach,
first used in 1960 to debug an {\mc ALGOL} compiler. The idea is to
construct a test file that is about as different from a typical user
application as could be imagined. Instead of testing things that people
normally want to do, the file tests complicated things that people would
never dare to think of, and it embeds these complexities in still
more arcane constructions. Instead of trying to make the compiler do the
right thing, the goal is to make it fail (until the bugs have all been found).

To write such a fiendish test routine, one simply gets into a nasty frame
of mind and tries to do everything in the unexpected way. Parameters
that are normally positive are set negative or zero; borderline cases
are pushed to the limit; deliberate errors are made in hopes that the
compiler will not be able to recover properly from them.

A user's application tends to exercise 50\%\ of a compiler's logic,
but my first fiendish tests tend to improve this to about 90\%. As the
next step I generally make use of frequency-counting software to identify
the instructions that have still not been called upon. Then I add ever more
fiendishness to the test routine, until more than 99\%\ of the code
has been used at least once. (The remaining bits are things that
can occur only if the source program is really huge, or if certain
fatal errors are detected; or they are cases so similar to other well-tested
things that there can be little doubt of their validity.)

Of course, this is not guaranteed to work. But my experience in 1960 was
that only two bugs were ever found in that {\mc ALGOL} compiler after it
correctly translated that original fiendish test. And one of those bugs
was actually present in the results of the test; I simply had failed to
notice that the output was incorrect. Similar experiences occurred later
during the 60s and 70s, with respect to a few assemblers, compilers,
and simulators that I wrote.

This method of debugging, combined with the methodology of structured
programming and informal proofs (otherwise known as careful desk checking),
leads to greater reliability of production software than any other
method I know. Therefore I have used it in developing \MF84, and the
main bulk of this report is simply a presentation of the test program
that was used to get the bugs out of \MF.

Such a test file is useful also after a program has been debugged, since
it can be used to give some assurance that subsequent modifications don't
mess things up.

The test file is called \.{TRAP.MF}, because of my warped sense of humor:
\MF's companion system, \TeX, has a similar test file called \.{TRIP}, and I
couldn't help thinking about Billy Goat Gruff and the story of ``trip,
trap, trip, trap.''

The contents of this test file are so remote from what people actually
do with \MF, I feel apologetic if I have to explain the correct
translation of \.{TRAP.MF}; nobody really cares about most of the
nitty-gritty rules that are involved. Yet I believe \.{TRAP} exemplifies
the sort of test program that has outstanding diagnostic ability, as
explained above.

If somebody claims to have a correct implementation of \MF, I will not
believe it until I see that \.{TRAP.MF} is translated properly.
I propose, in fact, that a program must meet two criteria before it
can justifiably be called \MF: (1)~The person who wrote it must be
happy with the way it works at his or her installation; and (2)~the
program must produce the correct results from \.{TRAP.MF}.

\MF\ is in the public domain, and its algorithms are published;
I've done this since I do not want to discourage its use by placing
proprietary restrictions on the software. However, I don't want
faulty imitations to masquerade as \MF\ processors, since users
want \MF\ to produce identical results on different machines.
Hence I am planning to do whatever I can to suppress any systems that
call themselves \MF\ without meeting conditions (1) and~(2).
I have copyrighted the programs so that I have some chance to forbid
unauthorized copies; I explicitly authorize copying of correct
\MF\ implementations, and not of incorrect ones!

The remainder of this report consists of appendices, whose contents ought
to be described briefly here:

Appendix A explains in detail how to carry out a test of \MF, given
a tape that contains copies of the other appendices.

Appendix B is \.{TRAP.MF}, the fiendish test file that has already
been mentioned. People who think that they understand \MF\ are challenged
to see if they know what \MF\ is supposed to do with this file.
People who know only a little about \MF\ might still find it
interesting to study Appendix~B, just to get some insights into the
methodology advocated here.

Appendix C is \.{TRAPIN.LOG}, a correct transcript file \.{TRAP.LOG}
that results if \.{INIMF} is applied to \.{TRAP.MF}. (\.{INIMF} is
the name of a version of \MF\ that does certain initializations;
this run of \.{INIMF} also creates a binary base file called \.{TRAP.BASE}.)

Appendix D is a correct transcript file \.{TRAP.LOG} that results if
\.{INIMF} or any other version of \MF\ is applied to \.{TRAP.MF}
with format \.{TRAP.FMT}.

Appendix E is \.{TRAP.TYP}, the symbolic version of a correct output
file \.{TRAP.72270GF} that was produced at the same time as the \.{TRAP.LOG}
file of Appendix~D.

Appendix F is \.{TRAP.PL}, the symbolic version of a correct output
file \.{TRAP.TFM} that was produced at the same time as the \.{TRAP.LOG}
file of Appendix~D.

Appendix G is \.{TRAP.FOT}, an abbreviated version of Appendix D that
appears on the user's terminal during the run that produces \.{TRAP.LOG},
\.{TRAP.72270GF}, and \.{TRAP.TFM}.

The debugging of \MF\ and the testing of the adequacy of \.{TRAP.MF}
could not have been done nearly as well as reported here except for
the magnificent software support provided by my colleague David R. Fuchs.
In particular, he extended our local Pascal compiler so that
frequency counting and a number of other important features were added
to its online debugging abilities.

The method of testing advocated here has one chief difficulty that deserves
comment: I had to verify by hand that \MF\ did the right things
to \.{TRAP.MF}. This took many hours, and perhaps I have missed
something (as I did in 1960); I must confess that I have not checked
every single number in Appendices D, E, and~F. However, I'm willing to pay
$\$$5.12 to the first finder of any remaining bug in \MF, and I will
be surprised if that bug doesn't show up also in Appendix~D.

\vfill\eject

\section Appendix A: How to test \MF.

\item{0.} Let's assume that you have a tape containing \.{TRAP.MF},
\.{TRAPIN.LOG}, \.{TRAP.LOG}, \.{TRAP.TYP}, \.{TRAP.PL}, and \.{TRAP.TYP},
as in Appendices B, C, D, E, F, and~G. Furthermore, let's suppose that you
have a working \.{WEB} system, and that you have working programs
\.{TFtoPL} and \.{GFtype}, as described in the \TeX ware and \MF ware reports.

\item{1.} Prepare a version of \.{INIMF}. (This means that your \.{WEB}
change file should have {\bf init} and {\bf tini} defined to be null.)
The {\bf debug} and {\bf gubed} macros should be null, in order to
activate special printouts that occur when $\\{tracingedges}>1.0$.
The {\bf stats} and {\bf tats} macros should also be null, so that
statistics are kept. Set \\{mem\_top} and \\{mem\_max} to 3000
(or to \\{mem\_min} plus 3000, if \\{mem\_min} isn't zero),
for purposes of this test version.
Also set $\\{error\_line}=64$, $\\{half\_error\_line}=32$,
$\\{max\_print\_line}=72$, $\\{screen\_width}=100$, and
$\\{screen\_depth}=200$; these parameters affect many of the lines of
the test output, so your job will be much easier if you use the same
settings that were used to produce Appendix~E. Also set
$\\{gf\_buf\_size}=8$, since this tests more parts of the program.
You probably should also use the ``normal'' settings of other parameters
found in \.{MF.WEB} (e.g., $\\{max\_internal}=50$, $\\{buf\_size}=500$,
etc.), since these show up in a few lines of the test output. Finally,
change \MF's screen-display routines by putting the following simple lines
in the change file:
$$\vbox{\halign{\tt#\hfil\cr
\char`\@x Screen routines:\cr
begin init\char`\_screen:=false;\cr
\char`\@y\cr
begin init\char`\_screen:=true;
 \char`\{screen instructions will be logged\char`\}\cr
\char`\@z\cr}}$$
None of the other screen routines (\\{update\_screen}, \\{blank\_rectangle},
\\{paint\_row}) should be changed in any way; the effect will be to have
\MF's actions recorded in the transcript files instead of on the screen,
in a machine-independent way.

\item{2.} Run the \.{INIMF} prepared in step 1. In response to the first
`\.{**}' prompt, type carriage return (thus getting another `\.{**}').
Then type `\.{\char`\\input trap}'. You should get an output that matches
the file \.{TRAPIN.LOG} (Appendix~C). Don't be alarmed by the error
messages that you see, unless they are different from those in Appendix~C.

\def\sp{{\char'40}}
\item{3.} Run \.{INIMF} again. This time type `\.{\sp\&trap\sp\sp trap\sp}'.
(The spaces in this input help to check certain parts of \MF\ that
aren't otherwise used.) You should get outputs \.{TRAP.LOG}, \.{TRAP.72270GF},
and \.{TRAP.TFM}.
Furthermore, your terminal should receive output that matches \.{TRAP.FOT}
(Appendix~G). During the middle part of this test, however, the terminal
will not be getting output, because \.{batchmode} is being
tested; don't worry if nothing seems to be happening for a while---nothing
is supposed to.

\item{4.} Compare the \.{TRAP.LOG} file from step 3 with the ``master''
\.{TRAP.LOG} file of step~0. (Let's hope you put that master file in a
safe place so that it wouldn't be clobbered.) There should be perfect
agreement between these files except in the following respects:

\itemitem{a)} The dates and possibly the file names will
naturally be different.

\itemitem{b)} If you had different values for \\{stack\_size}, \\{buf\_size},
etc., the corresponding capacity values will be different when they
are printed out at the end.

\itemitem{c)} Help messages may be different; indeed, the author encourages
non-English help messages in versions of \MF\ for people who don't
understand English as well as some other language.

\itemitem{d)} The total number and length of strings at the end and/or
``still untouched'' may well be different.

\itemitem{e)} If your \MF\ uses a different memory allocation or
packing scheme, the memory usage statistics may change.

\itemitem{f)} If you use a different storage allocation scheme, the
capsule numbers will probably be different, but the order of variables
should be unchaged when dependent variables are shown. \MF\ should also
choose the same variables to be dependent.

\itemitem{g)} If your computer handles integer division of negative operands
in a nonstandard way, you may get results that are rounded differently.
Although \TeX\ is careful to be machine-independent in this regard,
\MF\ is not, because integer divisions are present in so many places.

\item{5.} Use \.{GFtype} to convert your file \.{TRAP.72270GF} to a file
\.{TRAP.TYP}. (Both of \.{GFtype}'s options, i.e., mnemonic output and image
output, should be enabled so that you get the maximum amount of output.)
The resulting file should agree with the master \.{TRAP.TYP} file of step~0,
assuming that your \.{GFtype} has the ``normal'' values of compile-time
constants ($\\{top\_pixel}=69$, etc.).

\item{6.} Use \.{TFtoPL} to convert your file \.{TRAP.TFM} to a file
\.{TRAP.PL}. The resulting file should agree with the master \.{TRAP.PL}
file of step~0.

\item{7.} You might also wish to test \.{TRAP} with other versions of
\MF\ (i.e., \.{VIRMF} or a production version with another base file
preloaded). It should work unless \MF's primitives have been redefined in
the base file. However, this step isn't essential, since all the code of
\.{VIRMF} appears in \.{INIMF}; you probably won't catch any more errors
this way, unless they would already become obvious from normal use of
the~system.

\vfill\eject

\section Appendix B: The \.{TRAP.MF} file.
The contents of the test routine are prefixed here with line numbers, for
ease in comparing this file with the error messages printed later; the
line numbers aren't actually present.
\runninghead{APPENDIX B: \.{TRAP.MF} (CONTINUED)}

\vskip 8pt
\begingroup\count255=0
\everypar{\global\advance\count255 by 1
  \hbox to 20pt{\sevenrm\hfil\the\count255\ \ }}
\verbatim{trap.mf}
\endgroup
\vfill\eject

\section Appendix C: The \.{TRAPIN.LOG} file.
When \.{INIMF} makes the \.{TRAP.BASE} file, it also creates a file called
\.{TRAP.LOG} that looks like this.
\runninghead{APPENDIX C: \.{TRAPIN.LOG} (CONTINUED)}

\vskip8pt
\verbatim{trapin.log}
\vfill\eject

\section Appendix D: The \.{TRAP.LOG} file.
Here is the major output of the \.{TRAP} test; it is generated by running
\.{INIMF} and loading \.{TRAP.BASE}, then reading \.{TRAP.MF}.
\runninghead{APPENDIX D: \.{TRAP.LOG} (CONTINUED)}

{\let\tt=\eighttt\leftskip 1in\baselineskip 9pt plus .1pt minus .1pt
\vskip8pt
\verbatim{trap.log}
}
\vfill\eject

\section Appendix E: The \.{TRAP.TYP} file.
Here is another major component of the test. It shows the output of \.{GFtype}
applied to the file \.{TRAP.72270GF} that is created at the same time
Appendix D was produced.
\runninghead{APPENDIX E: \.{TRAP.TYP} (CONTINUED)}

{\let\tt=\eighttt\leftskip 1in\baselineskip 9pt plus .1pt minus .1pt
\vskip8pt
\verbatim{trap.typ}
}
\vfill\eject

\section Appendix F: The \.{TRAP.PL} file.
In this case we have the output of \.{TFtoPL}
applied to the file \.{TRAP.TFM} that is created at the same time
Appendix D was produced.
\runninghead{APPENDIX F: \.{TRAP.PL} (CONTINUED)}

{\let\tt=\eighttt\leftskip 1in\baselineskip 9pt plus .1pt minus .1pt
\vskip8pt
\verbatim{trap.pl}
}
\vfill\eject

\section Appendix G: The \.{TRAP.FOT} file.
This shows what appeared on the terminal while Appendix D was being produced.
\runninghead{APPENDIX G: \.{TRAP.FOT} (CONTINUED)}

\vskip8pt
\verbatim{trap.fot}

\vfill\end
