% CHANGES TO FASCICLE V4F6 OF THE ART OF COMPUTER PROGRAMMING
%
% Copyright (C) 2015, 2016, 2017 by Donald E. Knuth
% This file may be freely copied provided that no modifications are made.
% All other rights are reserved.
%
% Three levels of changes to the books are distinguished here:
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% "\bugonpage" introduces the correction of an error;
% "\amendpage" introduces new material for future editions;
% "\improvepage" introduces ameliorations of lesser importance.
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% (Changes introduced by \improvepage do not appear in the hardcopy listing.)
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%%%%%%%%%%%%%% opening remarks %%%%%%%%%%%%%%%%%%%%%%%%
\def\lhead{INTRODUCTION}
\let\rhead=\lhead
\titlepage
\volheadline{THE ART OF COMPUTER PROGRAMMING}
\bigskip
\volheadline{ERRATA TO VOLUME 4 FASCICLE 6}
\bigskip
\noindent This document is a transcript of the notes that I have been making
in my personal copy of {\sl The Art of Computer Programming}, Volume~4,
Fascicle~6, since it was first printed in 2015.
\ifall Four levels of updates\dash---``errors,'' ``amendments,'' ``plans,''
and ``improvements''\dash---appear, indicated by four
\else Three levels of updates\dash---``errors,'' ``amendments,'' and
``plans''\dash---appear, indicated by three \fi
different typographic conventions:
\begingroup\def\hundred{17}
\bugonpage 0.666 line 1 (76.07.04)
Technical or typographical errors (aka bugs) are the most critical items, so
they are flagged with a `\thinspace{\manfnt x}\thinspace' preceding the page
number. The date on which I first was told about the bug is shown; this is the
effective date on which I paid the finder's fee. The necessary
corrections are indicated in
a straightforward way. If,~for example, the book says
`$n$' where it should have said `$n+1$', the change is shown thus:
\smallskip
$n$ \becomes $n+1$
\endchange
\amendpage 0.666 line 2 (89.07.14)
Amendments to the text appear in the same format as bugs, but without
the~`\thinspace{\manfnt x}\thinspace'. These are things I wish I had known
about or thought of when I wrote the original text, so I added them later.
The date is the date I drafted the new text.
\endchange
\def\hundred{19}
\planforpage 0.666 line 3 (17.11.20)
Plans for the future represent a third kind of item. In such notes I~sketched
my intentions about things that I wasn't ready to flesh out further when
I~wrote them down. You can identify these items because they're written in
slanted type, and preceded by a bunch of dots
`\hbox to 6em{\leaders\hbox to 5pt{\hss.\hss}\hfill}' leading to the date on
which I recorded the plan in my files.
\endchange
\improvepage 0.666 line 4 (38.01.10)
The fourth and final category\dash---indicated by page and line number in
smaller, slanted type\dash---consists of minor corrections or improvements
that most readers don't want to know about, because they are so trivial.
You wouldn't even
be seeing these items if you hadn't specifically chosen to print the complete
errata list in all its gory details.
Are you sure you wanted to do that?
\endchange
\endgroup
\ifall\else\medskip\ninepoint
My personal file of updates also includes a fourth category of items, not
shown in this list. They are miscellaneous minor corrections or improvements
that most readers don't want to know about, because they are so trivial.
If you really want to see all of the gory details,
you can download the full list from Internet webpage
$$\.{http://www-cs-faculty.stanford.edu/\char`\~knuth/taocp.html}$$
by selecting the ``long form'' of the errata.
\fi
\medskip
\tenpoint
\beginconstruction
The ultimate, glorious, future editions of Volumes 1--5 are works in progress.
Please let me know of any improvements that you think I ought to make.
Send your comments either by snail mail to D.~E. Knuth, Computer
Science, Gates Building 4B, Stanford University, Stanford CA~94305-9045,
or by email to
{\tt taocp{\char`\@}cs.stanford.edu}. (Use email for book suggestions
only, please\dash---all
other correspondence is returned unread to the sender, or discarded,
because I have no time to
read ordinary email.) Although I'm working full time on
Volume~4 these days, I~will try to reply to all such messages
within a year of receipt. Current news about {\sl The Art of Computer
Programming\/} is posted on
$$\.{http://www-cs-faculty.stanford.edu/\char`~knuth/taocp.html}$$
and updated regularly.
\par\endconstruction
\rightline{\dash---Don Knuth, March 2006}
\bigskip
\bigskip
{\quoteformat
I thought there would be not much corrections.
I honestly wrote what I thought, but was most grievously mistaken.
I find the style incredibly bad, \& most difficult to make clear \& smooth.
.\thinspace.\thinspace. How I could have written so badly is quite inconceivable,
but I suppose it was owing to my whole attention being fixed
on general line of argument, \& not on details.
% "on general": sic
All I can say is that I am very sorry.
\author CHARLES DARWIN, letter to John Murray (14 June 1859)
% The Correspondence of Charles Darwin, vol 7 (Cambridge Univ Press 1991) p303
\bigskip
An opportunity to revise is a great luxury.
There's always something that needs correcting or improving.
And it's not just a matter of my own second thoughts.
Many of the most important changes and additions
start with letters from readers.
\author BRIAN HAYES (2017)
% American Scientist 105 (2017) 312
\vfill\eject
}
\def\today{\number\day\space\ifcase\month\or
January\or February\or March\or April\or May\or June\or
July\or August\or September\or October\or November\or December\fi
\space\number\year}
%%%%%%%%%%%%%%% CHANGES FOR VOLUME 4 FASCICLE 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\lhead{CHANGES TO V4F6: SATISFIABILITY}
\def\rhead{CHANGES TO V4F6: SATISFIABILITY}
\titlepage
\volheadline{SATISFIABILITY} % the fascicle title
\bigskip
\rightline{Copyright \copyright\ 2015, 2016, 2017
Addison\with Wesley}
\rightline{Last updated \today}
\bigskip
%\rightline{\sl Most of these corrections have already been made in
% recent printings.}
%\medskip
%\noindent Important note: The changes below include nearly every difference
%between the paperback fascicle and the first printing of Volume 4A,
%published in January 2011. All subsequent changes can be found in
%the errata list for that volume.
\smallskip
\let\defaultpointsize=\tenpoint
\bugonpage 4f6.i line $-2$ (16.05.11)
\ninepoint Sa\~o Paulo \becomes S\~ao Paulo
\endchange
\bugonpage 4f6.v line $-9$ (16.01.17)
\.{\char`\~/fasc5a} \becomes \.{\char`\~knuth/fasc5a}
\endchange
\amendpage 4f6.vii lines 4, 10, 11, 12 (17.05.31)
rows \becomes options\qquad[four changes]\nl
columns \becomes items\qquad[one change]
\endchange
\improvepage 4f6.4 line $-8$ (17.02.15)
{\sl equally spaced $1$s\/} \becomes
{\sl equally spaced $1$s\/} (or both)
\endchange
\amendpage 4f6.5 bottom and page 6 top (17.05.31)
row \becomes option\qquad[three changes]\nl
rows \becomes options\qquad[four changes]\nl
column \becomes item\qquad[three changes]\nl
columns \becomes items\qquad[two changes]
\endchange
\bugonpage 4f6.9 line $-18$ (16.04.10)
\rightline{\eightssi
it can be put into the conjunctive normal form ``piece by piece'',}
\endchange
\improvepage 4f6.19 line $-5$ (17.04.08)
predictable pattern \becomes repetitive pattern
\endchange
\bugonpage 4f6.32 line $-11$ (15.11.22)
$\bar5 \bar3 \bar4$ \becomes $\bar5 \bar3 \bar7$
\endchange
\bugonpage 4f6.33 replacement for Fig.~39 (16.02.10)
\centerline{\epsfbox{\figdir/ch7b.50}}
\endchange
\improvepage 4f6.34 line 2 after {\eq(60)} (16.07.05)
moves. \becomes moves (the 4s and the 5s).
\endchange
\amendpage 4f6.51 the line after {\eq(82)} (16.05.26)
[See A. C. Kaporis \becomes
[See M. Hajiaghayi and G. B. Sorkin, arXiv:math/0310193 [math.{\mc CO}];
A. C. Kaporis
\endchange
\bugonpage 4f6.55 replacement for Fig.~48 (16.02.10)
\centerline{\epsfbox{\figdir/ch7b.51}}
\endchange
\amendpage 4f6.66 new sentence at bottom of page (18.03.15)
The ``reason'' for literal~$l$'s current value is kept in variable~$R_l$.
\endchange
\bugonpage 4f6.68 line $-2$ (18.03.15)
$L_F\gets l'$, \becomes
$L_F\gets l'$,
$\.{VAL($\vert l'\vert$)}\gets 2d+(l'\band1)$,
\endchange
\amendpage 4f6.80 line 14 (17.05.17)
on variables that need \becomes on clauses whose variables need
\endchange
\amendpage 4f6.80 line 22 (17.05.17)
tackled \becomes tacked and when $N=\infty$.
\endchange
\bugonpage 4f6.86 bottom line (17.02.02)
(this is, \becomes (that is,
\endchange
\amendpage 4f6.118 line 27 (17.05.17)
tests, \becomes tests with $N=50n$ and $p=.4$,
\endchange
\amendpage 4f6.121 lines $-15$ through $-13$ (17.05.31)
the dancing links method of Section 7.2.2.1 \becomes
the backtrack method of exercise 7.2.2--21\nl
7.2 T$\mu$ \becomes 4 T$\mu$\nl
90 minutes \becomes 50 minutes
\endchange
\bugonpage 4f6.131 line $-14$ (16.07.09)
benchmark programs of 1992 \becomes benchmark problems of 1992
\endchange
\bugonpage 4f6.132 line 23 (16.08.23)
Independent Decreasing Sum \becomes Independent Decaying Sum
\endchange
\improvepage 4f6.134 better wording in exercise 15 (16.05.14)
that has an arbitrary order~$n\ge3$ \becomes
whose order is a given number~$n\ge3$
\endchange
\amendpage 4f6.135 new rating for exercise 24 (17.02.15)
{\it M32\/} \becomes {\it M34\/}
\endchange
\improvepage 4f6.136 end of exercise 38 (17.05.10)
each side. \becomes each side?
\endchange
\amendpage 4f6.136 line 3 of exercise 43 (18.01.21)
$(z_{m+n}\ldots z_1)_2=(z_1\ldots z_{m+n})_2$ \becomes
$(z_{2n}\ldots z_1)_2=(z_1\ldots z_{2n})_2$
\endchange
\amendpage 4f6.159 new rating for exercise 305 (17.05.17)
{\it M25\/} \becomes {\it \HM29\/}
\endchange
\bugonpage 4f6.165 line 7 of exercise 356 (17.08.17)
Theorems M and~K \becomes Theorems M and~S
\endchange
\bugonpage 4f6.173 line 2 of exercise 417 (16.02.23)
\ninepoint via~\eq(174) \becomes via~\eq(173)
\endchange
\amendpage 4f6.174 last line of exercise 428 (16.03.26)
\ninepoint $M$ and $N$ cannot be \becomes $M$ and $N$ cannot both be
\endchange
\improvepage 4f6.179 line 2 of exercise 481 (16.08.01)
\ninepoint $(x@{\xor}@y)y$ \becomes $(x@{\xor}@y)\,y$
\endchange
\bugonpage 4f6.183 row 16, column 39 of the big matrix in exercise 518 (16.01.11)
$\scriptscriptstyle0$ \becomes
$\scriptscriptstyle2$
\endchange
\improvepage 4f6.183 line 2 of exercise {518(a)} (16.01.11)
$r$ \becomes $p$\quad
$s$ \becomes $q$\quad
$t$ \becomes $r$
\endchange
\let\defaultpointsize=\ninepoint % get ready for answer pages
\bugonpage 4f6.185 line 6 of answer 6 (17.06.02)
By Theorem~J, \becomes By Theorem~L,
\endchange
\amendpage 4f6.186 replacement for lines 2--6 of answer 11 (16.02.04)
Furthermore, the only way to match $j'1$ is to choose one of the satisfiability
triples for clause~$j$. Suppose $\bar l_k j$ belongs to the chosen triple;
then we must also have chosen the true triples for literal $l_k$.
Thus a perfect matching implies satisfiable clauses.
\endchange
\amendpage 4f6.186 in answer 12 (17.05.31)
columns \becomes items\qquad[three changes]
\endchange
\amendpage 4f6.186 in answer 13 (17.05.31)
rows \becomes options\qquad[two changes]
\endchange
\amendpage 4f6.189 simplifying the formula for $f$ in answer 24 (17.02.15)
$r-lk$ we can set $v_j\gets\[d(s,v)\le j]$.\par
(e) $(s)\land\bigl(\bigwedge_{v\in V}\bigwedge_{w\in N(v)}(\bar v\lor w)\bigr)
\land(\bar t)$.\par
(f) Letting $s$ be any vertex, use
$(s)\land\bigl(\bigwedge_{v\in V}\bigwedge_{w\in N(v)}(\bar v\lor w)\bigr)
\land\bigl(\bigvee_{v\in V\setminus s}\bar v\bigr)$.\par
\endchange
\bugonpage 4f6.269 line 2 of answer 417 (16.02.23)
as in \eq(174) \becomes as in \eq(173)
\endchange
\bugonpage 4f6.269 line 3 of answer 418 (16.02.23)
via~\eq(174) \becomes via~\eq(173)
\endchange
\amendpage 4f6.269 line 1 of answer 423 (16.04.04)
[But a {\it forcing\/} \becomes
[But Ab{\'\i}o, Gange,
Mayer-Eichberger, and Stuckey have shown
[{\sl LNCS\/ \bf9676} (2016), 1--17]
that weak forcing is always achieved if $(\bar a_j\!\lor a_l\lor a_h)$ is
added to~\eq(173). Furthermore, a~{\it forcing\/}
\endchange
\amendpage 4f6.270 line 7 of answer 426 (16.04.05)
variable elimination \becomes the elimination of an auxiliary variable
\endchange
\amendpage 4f6.271 line 6 of answer 436 (16.4.04)
$\bigvee\{q_{k-1}\mid(q,a,q')\in T\}\bigr)$; together with (vi) \becomes
$\bigvee\{q_{k-1}\mid(q,a,q')\in T\}\bigr)$,
for $a\!\in\!A$, $q'\!\in\!Q$.
And (vi)
\endchange
\bugonpage 4f6.271 replacement for lines 18--23 of answer 436 (16.4.04)
For example, the language $L_2$ of exercise 434 yields $20n+4$ clauses with
$8n+3$ auxiliary variables:
$F=\bigwedge_{k=1}^n\bigl(
(\bar t_{k00}\!\lor\bar x_k)\land(\bar t_{k00}\!\lor 0_k)\land
(\bar t_{k11}\!\lor x_k)\land(\bar t_{k11}\!\lor 1_k)\land
(\bar t_{k12}\!\lor x_k)\land(\bar t_{k12}\!\lor 2_k)\land
(\bar t_{k02}\!\lor\bar x_k)\land(\bar t_{k02}\!\lor 2_k)\land
(\bar 0_{k-1}\!\lor t_{k00}\lor t_{k11})\land
(\bar 1_{k-1}\!\lor t_{k12})\land
(\bar 2_{k-1}\!\lor t_{k02})\land
(\bar 0_k\!\lor t_{k00})\land
(\bar 1_k\!\lor t_{k11})\land
(\bar 2_k\!\lor t_{k02}\lor t_{k12})\land
(x_k\!\lor t_{k00}\lor t_{k02})\land
(\bar x_k\!\lor t_{k11}\lor t_{k12})\land
(\bar t_{k00}\!\lor 0_{k-1})\land
(\bar t_{k11}\!\lor 0_{k-1})\land
(\bar t_{k12}\!\lor 1_{k-1})\land
(\bar t_{k02}\!\lor 2_{k-1})\bigr)\land
(\bar 1_0)\land(\bar 2_0)\land(\bar 0_n)\land(\bar 1_n)$.
%(Unit~propagation will immediately assign values to 12 of the $8n+3$ variables,
%thereby satisfying 26 of these clauses, when $n\ge3$.
%For example, $\bar t_{112}$, $\bar t_{n11}$, $\bar 0_{n-1}$ are~forced.)\par
\endchange
\bugonpage 4f6.272 lines 6 and 10 of answer 440 (16.04.03)
$QR_{hik}$ \becomes $QP_{hik}$ \quad(twice)
\endchange
\bugonpage 4f6.275 line 1 of answer 455 (17.01.10)
1101 \becomes 1011
\endchange
\amendpage 4f6.283 last lines of answer 488 (15.12.27)
F.~R.~K. Chung and R.~L. Graham conjecture that \becomes
D.~M. Kane has proved, among other things, that\nl
$O(1)$.] \becomes $O(1)$. (To appear.)]
\endchange
\improvepage 4f6.288 lines 3 and 7 (16.02.09)
$S_{1kk}\land\bar Z_{11}\land Z_{12}$ becomes
$(S_{1kk})\land(\bar Z_{11})\land(Z_{12})$\quad and\quad
$S_{553}\land\bar Z_{17}$ \becomes
$(S_{553})\land(\bar Z_{17})$
\endchange
\amendpage 4f6.288 replacement for answer 517{(a)} (16.01.11)
\ans517. (a) (Solution by G\"unter Rote.) Replace the $j$th ternary clause
$(l_j\!\lor l'_j\!\lor l''_j)$ by three ternary equations
$l_j+a_j+c_j=1$, $\bar l'_j+a_j+b_j=1$, $\bar l''_j+c_j+d_j=1$,
where $a_j$, $b_j$, $c_j$, and $d_j$ are new variables.
\endchange
\expandafter\ifx\csname indexeject\endcsname\relax\else\vfill\eject\fi
\amendpage 4f6.292 and following (06.03.19)
Miscellaneous changes to the existing index of Volume~4 Fascicle~6
are collected here,
including corrections and amendments to the old entries as well as new entries
that are occasioned by the new material. Thus, the lines of the full index
that have changed serve also as an index to the present document. However,
when a correction or amendment has caused an old index entry to be deleted,
the deletion is usually not indicated.
\input exotic
\begindoublecolumns
\indexformat
\gdef\Uni1.08:{\bitmap24:1.08:}
\hangindent2em
1-in-3 {\mc SAT}, 183. % 2nd
Ab{\'\i}o Roig, Ignasi, 269. % 2nd
Calabro, Christopher Matthew, 288. % 2nd
Connection puzzles, 114, 170. % 2nd
Dancing links, 5, 134, 208, 288, 291. % 2nd
Exact cover problems, vii, 2, 5--6, 28, 134, 183, 186, 219, 225, 290. % 2nd
Exact (one-per-clause) satisfiability, 183. % 2nd
Gange, Graeme Keith, 269. % 2nd
Graham, Ronald Lewis (\GC72:74:-4:61% G2480
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), 185. % 2nd
Hajiaghayi, MohammadTaghi\indexbreak ({\arab\Afam/\Aihy/\Afa/\Aiq/\Aamd/ \Afam/\Aij/\Afa/\Aihh/ \Afam/\Amq/\Ait/ \Afd/\Amm/\Amhh/\Aim/}), 51. % 2nd
{\mc ITE}, \see If-then-else operation. % 2nd
{\sl JRM\/}: {\sl Journal of Recreational Mathematics}, published 1970--2014. % 2nd
Kane, Daniel Mertz, 283. % 2nd
Linear hypergraphs, \see Quad-free matrices. % 2nd
Matchings, three-dimensional, \see {\mc 3D~MATCHING} problem. % 2nd
Mayer-Eichberger, Valentin Christian Johannes Kaspar, 269. % 2nd
Megamem (M$\mu$): One million memory accesses, 98, 123. % 2nd
Monus operation ($x@{\dotminus}@y=\max\{0,x{-}y\}$), vi, 92, 189, 247, 268. % 2nd
Nonprimary items, 186. % 2nd
Notations, $x@{\dotminus}@y$ (monus), 92, 189, 247, 268. % 2nd
Peaceable queens, 180. % 2nd
Patents, 132. % 2nd
Paturi, Ramamohan ({\tl
\|2103|\C129\|1128|\C101\\1\C238\|0128|\C111\\{-2.5}\|1148|\C90\
\|0129|\C209\\4\C86\C136\\1\C221}), 288. % 2nd
Prins, Christian, 267. % 2nd
Putnam, Hilary Whitehall, 9, 32, 130, 298. % 2nd
Reasons, 62--63, 66, 72, 157, 165, 233. % 2nd
Rote, G\"unter (= Rothe, G\"unther Alfred Heinrich), 288. % 2nd
Rectangle-free grids, \see Quad-free matrices. % 2nd
Set splitting, \see Not-all-equal {\mc SAT}. % 2nd
Socrates, son of Sophroniscus of Alopece ({\grk Swkr'aths Swfron'iskou >Alwpek=hjen}), 129. % 2nd
Sorkin, Gregory Bret, 51. % 2nd
Stuckey, Peter James, 269. % 2nd
Sudoku, 183. % 2nd
Tseytin, Gregory Samuelovich ({\rus Tsei0tin, Grigorii0 Samuilovich}), 9, 59--60, 71, 133, 152, 154, 168, 178, 215, 231, 290. % 2nd
{\mc XSAT}, \see $\,$Exact (one-per-clause) satisfiability. % 2nd
\vfill
\enddoublecolumns
\endchange
\bye
[The next printing will be the 2nd.]
not listed above:
the grid lines will be made darker on illustrations that appear on pages
19, 20, 25, 139, 197--201, 206, 207.
page 20, in line after (39), say "that was"
page 149, better spacing in ex 199(b)
page 192, "makes a total" -> "makes a total of" in answer 41
page 192, JRM abbreviated in answer 46
page 193, line 2 of answer 48, sneak the period inside the quotes
page 224, line 5 of answer 210, better spacing
page 275, JRM abbreviated in answer 451
page 288, JCSS expanded in answer 516
page 291, JCSS expanded in answer 526
Fan Chung leaves the index
index entries for \mu made singular not plural
delete "Chiral symmetry" from the index
ARTICLES "TO APPEAR" THAT ARE STILL PENDING:
Kane on nonattacking queens: answer to exercise 7.2.2.2--488.