EXTRA_DIST = \
basis_reduction_templ.c \
isl_pw_templ.c \
+ doc/chicago.bst \
+ doc/chicago.sty \
+ doc/implementation.tex \
+ doc/isl.bib \
doc/mypod2latex \
doc/manual.tex \
doc/user.pod \
if GENERATE_DOC
+export TEXINPUTS := $(srcdir):$(TEXINPUTS)
+export BIBINPUTS := $(srcdir):$(BIBINPUTS)
+export BSTINPUTS := $(srcdir):$(BSTINPUTS)
+
user.tex: user.pod
$(PERL) $(srcdir)/mypod2latex $< $@
-manual.pdf: manual.tex user.tex
+manual.pdf: manual.tex user.tex $(srcdir)/implementation.tex
+ $(PDFLATEX) $<
+ bibtex manual
$(PDFLATEX) $<
$(PDFLATEX) $<
user.html: user.pod
--- /dev/null
+%%% ====================================================================
+%%% @BibTeX-style-file{
+%%% author = "Glenn Paulley",
+%%% version = "4",
+%%% date = "28 August 1992",
+%%% time = "10:23:39 199",
+%%% filename = "chicago.bst",
+%%% address = "Data Structuring Group
+%%% Department of Computer Science
+%%% University of Waterloo
+%%% Waterloo, Ontario, Canada
+%%% N2L 3G1",
+%%% telephone = "(519) 885-1211",
+%%% FAX = "(519) 885-1208",
+%%% checksum = "26323 1654 5143 37417",
+%%% email = "gnpaulle@bluebox.uwaterloo.ca",
+%%% codetable = "ISO/ASCII",
+%%% keywords = "",
+%%% supported = "yes",
+%%% abstract = "A BibTeX bibliography style that follows the
+%%% `B' reference style of the 13th Edition of
+%%% the Chicago Manual of Style. A detailed
+%%% feature list is given below.",
+%%% docstring = "The checksum field above contains a CRC-16
+%%% checksum as the first value, followed by the
+%%% equivalent of the standard UNIX wc (word
+%%% count) utility output of lines, words, and
+%%% characters. This is produced by Robert
+%%% Solovay's checksum utility.",
+%%% }
+%%% ====================================================================
+%
+% "Chicago" BibTeX style, chicago.bst
+% ===================================
+%
+% BibTeX `chicago' style file for BibTeX version 0.99c, LaTeX version 2.09
+% Place it in a file called chicago.bst in the BibTeX search path.
+% You need to include chicago.sty as a \documentstyle option.
+% (Placing it in the same directory as the LaTeX document should also work.)
+% This "chicago" style is based on newapa.bst (American Psych. Assoc.)
+% found at ymir.claremont.edu.
+%
+% Citation format: (author-last-name year)
+% (author-last-name and author-last-name year)
+% (author-last-name, author-last-name, and author-last-name year)
+% (author-last-name et al. year)
+% (author-last-name)
+% author-last-name (year)
+% (author-last-name and author-last-name)
+% (author-last-name et al.)
+% (year) or (year,year)
+% year or year,year
+%
+% Reference list ordering: alphabetical by author or whatever passes
+% for author in the absence of one.
+%
+% This BibTeX style has support for abbreviated author lists and for
+% year-only citations. This is done by having the citations
+% actually look like
+%
+% \citeauthoryear{full-author-info}{abbrev-author-info}{year}
+%
+% The LaTeX style has to have the following (or similar)
+%
+% \let\@internalcite\cite
+% \def\fullcite{\def\citeauthoryear##1##2##3{##1, ##3}\@internalcite}
+% \def\fullciteA{\def\citeauthoryear##1##2##3{##1}\@internalcite}
+% \def\shortcite{\def\citeauthoryear##1##2##3{##2, ##3}\@internalcite}
+% \def\shortciteA{\def\citeauthoryear##1##2##3{##2}\@internalcite}
+% \def\citeyear{\def\citeauthoryear##1##2##3{##3}\@internalcite}
+%
+% These TeX macro definitions are found in chicago.sty. Additional
+% commands to manipulate different components of a citation can be defined
+% so that, for example, you can list author's names without parentheses
+% if using a citation as a noun or object in a sentence.
+%
+% This file was originally copied from newapa.bst at ymir.claremont.edu.
+%
+% Features of chicago.bst:
+% =======================
+%
+% - full names used in citations, but abbreviated citations are available
+% (see above)
+% - if an entry has a "month", then the month and year are also printed
+% as part of that bibitem.
+% - all conjunctions use "and" instead of "\&"
+% - major modification from Chicago Manual of Style (13th ed.) is that
+% only the first author in a reference appears last name first-
+% additional authors appear as J. Q. Public.
+% - pages are listed as "pp. xx-xx" in all entry types except
+% article entries.
+% - book, inbook, and manual use "location: publisher" (or organization)
+% for address and publisher. All other types list publishers separately.
+% - "pp." are used to identify page numbers for all entry types except
+% articles.
+% - organization is used as a citation label if neither author nor editor
+% is present (for manuals).
+% - "et al." is used for long author and editor lists, or when "others"
+% is used.
+%
+% Modifications and bug fixes from newapa.bst:
+% ===========================================
+%
+% - added month, year to bib entries if month is present
+% - fixed bug with In proceedings, added necessary comma after title
+% - all conjunctions changed to "and" from "\&"
+% - fixed bug with author labels in my.full.label: "et al." now is
+% generated when "others" is an author name
+% - major modification from Chicago Manual of Style (13th ed.) is that
+% only the first author in a reference appears last name first-
+% additional authors appear as J. Q. Public.
+% - pages are listed as "pp. xx-xx" in all entry types except
+% article entries. Unnecessary (IMHO) "()" around page numbers
+% were removed, and page numbers now don't end with a period.
+% - created chicago.sty for use with this bibstyle (required).
+% - fixed bugs in FUNCTION {format.vol.num.pages} for missing volume,
+% number, and /or pages. Renamed to format.jour.vol.
+% - fixed bug in formatting booktitles: additional period an error if
+% book has a volume.
+% - fixed bug: editors usually given redundant period before next clause
+% (format.editors.dot) removed.
+% - added label support for organizations, if both author and editor
+% are missing (from alpha.bst). If organization is too long, then
+% the key field is used for abbreviated citations.
+% - In proceedings or books of several volumes, no comma was written
+% between the "Volume x" and the page numbers (this was intentional
+% in newapa.bst). Fixed.
+% - Some journals may not have volumes/numbers, only month/year (eg.
+% IEEE Computer). Fixed bug in article style that assumed volume/number
+% was always present.
+%
+% Original documentation for newapa.sty:
+% =====================================
+%
+% This version was made by modifying the master file made by
+% Oren Patashnik (PATASHNIK@SCORE.STANFORD.EDU), and the 'named' BibTeX
+% style of Peter F. Patel-Schneider.
+%
+% Copyright (C) 1985, all rights reserved.
+% Copying of this file is authorized only if either
+% (1) you make absolutely no changes to your copy, including name, or
+% (2) if you do make changes, you name it something other than 'newapa.bst'.
+% There are undoubtably bugs in this style. If you make bug fixes,
+% improvements, etc. please let me know. My e-mail address is:
+% spencer@cgrg.ohio.state.edu or 71160.3141@compuserve.com
+%
+% This style was made from 'plain.bst', 'named.bst', and 'apalike.bst',
+% with lots of tweaking to make it look like APA style, along with tips
+% from Young Ryu and Brian Reiser's modifications of 'apalike.bst'.
+
+ENTRY
+ { address
+ author
+ booktitle
+ chapter
+ edition
+ editor
+ fjournal
+ howpublished
+ institution
+ journal
+ key
+ month
+ note
+ number
+ organization
+ pages
+ publisher
+ school
+ series
+ title
+ type
+ volume
+ year
+ }
+ {}
+ { label.year extra.label sort.year sort.label }
+
+INTEGERS { output.state before.all mid.sentence after.sentence after.block }
+
+FUNCTION {init.state.consts}
+{ #0 'before.all :=
+ #1 'mid.sentence :=
+ #2 'after.sentence :=
+ #3 'after.block :=
+}
+
+STRINGS { s t u }
+
+FUNCTION {output.nonnull}
+{ 's :=
+ output.state mid.sentence =
+ { ", " * write$ }
+ { output.state after.block =
+ { add.period$ write$
+ newline$
+ "\newblock " write$
+ }
+ { output.state before.all =
+ 'write$
+ { add.period$ " " * write$ }
+ if$
+ }
+ if$
+ mid.sentence 'output.state :=
+ }
+ if$
+ s
+}
+
+% Use a colon to separate output. Used only for address/publisher
+% combination in book/inbook types, address/institution for manuals,
+% and organization:publisher for proceedings (inproceedings).
+%
+FUNCTION {output.nonnull.colon}
+{ 's :=
+ output.state mid.sentence =
+ { ": " * write$ }
+ { output.state after.block =
+ { add.period$ write$
+ newline$
+ "\newblock " write$
+ }
+ { output.state before.all =
+ 'write$
+ { add.period$ " " * write$ }
+ if$
+ }
+ if$
+ mid.sentence 'output.state :=
+ }
+ if$
+ s
+}
+
+FUNCTION {output}
+{ duplicate$ empty$
+ 'pop$
+ 'output.nonnull
+ if$
+}
+
+FUNCTION {output.colon}
+{ duplicate$ empty$
+ 'pop$
+ 'output.nonnull.colon
+ if$
+}
+
+FUNCTION {output.check}
+{ 't :=
+ duplicate$ empty$
+ { pop$ "empty " t * " in " * cite$ * warning$ }
+ 'output.nonnull
+ if$
+}
+
+FUNCTION {output.check.colon}
+{ 't :=
+ duplicate$ empty$
+ { pop$ "empty " t * " in " * cite$ * warning$ }
+ 'output.nonnull.colon
+ if$
+}
+
+FUNCTION {output.year.check}
+{ year empty$
+ { "empty year in " cite$ * warning$ }
+ { write$
+ " (" year * extra.label *
+ month empty$
+ { ")" * }
+ { ", " * month * ")" * }
+ if$
+ mid.sentence 'output.state :=
+ }
+ if$
+}
+
+
+FUNCTION {fin.entry}
+{ add.period$
+ write$
+ newline$
+}
+
+FUNCTION {new.block}
+{ output.state before.all =
+ 'skip$
+ { after.block 'output.state := }
+ if$
+}
+
+FUNCTION {new.sentence}
+{ output.state after.block =
+ 'skip$
+ { output.state before.all =
+ 'skip$
+ { after.sentence 'output.state := }
+ if$
+ }
+ if$
+}
+
+FUNCTION {not}
+{ { #0 }
+ { #1 }
+ if$
+}
+
+FUNCTION {and}
+{ 'skip$
+ { pop$ #0 }
+ if$
+}
+
+FUNCTION {or}
+{ { pop$ #1 }
+ 'skip$
+ if$
+}
+
+FUNCTION {new.block.checka}
+{ empty$
+ 'skip$
+ 'new.block
+ if$
+}
+
+FUNCTION {new.block.checkb}
+{ empty$
+ swap$ empty$
+ and
+ 'skip$
+ 'new.block
+ if$
+}
+
+FUNCTION {new.sentence.checka}
+{ empty$
+ 'skip$
+ 'new.sentence
+ if$
+}
+
+FUNCTION {new.sentence.checkb}
+{ empty$
+ swap$ empty$
+ and
+ 'skip$
+ 'new.sentence
+ if$
+}
+
+FUNCTION {field.or.null}
+{ duplicate$ empty$
+ { pop$ "" }
+ 'skip$
+ if$
+}
+
+%
+% Emphasize the top string on the stack.
+%
+FUNCTION {emphasize}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "{\em " swap$ * "}" * }
+ if$
+}
+
+%
+% Emphasize the top string on the stack, but add a trailing space.
+%
+FUNCTION {emphasize.space}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "{\em " swap$ * "\/}" * }
+ if$
+}
+
+INTEGERS { nameptr namesleft numnames }
+%
+% Format bibliographical entries with the first author last name first,
+% and subsequent authors with initials followed by last name.
+% All names are formatted in this routine.
+%
+FUNCTION {format.names}
+{ 's :=
+ #1 'nameptr := % nameptr = 1;
+ s num.names$ 'numnames := % numnames = num.name$(s);
+ numnames 'namesleft :=
+ { namesleft #0 > }
+
+ { nameptr #1 =
+ {s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ 't := }
+ {s nameptr "{f.~}{vv~}{ll}{, jj}" format.name$ 't := }
+ if$
+ nameptr #1 >
+ { namesleft #1 >
+ { ", " * t * }
+ { numnames #2 >
+ { "," * }
+ 'skip$
+ if$
+ t "others" =
+ { " et~al." * }
+ { " and " * t * } % from Chicago Manual of Style
+ if$
+ }
+ if$
+ }
+ 't
+ if$
+ s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ 't :=
+ "\protect \index {" * t * "|hyperemph}" *
+ nameptr #1 + 'nameptr := % nameptr += 1;
+ namesleft #1 - 'namesleft := % namesleft =- 1;
+ }
+ while$
+}
+
+FUNCTION {my.full.label}
+{ 's :=
+ #1 'nameptr := % nameptr = 1;
+ s num.names$ 'numnames := % numnames = num.name$(s);
+ numnames 'namesleft :=
+ { namesleft #0 > }
+
+ { s nameptr "{vv~}{ll}" format.name$ 't := % get the next name
+ nameptr #1 >
+ { namesleft #1 >
+ { ", " * t * }
+ { numnames #2 >
+ { "," * }
+ 'skip$
+ if$
+ t "others" =
+ { " et~al." * }
+ { " and " * t * } % from Chicago Manual of Style
+ if$
+ }
+ if$
+ }
+ 't
+ if$
+ s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ 't :=
+ "\protect \index {" * t * "|bold}" *
+ nameptr #1 + 'nameptr := % nameptr += 1;
+ namesleft #1 - 'namesleft := % namesleft =- 1;
+ }
+ while$
+
+}
+
+FUNCTION {format.names.fml}
+%
+% Format names in "familiar" format, with first initial followed by
+% last name. Like format.names, ALL names are formatted.
+%
+{ 's :=
+ #1 'nameptr := % nameptr = 1;
+ s num.names$ 'numnames := % numnames = num.name$(s);
+ numnames 'namesleft :=
+ { namesleft #0 > }
+
+ { s nameptr "{f.~}{vv~}{ll}{, jj}" format.name$ 't :=
+
+ nameptr #1 >
+ { namesleft #1 >
+ { ", " * t * }
+ { numnames #2 >
+ { "," * }
+ 'skip$
+ if$
+ t "others" =
+ { " et~al." * }
+ { " and " * t * }
+% { " \& " * t * }
+ if$
+ }
+ if$
+ }
+ 't
+ if$
+ nameptr #1 + 'nameptr := % nameptr += 1;
+ namesleft #1 - 'namesleft := % namesleft =- 1;
+ }
+ while$
+}
+
+FUNCTION {format.authors}
+{ author empty$
+ { "" }
+ { author format.names }
+ if$
+}
+
+FUNCTION {format.key}
+{ empty$
+ { key field.or.null }
+ { "" }
+ if$
+}
+
+%
+% Format editor names for use in the "in" types: inbook, incollection,
+% inproceedings: first initial, then last names. When editors are the
+% LABEL for an entry, then format.editor is used which lists editors
+% by last name first.
+%
+FUNCTION {format.editors.fml}
+{ editor empty$
+ { "" }
+ { editor format.names.fml
+ editor num.names$ #1 >
+ { " (Eds.)" * }
+ { " (Ed.)" * }
+ if$
+ }
+ if$
+}
+
+%
+% Format editor names for use in labels, last names first.
+%
+FUNCTION {format.editors}
+{ editor empty$
+ { "" }
+ { editor format.names
+ editor num.names$ #1 >
+ { " (Eds.)" * }
+ { " (Ed.)" * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.title}
+{ title empty$
+ { "" }
+ { title "t" change.case$ }
+ if$
+}
+
+% Note that the APA style requres case changes
+% in article titles. The following does not
+% change cases. If you perfer it, uncomment the
+% following and comment out the above.
+
+%FUNCTION {format.title}
+%{ title empty$
+% { "" }
+% { title }
+% if$
+%}
+
+FUNCTION {n.dashify}
+{ 't :=
+ ""
+ { t empty$ not }
+ { t #1 #1 substring$ "-" =
+ { t #1 #2 substring$ "--" = not
+ { "--" *
+ t #2 global.max$ substring$ 't :=
+ }
+ { { t #1 #1 substring$ "-" = }
+ { "-" *
+ t #2 global.max$ substring$ 't :=
+ }
+ while$
+ }
+ if$
+ }
+ { t #1 #1 substring$ *
+ t #2 global.max$ substring$ 't :=
+ }
+ if$
+ }
+ while$
+}
+
+FUNCTION {format.btitle}
+{ edition empty$
+ { title emphasize }
+ { title empty$
+ { title emphasize }
+ { volume empty$ % gnp - check for volume, then don't need period
+ { "{\em " title * "\/} (" * edition * " ed.)" * "." * }
+ { "{\em " title * "\/} (" * edition * " ed.)" * }
+ if$
+ }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.emphasize.booktitle}
+{ edition empty$
+ { booktitle emphasize }
+ { booktitle empty$
+ { booktitle emphasize }
+ { volume empty$ % gnp - extra period an error if book has a volume
+ { "{\em " booktitle * "\/} (" * edition * " ed.)" * "." *}
+ { "{\em " booktitle * "\/} (" * edition * " ed.)" * }
+ if$
+ }
+ if$
+ }
+ if$
+ }
+
+
+FUNCTION {tie.or.space.connect}
+{ duplicate$ text.length$ #3 <
+ { "~" }
+ { " " }
+ if$
+ swap$ * *
+}
+
+FUNCTION {either.or.check}
+{ empty$
+ 'pop$
+ { "can't use both " swap$ * " fields in " * cite$ * warning$ }
+ if$
+}
+
+FUNCTION {format.bvolume}
+{ volume empty$
+ { "" }
+ { "Volume" volume tie.or.space.connect % gnp - changed to mixed case
+ series empty$
+ 'skip$
+ { " of " * series emphasize * }
+ if$
+ "volume and number" number either.or.check
+ }
+ if$
+}
+
+FUNCTION {format.number.series}
+{ volume empty$
+ { number empty$
+ { series field.or.null }
+ { output.state mid.sentence =
+ { "Number" } % gnp - changed to mixed case always
+ { "Number" }
+ if$
+ number tie.or.space.connect
+ series empty$
+ { "there's a number but no series in " cite$ * warning$ }
+ { " in " * series * }
+ if$
+ }
+ if$
+ }
+ { "" }
+ if$
+}
+
+INTEGERS { multiresult }
+
+FUNCTION {multi.page.check}
+{ 't :=
+ #0 'multiresult :=
+ { multiresult not
+ t empty$ not
+ and
+ }
+ { t #1 #1 substring$
+ duplicate$ "-" =
+ swap$ duplicate$ "," =
+ swap$ "+" =
+ or or
+ { #1 'multiresult := }
+ { t #2 global.max$ substring$ 't := }
+ if$
+ }
+ while$
+ multiresult
+}
+
+FUNCTION {format.pages}
+{ pages empty$
+ { "" }
+ { pages multi.page.check
+ { "pp.\ " pages n.dashify tie.or.space.connect } % gnp - removed ()
+ { "pp.\ " pages tie.or.space.connect }
+ if$
+ }
+ if$
+}
+
+% By Young (and Spencer)
+% GNP - fixed bugs with missing volume, number, and/or pages
+%
+% Format journal, volume, number, pages for article types.
+%
+FUNCTION {format.jour.vol}
+{ fjournal empty$
+ { journal empty$
+ { "no journal in " cite$ * warning$
+ "" }
+ { journal emphasize.space }
+ if$
+ }
+ { fjournal emphasize.space }
+ if$
+ number empty$
+ { volume empty$
+ { "no number and no volume in " cite$ * warning$
+ "" * }
+ { "~{\em " * Volume * "}" * }
+ if$
+ }
+ { volume empty$
+ {"no volume for " cite$ * warning$
+ "~(" * number * ")" * }
+ { "~" *
+ volume emphasize.space
+ "(" * number * ")" * * }
+ if$
+ }
+ if$
+ pages empty$
+ {"page numbers missing in " cite$ * warning$
+ "" * } % gnp - place a null string on the stack for output
+ { duplicate$ empty$
+ { pop$ format.pages }
+ { ", " * pages n.dashify * } % gnp - removed pp. for articles
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.chapter.pages}
+{ chapter empty$
+ 'format.pages
+ { type empty$
+ { "Chapter" } % gnp - changed to mixed case
+ { type "t" change.case$ }
+ if$
+ chapter tie.or.space.connect
+ pages empty$
+ {"page numbers missing in " cite$ * warning$} % gnp - added check
+ { ", " * format.pages * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.in.ed.booktitle}
+{ booktitle empty$
+ { "" }
+ { editor empty$
+ { "In " format.emphasize.booktitle * }
+ { "In " format.editors.fml * ", " * format.emphasize.booktitle * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.thesis.type}
+{ type empty$
+ 'skip$
+ { pop$
+ type "t" change.case$
+ }
+ if$
+}
+
+FUNCTION {format.tr.number}
+{ type empty$
+ { "Technical Report" }
+ 'type
+ if$
+ number empty$
+ { "t" change.case$ }
+ { number tie.or.space.connect }
+ if$
+}
+
+FUNCTION {format.article.crossref}
+{ "See"
+ "\citeN{" * crossref * "}" *
+}
+
+FUNCTION {format.crossref.editor}
+{ editor #1 "{vv~}{ll}" format.name$
+ editor num.names$ duplicate$
+ #2 >
+ { pop$ " et~al." * }
+ { #2 <
+ 'skip$
+ { editor #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =
+ { " et~al." * }
+ { " and " * editor #2 "{vv~}{ll}" format.name$ * }
+ if$
+ }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.book.crossref}
+{ volume empty$
+ { "empty volume in " cite$ * "'s crossref of " * crossref * warning$
+ "In "
+ }
+ { "Volume" volume tie.or.space.connect % gnp - changed to mixed case
+ " of " *
+ }
+ if$
+ editor empty$
+ editor field.or.null author field.or.null =
+ or
+ { key empty$
+ { series empty$
+ { "need editor, key, or series for " cite$ * " to crossref " *
+ crossref * warning$
+ "" *
+ }
+ { "{\em " * series * "\/}" * }
+ if$
+ }
+ { key * }
+ if$
+ }
+ { format.crossref.editor * }
+ if$
+ " \citeN{" * crossref * "}" *
+}
+
+FUNCTION {format.incoll.inproc.crossref}
+{ "See"
+ " \citeN{" * crossref * "}" *
+}
+
+% format.lab.names:
+%
+% determines "short" names for the abbreviated author information.
+% "Long" labels are created in calc.label, using the routine my.full.label
+% to format author and editor fields.
+%
+% There are 4 cases for labels. (n=3 in the example)
+% a) one author Foo
+% b) one to n Foo, Bar and Baz
+% c) use of "and others" Foo, Bar et al.
+% d) more than n Foo et al.
+%
+FUNCTION {format.lab.names}
+{ 's :=
+ s num.names$ 'numnames :=
+ numnames #2 > % change number to number of others allowed before
+ % forcing "et al".
+ { s #1 "{vv~}{ll}" format.name$
+ "\protect \index {" *
+ s #1 "{vv~}{ll}{, jj}{, f.}" format.name$ *
+ "}" *
+ "\protect\chicagoetal/" * }
+ {
+ numnames #1 - 'namesleft :=
+ #2 'nameptr :=
+ s #1 "{vv~}{ll}" format.name$
+ "\protect \index {" *
+ s #1 "{vv~}{ll}{, jj}{, f.}" format.name$ *
+ "}" *
+ { namesleft #0 > }
+ { nameptr numnames =
+ { s nameptr "{ff }{vv }{ll}{ jj}" format.name$ "others" =
+ { "\protect\chicagoetal/" * }
+ { "\protect\chicagoand/" * s nameptr "{vv~}{ll}" format.name$ *
+ "\protect \index {" *
+ s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ *
+ "}" *
+ }
+ if$
+ }
+ { ", " * s nameptr "{vv~}{ll}" format.name$ * }
+ if$
+ nameptr #1 + 'nameptr :=
+ namesleft #1 - 'namesleft :=
+ }
+ while$
+ }
+ if$
+}
+
+FUNCTION {author.key.label}
+{ author empty$
+ { key empty$
+ { "no key, author in " cite$ * warning$
+ cite$ #1 #3 substring$ }
+ 'key
+ if$
+ }
+ { author format.lab.names }
+ if$
+}
+
+FUNCTION {editor.key.label}
+{ editor empty$
+ { key empty$
+ { "no key, editor in " cite$ * warning$
+ cite$ #1 #3 substring$ }
+ 'key
+ if$
+ }
+ { editor format.lab.names }
+ if$
+}
+
+FUNCTION {author.key.organization.label}
+%
+% added - gnp. Provide label formatting by organization if author is null.
+%
+{ author empty$
+ { organization empty$
+ { key empty$
+ { "no key, author or organization in " cite$ * warning$
+ cite$ #1 #3 substring$ }
+ 'key
+ if$
+ }
+ { organization }
+ if$
+ }
+ { author format.lab.names }
+ if$
+}
+
+FUNCTION {editor.key.organization.label}
+%
+% added - gnp. Provide label formatting by organization if editor is null.
+%
+{ editor empty$
+ { organization empty$
+ { key empty$
+ { "no key, editor or organization in " cite$ * warning$
+ cite$ #1 #3 substring$ }
+ 'key
+ if$
+ }
+ { organization }
+ if$
+ }
+ { editor format.lab.names }
+ if$
+}
+
+FUNCTION {author.editor.key.label}
+{ author empty$
+ { editor empty$
+ { key empty$
+ { "no key, author, or editor in " cite$ * warning$
+ cite$ #1 #3 substring$ }
+ 'key
+ if$
+ }
+ { editor format.lab.names }
+ if$
+ }
+ { author format.lab.names }
+ if$
+}
+
+FUNCTION {calc.label.orig}
+%
+% Changed - GNP. See also author.organization.sort, editor.organization.sort
+% Form label for BibTeX entry. The classification of which fields are used
+% for which type of entry (book, inbook, etc.) are taken from alpha.bst.
+% The change here from newapa is to also include organization as a
+% citation label if author or editor is missing.
+%
+{ type$ "book" =
+ type$ "inbook" =
+ or
+ 'author.editor.key.label
+ { type$ "proceedings" =
+ 'editor.key.organization.label
+ { type$ "manual" =
+ 'author.key.organization.label
+ 'author.key.label
+ if$
+ }
+ if$
+ }
+ if$
+
+ author empty$ % generate the full label citation information.
+ { editor empty$
+ { organization empty$
+ { "no author, editor, or organization in " cite$ * warning$
+ "??" }
+ { organization }
+ if$
+ }
+ { editor my.full.label }
+ if$
+ }
+ { author.key.label }
+ if$
+
+% leave label on the stack, to be popped when required.
+
+ "}{" * swap$ *
+% year field.or.null purify$ #-1 #4 substring$ *
+%
+% save the year for sort processing afterwards (adding a, b, c, etc.)
+%
+ year field.or.null purify$ #-1 #4 substring$
+ 'label.year :=
+}
+
+FUNCTION {calc.label}
+%
+% Changed - GNP. See also author.organization.sort, editor.organization.sort
+% Form label for BibTeX entry. The classification of which fields are used
+% for which type of entry (book, inbook, etc.) are taken from alpha.bst.
+% The change here from newapa is to also include organization as a
+% citation label if author or editor is missing.
+%
+{ type$ "book" =
+ type$ "inbook" =
+ or
+ 'author.editor.key.label
+ { type$ "proceedings" =
+ 'editor.key.organization.label
+ { type$ "manual" =
+ 'author.key.organization.label
+ 'author.key.label
+ if$
+ }
+ if$
+ }
+ if$
+
+ author empty$ % generate the full label citation information.
+ { editor empty$
+ { organization empty$
+ { "no author, editor, or organization in " cite$ * warning$
+ "??" }
+ { organization }
+ if$
+ }
+ { editor my.full.label }
+ if$
+ }
+ { author my.full.label }
+ if$
+
+% leave label on the stack, to be popped when required.
+
+ "}{" * swap$ * "}{" * title * "}{" *
+% year field.or.null purify$ #-1 #4 substring$ *
+%
+% save the year for sort processing afterwards (adding a, b, c, etc.)
+%
+ year field.or.null purify$ #-1 #4 substring$
+ 'label.year :=
+}
+
+FUNCTION {output.bibitem}
+{ newline$
+
+ "\bibitem[\protect\citeauthortitleyear{" write$
+ calc.label write$
+ sort.year write$
+ "}]{" write$
+
+ cite$ write$
+ "}" write$
+ newline$
+ ""
+ before.all 'output.state :=
+}
+
+FUNCTION {article}
+{ output.bibitem
+ format.authors
+ "author" output.check
+ author format.key output % added
+ output.year.check % added
+ new.block
+ format.title
+ "title" output.check
+ new.block
+ crossref missing$
+ { format.jour.vol output
+ }
+ { format.article.crossref output.nonnull
+ format.pages output
+ }
+ if$
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {book}
+{ output.bibitem
+ author empty$
+ { format.editors
+ "author and editor" output.check }
+ { format.authors
+ output.nonnull
+ crossref missing$
+ { "author and editor" editor either.or.check }
+ 'skip$
+ if$
+ }
+ if$
+ output.year.check % added
+ new.block
+ format.btitle
+ "title" output.check
+ crossref missing$
+ { format.bvolume output
+ new.block
+ format.number.series output
+ new.sentence
+ address output
+ publisher "publisher" output.check.colon
+ }
+ { new.block
+ format.book.crossref output.nonnull
+ }
+ if$
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {booklet}
+{ output.bibitem
+ format.authors output
+ author format.key output % added
+ output.year.check % added
+ new.block
+ format.title
+ "title" output.check
+ new.block
+ howpublished output
+ address output
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {inbook}
+{ output.bibitem
+ author empty$
+ { format.editors
+ "author and editor" output.check
+ }
+ { format.authors output.nonnull
+ crossref missing$
+ { "author and editor" editor either.or.check }
+ 'skip$
+ if$
+ }
+ if$
+ output.year.check % added
+ new.block
+ format.btitle
+ "title" output.check
+ crossref missing$
+ { format.bvolume output
+ format.chapter.pages
+ "chapter and pages" output.check
+ new.block
+ format.number.series output
+ new.sentence
+ address output
+ publisher
+ "publisher" output.check.colon
+ }
+ { format.chapter.pages "chapter and pages" output.check
+ new.block
+ format.book.crossref output.nonnull
+ }
+ if$
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {incollection}
+{ output.bibitem
+ format.authors
+ "author" output.check
+ author format.key output % added
+ output.year.check % added
+ new.block
+ format.title
+ "title" output.check
+ new.block
+ crossref missing$
+ { format.in.ed.booktitle
+ "booktitle" output.check
+ format.bvolume output
+ format.number.series output
+ format.chapter.pages output % gnp - was special.output.nonnull
+% left out comma before page numbers
+ new.sentence
+ address output
+ publisher "publisher" output.check.colon
+ }
+ { format.incoll.inproc.crossref
+ output.nonnull
+ format.chapter.pages output
+ }
+ if$
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {inproceedings}
+{ output.bibitem
+ format.authors
+ "author" output.check
+ author format.key output % added
+ output.year.check % added
+ new.block
+ format.title
+ "title" output.check
+ new.block
+ crossref missing$
+ { format.in.ed.booktitle
+ "booktitle" output.check
+ format.bvolume output
+ format.number.series output
+ address output
+ format.pages output
+ new.sentence
+ organization output
+ publisher output.colon
+ }
+ { format.incoll.inproc.crossref output.nonnull
+ format.pages output
+ }
+ if$
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {conference} { inproceedings }
+
+FUNCTION {manual}
+{ output.bibitem
+ author empty$
+ { editor empty$
+ { organization "organization" output.check
+ organization format.key output } % if all else fails, use key
+ { format.editors "author and editor" output.check }
+ if$
+ }
+ { format.authors output.nonnull }
+ if$
+ output.year.check % added
+ new.block
+ format.btitle
+ "title" output.check
+ organization address new.block.checkb
+% Reversed the order of "address" and "organization", added the ":".
+ address output
+ organization "organization" output.check.colon
+% address output
+% ":" output
+% organization output
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {mastersthesis}
+{ output.bibitem
+ format.authors
+ "author" output.check
+ author format.key output % added
+ output.year.check % added
+ new.block
+ format.title
+ "title" output.check
+ new.block
+ "Master's thesis" format.thesis.type output.nonnull
+ school "school" output.check
+ address output
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {misc}
+{ output.bibitem
+ format.authors output
+ author format.key output % added
+ output.year.check % added
+ title howpublished new.block.checkb
+ format.title output
+ new.block
+ howpublished output
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {phdthesis}
+{ output.bibitem
+ format.authors
+ "author" output.check
+ author format.key output % added
+ output.year.check % added
+ new.block
+ format.btitle
+ "title" output.check
+ new.block
+ "Ph.\ D. thesis" format.thesis.type output.nonnull
+ school "school" output.check
+ address output
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {proceedings}
+{ output.bibitem
+ editor empty$
+ { organization output
+ organization format.key output } % gnp - changed from author format.key
+ { format.editors output.nonnull }
+ if$
+% author format.key output % gnp - removed (should be either
+% editor or organization
+ output.year.check % added (newapa)
+ new.block
+ format.btitle
+ "title" output.check
+ format.bvolume output
+ format.number.series output
+ address output
+ new.sentence
+ organization output
+ publisher output.colon
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {techreport}
+{ output.bibitem
+ format.authors
+ "author" output.check
+ author format.key output % added
+ output.year.check % added
+ new.block
+ format.title
+ "title" output.check
+ new.block
+ format.tr.number output.nonnull
+ institution
+ "institution" output.check
+ address output
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {unpublished}
+{ output.bibitem
+ format.authors
+ "author" output.check
+ author format.key output % added
+ output.year.check % added
+ new.block
+ format.title
+ "title" output.check
+ new.block
+ note "note" output.check
+ fin.entry
+}
+
+FUNCTION {default.type} { misc }
+
+MACRO {jan} {"January"}
+
+MACRO {feb} {"February"}
+
+MACRO {mar} {"March"}
+
+MACRO {apr} {"April"}
+
+MACRO {may} {"May"}
+
+MACRO {jun} {"June"}
+
+MACRO {jul} {"July"}
+
+MACRO {aug} {"August"}
+
+MACRO {sep} {"September"}
+
+MACRO {oct} {"October"}
+
+MACRO {nov} {"November"}
+
+MACRO {dec} {"December"}
+
+MACRO {acmcs} {"ACM Computing Surveys"}
+
+MACRO {acta} {"Acta Informatica"}
+
+MACRO {ai} {"Artificial Intelligence"}
+
+MACRO {cacm} {"Communications of the ACM"}
+
+MACRO {ibmjrd} {"IBM Journal of Research and Development"}
+
+MACRO {ibmsj} {"IBM Systems Journal"}
+
+MACRO {ieeese} {"IEEE Transactions on Software Engineering"}
+
+MACRO {ieeetc} {"IEEE Transactions on Computers"}
+
+MACRO {ieeetcad}
+ {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"}
+
+MACRO {ipl} {"Information Processing Letters"}
+
+MACRO {jacm} {"Journal of the ACM"}
+
+MACRO {jcss} {"Journal of Computer and System Sciences"}
+
+MACRO {scp} {"Science of Computer Programming"}
+
+MACRO {sicomp} {"SIAM Journal on Computing"}
+
+MACRO {tocs} {"ACM Transactions on Computer Systems"}
+
+MACRO {tods} {"ACM Transactions on Database Systems"}
+
+MACRO {tog} {"ACM Transactions on Graphics"}
+
+MACRO {toms} {"ACM Transactions on Mathematical Software"}
+
+MACRO {toois} {"ACM Transactions on Office Information Systems"}
+
+MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"}
+
+MACRO {tcs} {"Theoretical Computer Science"}
+
+READ
+
+FUNCTION {sortify}
+{ purify$
+ "l" change.case$
+}
+
+INTEGERS { len }
+
+FUNCTION {chop.word}
+{ 's :=
+ 'len :=
+ s #1 len substring$ =
+ { s len #1 + global.max$ substring$ }
+ 's
+ if$
+}
+
+
+
+FUNCTION {sort.format.names}
+{ 's :=
+ #1 'nameptr :=
+ ""
+ s num.names$ 'numnames :=
+ numnames 'namesleft :=
+ { namesleft #0 > }
+ { nameptr #2 =
+ { year field.or.null purify$ #-1 #4 substring$ * }
+ 'skip$
+ if$
+ nameptr #1 >
+ { " " * }
+ 'skip$
+ if$
+ s nameptr "{vv{ } }{ll{ }}{ f{ }}{ jj{ }}" format.name$ 't :=
+ nameptr numnames = t "others" = and
+ { " et~al" * }
+ { t sortify * }
+ if$
+ nameptr #1 + 'nameptr :=
+ namesleft #1 - 'namesleft :=
+ }
+ while$
+}
+
+FUNCTION {sort.format.title}
+{ 't :=
+ "A " #2
+ "An " #3
+ "The " #4 t chop.word
+ chop.word
+ chop.word
+ sortify
+ #1 global.max$ substring$
+}
+
+FUNCTION {author.sort}
+{ author empty$
+ { key empty$
+ { "to sort, need author or key in " cite$ * warning$
+ "" }
+ { key sortify }
+ if$
+ }
+ { author sort.format.names }
+ if$
+}
+
+FUNCTION {editor.sort}
+{ editor empty$
+ { key empty$
+ { "to sort, need editor or key in " cite$ * warning$
+ ""
+ }
+ { key sortify }
+ if$
+ }
+ { editor sort.format.names }
+ if$
+}
+
+FUNCTION {author.editor.sort}
+{ author empty$
+ { "missing author in " cite$ * warning$
+ editor empty$
+ { key empty$
+ { "to sort, need author, editor, or key in " cite$ * warning$
+ ""
+ }
+ { key sortify }
+ if$
+ }
+ { editor sort.format.names }
+ if$
+ }
+ { author sort.format.names }
+ if$
+}
+
+FUNCTION {author.organization.sort}
+%
+% added - GNP. Stack author or organization for sorting (from alpha.bst).
+% Unlike alpha.bst, we need entire names, not abbreviations
+%
+{ author empty$
+ { organization empty$
+ { key empty$
+ { "to sort, need author, organization, or key in " cite$ * warning$
+ ""
+ }
+ { key sortify }
+ if$
+ }
+ { organization sortify }
+ if$
+ }
+ { author sort.format.names }
+ if$
+}
+
+FUNCTION {editor.organization.sort}
+%
+% added - GNP. Stack editor or organization for sorting (from alpha.bst).
+% Unlike alpha.bst, we need entire names, not abbreviations
+%
+{ editor empty$
+ { organization empty$
+ { key empty$
+ { "to sort, need editor, organization, or key in " cite$ * warning$
+ ""
+ }
+ { key sortify }
+ if$
+ }
+ { organization sortify }
+ if$
+ }
+ { editor sort.format.names }
+ if$
+}
+
+FUNCTION {presort}
+%
+% Presort creates the bibentry's label via a call to calc.label, and then
+% sorts the entries based on entry type. Chicago.bst adds support for
+% including organizations as the sort key; the following is stolen from
+% alpha.bst.
+%
+{ %calc.label sortify % recalculate bibitem label
+ %year field.or.null purify$ #-1 #4 substring$ * % add year
+ %duplicate$ warning$
+ %" "
+ %*
+ type$ "book" =
+ type$ "inbook" =
+ or
+ 'author.editor.sort
+ { type$ "proceedings" =
+ 'editor.organization.sort
+ { type$ "manual" =
+ 'author.organization.sort
+ 'author.sort
+ if$
+ }
+ if$
+ }
+ if$
+ #1 entry.max$ substring$ % added for newapa
+ 'sort.label := % added for newapa
+ sort.label % added for newapa
+ %*
+ " "
+ *
+ title field.or.null
+ sort.format.title
+ *
+ #1 entry.max$ substring$
+ 'sort.key$ :=
+}
+
+ITERATE {presort}
+
+SORT % by label, year, author/editor, title
+
+STRINGS { last.label next.extra }
+
+INTEGERS { last.extra.num }
+
+FUNCTION {initialize.extra.label.stuff}
+{ #0 int.to.chr$ 'last.label :=
+ "" 'next.extra :=
+ #0 'last.extra.num :=
+}
+
+FUNCTION {forward.pass}
+%
+% Pass through all entries, comparing current entry to last one.
+% Need to concatenate year to the stack (done by calc.label) to determine
+% if two entries are the same (see presort)
+%
+{ last.label
+ calc.label.orig year field.or.null purify$ #-1 #4 substring$ * % add year
+ #1 entry.max$ substring$ = % are they equal?
+ { last.extra.num #1 + 'last.extra.num :=
+ last.extra.num int.to.chr$ 'extra.label :=
+ }
+ { "a" chr.to.int$ 'last.extra.num :=
+ "" 'extra.label :=
+ calc.label.orig year field.or.null purify$ #-1 #4 substring$ * % add year
+ #1 entry.max$ substring$ 'last.label := % assign to last.label
+ }
+ if$
+}
+
+FUNCTION {reverse.pass}
+{ next.extra "b" =
+ { "a" 'extra.label := }
+ 'skip$
+ if$
+ label.year extra.label * 'sort.year :=
+ extra.label 'next.extra :=
+}
+
+EXECUTE {initialize.extra.label.stuff}
+
+ITERATE {forward.pass}
+
+REVERSE {reverse.pass}
+
+FUNCTION {bib.sort.order}
+{ sort.label
+ " "
+ *
+ year field.or.null sortify
+ *
+ " "
+ *
+ title field.or.null
+ sort.format.title
+ *
+ #1 entry.max$ substring$
+ 'sort.key$ :=
+}
+
+ITERATE {bib.sort.order}
+
+SORT % by sort.label, year, title --- giving final bib. order.
+
+FUNCTION {begin.bib}
+
+{ preamble$ empty$
+ 'skip$
+ { preamble$ write$ newline$ }
+ if$
+ "\begin{thebibliography}{}" write$ newline$
+}
+
+
+EXECUTE {begin.bib}
+
+EXECUTE {init.state.consts}
+
+ITERATE {call.type$}
+
+FUNCTION {end.bib}
+{ newline$
+ "\end{thebibliography}" write$ newline$
+}
+
+EXECUTE {end.bib}
+
--- /dev/null
+% -*- LaTeX -*-
+%%% ====================================================================
+%%% @LaTeX-style-file{
+%%% author = "Glenn Paulley",
+%%% version = "4",
+%%% date = "31 August 1992",
+%%% time = "09:42:44 199",
+%%% filename = "chicago.sty",
+%%% address = "Data Structuring Group
+%%% Department of Computer Science
+%%% University of Waterloo
+%%% Waterloo, Ontario, Canada
+%%% N2L 3G1",
+%%% telephone = "(519) 885-1211",
+%%% FAX = "(519) 885-1208",
+%%% checksum = "44674 264 1050 10394",
+%%% email = "gnpaulle@bluebox.uwaterloo.ca",
+%%% codetable = "ISO/ASCII",
+%%% keywords = "",
+%%% supported = "yes",
+%%% abstract = "Contains the LaTeX style command definitions
+%%% for the Chicago BibTeX styles chicago.bst and
+%%% chicagoa.bst. For details, see below.",
+%%% docstring = "The checksum field above contains a CRC-16
+%%% checksum as the first value, followed by the
+%%% equivalent of the standard UNIX wc (word
+%%% count) utility output of lines, words, and
+%%% characters. This is produced by Robert
+%%% Solovay's checksum utility.",
+%%% }
+%%% ====================================================================
+%
+% chicago.sty: Style file for use with bibtex style chicago.bst, for
+% bibliographies formatted according to the 13th Edition of the Chicago
+% Manual of Style.
+%
+% 'newapa.bst' was made from 'plain.bst', 'named.bst', and 'apalike.bst',
+% with lots of tweaking to make it look like APA style, along with tips
+% from Young Ryu and Brian Reiser's modifications of 'apalike.bst'.
+% newapa.sty formed the basis of this style, chicago.sty. Author-date
+% references in newapa.bst formed the basis for chicago.bst. Chicagoa.bst
+% supports annotations.
+%
+% Version 4 (August, 1992):
+% - fixed chicago.bst and chicagoa.bst to handle long author lists in
+% sorting
+% - fixed chicago.bst and chicagoa.bst so that missing page numbers in
+% ``article'' entries are handled correctly
+% - modified chicago.sty to format entries with 2nd and subsequent lines
+% indented.
+%
+% Citation format: (author-last-name year)
+% (author-last-name and author-last-name year)
+% (author-last-name et al. year)
+% (author-last-name)
+% author-last-name
+% author-last-name (year)
+% (author-last-name and author-last-name)
+% (author-last-name et al.)
+% (year) or (year,year)
+% year or year,year
+%
+% Reference list ordering: alphabetical by author or whatever passes
+% for author in the absence of one.
+%
+% This BibTeX style has support for abbreviated author lists and for
+% year-only citations. This is done by having the citations
+% actually look like
+%
+% \citeauthoryear{full-author-info}{abbrev-author-info}{year}
+%
+% The LaTeX style has to have the following (or similar)
+%
+% \let\@internalcite\cite
+% \def\fullcite{\def\citeauthoryear##1##2##3{##1, ##3}\@internalcite}
+% \def\fullciteA{\def\citeauthoryear##1##2##3{##1}\@internalcite}
+% \def\shortcite{\def\citeauthoryear##1##2##3{##2, ##3}\@internalcite}
+% \def\shortciteA{\def\citeauthoryear##1##2##3{##2}\@internalcite}
+% \def\citeyear{\def\citeauthoryear##1##2##3{##3}\@internalcite}
+%
+% -------------------------------------------------------------------------
+% This file implements citations for the ``chicago'' bibliography style.
+% Place it in a file called chicago.sty in the TeX search path.
+%(Placing it in the same directory as the LaTeX document should also work.)
+%
+% This file is a modification of the ``newapa'' LaTeX style,
+% originally adapted by Steven Spencer from the ``apalike'' LaTeX style.
+% It was originally modified by Stephen N. Spencer, with further
+% modifications by Young U. Ryu.
+%
+% The ``chicago'' BibTeX bibliography style creates citations with labels:
+% \citeauthoryear{author-info}{abbrev. author-info}{year}
+%
+% These labels are processed by the following LaTeX commands:
+%
+% \cite{key}
+% which produces citations with full author list and year.
+% eg. (Brown 1978; Jarke, Turner, Stohl, et al. 1985)
+% \citeNP{key}
+% which produces citations with full author list and year, but without
+% enclosing parentheses:
+% eg. Brown 1978; Jarke, Turner and Stohl 1985
+% \citeA{key}
+% which produces citations with only the full author list.
+% eg. (Brown; Jarke, Turner and Stohl)
+% \citeANP{key}
+% which produces citations with only the full author list, without
+% parentheses eg. Brown; Jarke, Turner and Stohl
+% \citeN{key}
+% which produces citations with the full author list and year, but
+% can be used as nouns in a sentence; no parentheses appear around
+% the author names, but only around the year.
+% eg. Shneiderman (1978) states that......
+% \citeN should only be used for a single citation.
+% \shortcite{key}
+% which produces citations with abbreviated author list and year.
+% \shortciteNP{key}
+% which produces citations with abbreviated author list and year.
+% \shortciteA{key}
+% which produces only the abbreviated author list.
+% \shortciteANP{key}
+% which produces only the abbreviated author list.
+% \shortciteN{key}
+% which produces the abbreviated author list and year, with only the
+% year in parentheses. Use with only one citation.
+% \citeyear{key}
+% which produces the year information only, within parentheses.
+% \citeyearNP{key}
+% which produces the year information only.
+%
+% Abbreviated author lists use the ``et al.'' construct.
+%
+% `NP' means `no parentheses'.
+%
+% This LaTeX style file must be used with the ``chicago'' or ``chicagoa''
+% (annotated chicago style) BibTeX styles.
+%
+\typeout{Using Chicago Manual of Style bibliography: 31 August 1992}
+%
+% -------------------------------------------------------------------------
+%
+% Citation macros.
+%
+\def\chicagoand/{ and }
+\def\chicagoetal/{ et~al.}
+%
+\let\@internalcite\cite
+%
+\def\cite{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{(##1\if@tempswa , ##2\fi)}%
+ \def\citeauthortitleyear##1##2##3##4{##1\ ##4}\@internalcite}
+\def\citeNP{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{##1\if@tempswa , ##2\fi}%
+ \def\citeauthortitleyear##1##2##3##4{##1\ ##4}\@internalcite}
+\def\citetitleN{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{##1\if@tempswa , ##2)\else{)}\fi}%
+ \def\citeauthortitleyear##1##2##3##4{##3\ (##1; ##4}\@citedata}
+\def\citeN{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{##1\if@tempswa , ##2)\else{)}\fi}%
+ \def\citeauthortitleyear##1##2##3##4{##1\ (##4}\@citedata}
+\def\citeA{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{(##1\if@tempswa , ##2\fi)}%
+ \def\citeauthortitleyear##1##2##3##4{##1}\@internalcite}
+\def\citeANP{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{##1\if@tempswa , ##2\fi}%
+ \def\citeauthortitleyear##1##2##3##4{##1}\@internalcite}
+%
+\def\shortcite{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{(##1\if@tempswa , ##2\fi)}%
+ \def\citeauthortitleyear##1##2##3##4{##2\ ##4}\@internalcite}
+\def\shortciteNP{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{##1\if@tempswa , ##2\fi}%
+ \def\citeauthortitleyear##1##2##3##4{##2\ ##4}\@internalcite}
+\def\shortciteN{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{##1\if@tempswa , ##2)\else{)}\fi}%
+ \def\citeauthortitleyear##1##2##3##4{##2\ (##4}\@citedata}
+\def\shortciteA{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{(##1\if@tempswa , ##2\fi)}%
+ \def\citeauthortitleyear##1##2##3##4{##2}\@internalcite}
+\def\shortciteANP{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{##1\if@tempswa , ##2\fi}%
+ \def\citeauthortitleyear##1##2##3##4{##2}\@internalcite}
+%
+\def\citeyear{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{(##1\if@tempswa , ##2\fi)}%
+ \def\citeauthortitleyear##1##2##3##4{##4}\@citedata}
+\def\citeyearNP{\def\@citeseppen{-1000}%
+ \def\@cite##1##2{##1\if@tempswa , ##2\fi}%
+ \def\citeauthortitleyear##1##2##3##4{##4}\@citedata}
+
+%
+% \@citedata and \@citedatax:
+%
+% Place commas in-between citations in the same \citeyear, \citeyearNP,
+% \citeN, or \shortciteN command.
+% Use something like \citeN{ref1,ref2,ref3} and \citeN{ref4} for a list.
+%
+\def\@citedata{%
+ \@ifnextchar [{\@tempswatrue\@citedatax}%
+ {\@tempswafalse\@citedatax[]}%
+}
+
+\def\@citedatax[#1]#2{%
+\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi%
+ \def\@citea{}\@cite{\@for\@citeb:=#2\do%
+ {\@citea\def\@citea{), }\@ifundefined% by Young
+ {b@\@citeb}{{\bf ?}%
+ \@warning{Citation `\@citeb' on page \thepage \space undefined}}%
+{\csname b@\@citeb\endcsname}}}{#1}}%
+
+\@ifpackageloaded{hyperref}{%
+ \let\BRorg@citedatax\@citedatax
+ \def\@citedatax[#1]#2{%
+ \BRorg@citedatax[#1]{#2}%
+ \Hy@backout{#2}%
+ }%
+}{}
+\@ifpackageloaded{hyperref}{%
+\def\hyperemph#1{{\em\hyperpage{#1}}}%
+\def\bold#1{{\bf\hyperpage{#1}}}%
+}{%
+\def\hyperemph#1{{\em #1}}%
+\def\bold#1{{\bf #1}}%
+}
+
+\def\BR@@lbibitem[#1]#2#3\par{%
+ \BRorg@bibitem[#1]{#2}#3\hfill\penalty100\hbox{}
+ \newblock
+ \backref\hfill[{\csname br@#2\endcsname}%
+ ]\parskip=-10pt\penalty-10000\hbox{}\nobreak\par
+}%
+\def\BR@@bibitem#1#2\par{%
+ \BRorg@bibitem{#1}#2
+ \newblock
+ \backref\penalty-100\hbox{}\nobreak\hfill[\hbox{\csname br@#2\endcsname}%
+ ]\par
+}
+\def\thepageorcolor{\thepage}
+\def\Hy@backout#1{%
+ \@bsphack
+ \ifx\@empty\@currentlabel
+ \protected@write\@auxout{}{%
+ \string\@writefile{brf}{%
+ \string\backcite{#1}{{\thepageorcolor}{(document)}{Doc-Start}}%
+ }%
+ }%
+ \else
+ \protected@write\@auxout{}{%
+ \string\@writefile{brf}{%
+ \string\backcite{#1}{{\thepageorcolor}{\@currentlabel}{\@currentHref}}%
+ }%
+ }%
+ \fi
+ \@esphack
+}
+
+% don't box citations, separate with ; and a space
+% also, make the penalty between citations negative: a good place to break.
+%
+\def\@citex[#1]#2{%
+\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi%
+ \def\@citea{}\@cite{\@for\@citeb:=#2\do%
+ {\@citea\def\@citea{; }\@ifundefined% by Young
+ {b@\@citeb}{{\bf ?}%
+ \@warning{Citation `\@citeb' on page \thepage \space undefined}}%
+{\csname b@\@citeb\endcsname}}}{#1}}%
+
+% (from apalike.sty)
+% No labels in the bibliography.
+%
+\def\@biblabel#1{}
+
+% (from apalike.sty)
+% Set length of hanging indentation for bibliography entries.
+%
+\newlength{\bibhang}
+\setlength{\bibhang}{2em}
+
+% Indent second and subsequent lines of bibliographic entries. Stolen
+% from openbib.sty: \newblock is set to {}.
+
+\newdimen\bibindent
+\bibindent=1.5em
+\@ifundefined{refname}%
+ {\@ifundefined{chapter}%
+ {\newcommand{\refname}{References}}%
+ {\newcommand{\refname}{Bibliography}}%
+ }%
+ {}%
+\@ifundefined{chapter}%
+ {\def\thebibliography#1{\section*{\refname\@mkboth
+ {\uppercase{\refname}}{\uppercase{\refname}}}
+ \addcontentsline{toc}{section}{References}
+ \list
+ {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}
+ \leftmargin\labelwidth
+ \advance\leftmargin\labelsep
+ \advance\leftmargin\bibindent
+ \itemindent -\bibindent
+ \listparindent \itemindent
+ \parsep \z@
+ \usecounter{enumi}}
+ \def\newblock{}
+ \sloppy
+ \sfcode`\.=1000\relax}}
+ {\def\thebibliography#1{\chapter*{\refname\@mkboth
+ {\refname}{\refname}}
+ \addcontentsline{toc}{chapter}{References}
+ \list
+ {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}
+ \leftmargin\labelwidth
+ \advance\leftmargin\labelsep
+ \advance\leftmargin\bibindent
+ \itemindent -\bibindent
+ \listparindent \itemindent
+ \parsep \z@
+ \usecounter{enumi}}
+ \def\newblock{}
+ \sloppy
+ \sfcode`\.=1000\relax}}
--- /dev/null
+\section{Sets and Relations}
+
+\begin{definition}[Polyhedral Set]
+A {\em polyhedral set}\index{polyhedral set} $S$ is a finite union of basic sets
+$S = \bigcup_i S_i$, each of which can be represented using affine
+constraints
+$$
+S_i : \Q^n \to 2^{\Q^d} : \vec s \mapsto
+S_i(\vec s) =
+\{\, \vec x \in \Z^d \mid \exists \vec z \in \Z^e :
+A \vec x + B \vec s + D \vec z + \vec c \geq \vec 0 \,\}
+,
+$$
+with $A \in \Z^{m \times d}$,
+$B \in \Z^{m \times n}$,
+$D \in \Z^{m \times e}$
+and $\vec c \in \Z^m$.
+\end{definition}
+
+\begin{definition}[Parameter Domain of a Set]
+Let $S \in \Q^n \to 2^{\Q^d}$ be a set.
+The {\em parameter domain} of $S$ is the set
+$$\pdom S \coloneqq \{\, \vec s \in \Z^n \mid S(\vec s) \ne \emptyset \,\}.$$
+\end{definition}
+
+\begin{definition}[Polyhedral Relation]
+A {\em polyhedral relation}\index{polyhedral relation}
+$R$ is a finite union of basic relations
+$R = \bigcup_i R_i$ of type
+$\Q^n \to 2^{\Q^{d_1+d_2}}$,
+each of which can be represented using affine
+constraints
+$$
+R_i = \vec s \mapsto
+R_i(\vec s) =
+\{\, \vec x_1 \to \vec x_2 \in \Z^{d_1} \times \Z^{d_2}
+\mid \exists \vec z \in \Z^e :
+A_1 \vec x_1 + A_2 \vec x_2 + B \vec s + D \vec z + \vec c \geq \vec 0 \,\}
+,
+$$
+with $A_i \in \Z^{m \times d_i}$,
+$B \in \Z^{m \times n}$,
+$D \in \Z^{m \times e}$
+and $\vec c \in \Z^m$.
+\end{definition}
+
+\begin{definition}[Parameter Domain of a Relation]
+Let $R \in \Q^n \to 2^{\Q^{d+d}}$ be a relation.
+The {\em parameter domain} of $R$ is the set
+$$\pdom R \coloneqq \{\, \vec s \in \Z^n \mid R(\vec s) \ne \emptyset \,\}.$$
+\end{definition}
+
+\begin{definition}[Domain of a Relation]
+Let $R \in \Q^n \to 2^{\Q^{d+d}}$ be a relation.
+The {\em domain} of $R$ is the polyhedral set
+$$\domain R \coloneqq \vec s \mapsto
+\{\, \vec x_1 \in \Z^{d_1} \mid \exists \vec x_2 \in \Z^{d_2} :
+(\vec x_1, \vec x_2) \in R(\vec s) \,\}
+.
+$$
+\end{definition}
+
+\begin{definition}[Range of a Relation]
+Let $R \in \Q^n \to 2^{\Q^{d+d}}$ be a relation.
+The {\em range} of $R$ is the polyhedral set
+$$
+\range R \coloneqq \vec s \mapsto
+\{\, \vec x_2 \in \Z^{d_2} \mid \exists \vec x_1 \in \Z^{d_1} :
+(\vec x_1, \vec x_2) \in R(\vec s) \,\}
+.
+$$
+\end{definition}
+
+\begin{definition}[Composition of Relations]
+Let $R \in \Q^n \to 2^{\Q^{d_1+d_2}}$ and
+$S \in \Q^n \to 2^{\Q^{d_2+d_3}}$ be two relations,
+then the composition of
+$R$ and $S$ is defined as
+$$
+S \circ R \coloneqq
+\vec s \mapsto
+\{\, \vec x_1 \to \vec x_3 \in \Z^{d_1} \times \Z^{d_3}
+\mid \exists \vec x_2 \in \Z^{d_2} :
+\vec x_1 \to \vec x_2 \in R(\vec s) \wedge
+\vec x_2 \to \vec x_3 \in S(\vec s)
+\,\}
+.
+$$
+\end{definition}
+
+\begin{definition}[Difference Set of a Relation]
+Let $R \in \Q^n \to 2^{\Q^{d+d}}$ be a relation.
+The difference set ($\Delta \, R$) of $R$ is the set
+of differences between image elements and the corresponding
+domain elements,
+$$
+\Delta \, R \coloneqq
+\vec s \mapsto
+\{\, \vec \delta \in \Z^{d} \mid \exists \vec x \to \vec y \in R :
+\vec \delta = \vec y - \vec x
+\,\}
+$$
+\end{definition}
+
+\section{Transitive Closure}
+
+\subsection{Introduction}
+
+\begin{definition}[Power of a Relation]
+Let $R \in \Q^n \to 2^{\Q^{d+d}}$ be a relation and
+$k \in \Z_{\ge 1}$
+a positive number, then power $k$ of relation $R$ is defined as
+\begin{equation}
+\label{eq:transitive:power}
+R^k \coloneqq
+\begin{cases}
+R & \text{if $k = 1$}
+\\
+R \circ R^{k-1} & \text{if $k \ge 2$}
+.
+\end{cases}
+\end{equation}
+\end{definition}
+
+\begin{definition}[Transitive Closure of a Relation]
+Let $R \in \Q^n \to 2^{\Q^{d+d}}$ be a relation,
+then the transitive closure $R^+$ of $R$ is the union
+of all positive powers of $R$,
+$$
+R^+ \coloneqq \bigcup_{k \ge 1} R^k
+.
+$$
+\end{definition}
+Alternatively, the transitive closure may be defined
+inductively as
+\begin{equation}
+\label{eq:transitive:inductive}
+R^+ \coloneqq R \cup \left(R \circ R^+\right)
+.
+\end{equation}
+
+Since the transitive closure of a polyhedral relation
+may no longer be a polyhedral relation \shortcite{Kelly1996closure},
+we can, in the general case, only compute an approximation
+of the transitive closure.
+Whereas \shortciteN{Kelly1996closure} compute underapproximations,
+we, like \shortciteN{Beletska2009}, compute overapproximations.
+That is, given a relation $R$, we will compute a relation $T$
+such that $R^+ \subseteq T$. Of course, we want this approximation
+to be as close as possible to the actual transitive closure
+$R^+$ and we want to detect the cases where the approximation is
+exact, i.e., where $T = R^+$.
+
+For computing an approximation of the transitive closure of $R$,
+we follow the same general strategy as \shortciteN{Beletska2009}
+and first compute an approximation of $R^k$ for $k \ge 1$ and then project
+out the parameter $k$ from the resulting relation.
+
+\begin{example}
+As a trivial example, consider the relation
+$R = \{\, x \to x + 1 \,\}$. The $k$th power of this map
+for arbitrary $k$ is
+$$
+R^k = k \mapsto \{\, x \to x + k \mid k \ge 1 \,\}
+.
+$$
+The transitive closure is then
+$$
+\begin{aligned}
+R^+ & = \{\, x \to y \mid \exists k \in \Z_{\ge 1} : y = x + k \,\}
+\\
+& = \{\, x \to y \mid y \ge x + 1 \,\}
+.
+\end{aligned}
+$$
+\end{example}
+
+\subsection{Computing an Approximation of $R^k$}
+
+In general, it is impossible to construct a closed form
+of $R^k$ as a polyhedral relation.
+We will therefore need to make some approximations.
+As a first approximations, we will consider each of the basic
+relations in $R$ as simply adding one or more offsets to a domain element
+to arrive at an image element and ignore the fact that some of these
+offsets may only be applied to some of the domain elements.
+That is, we will only consider the difference set $\Delta\,R$ of the relation.
+In particular, we will first construct a collection $P$ of paths
+that move through
+a total of $k$ offsets and then intersect domain and range of this
+collection with those of $R$.
+That is,
+\begin{equation}
+\label{eq:transitive:approx}
+K = P \cap \left(\domain R \to \range R\right)
+,
+\end{equation}
+with
+\begin{equation}
+\label{eq:transitive:path}
+P = \vec s \mapsto \{\, \vec x \to \vec y \mid
+\exists k_i \in \Z_{\ge 0} :
+\vec y = \vec x + \sum_i k_i \, \Delta_i(\vec s)
+\wedge
+\sum_i k_i = k > 0
+\,\}
+\end{equation}
+and with $\Delta_i$ the basic sets that compose
+the difference set $\Delta\,R$.
+Note that the number of basic sets $\Delta_i$ need not be
+the same as the number of basic relations in $R$.
+Also note that since addition is commutative, it does not
+matter in which order we add the offsets and so we are allowed
+to group them as we did in \eqref{eq:transitive:path}.
+
+If all the $\Delta_i$s are singleton sets
+$\Delta_i = \{\, \vec \delta_i \,\}$ with $\vec \delta_i \in \Z^d$,
+then \eqref{eq:transitive:path} simplifies to
+\begin{equation}
+\label{eq:transitive:singleton}
+P = \{\, \vec x \to \vec y \mid
+\exists k_i \in \Z_{\ge 0} :
+\vec y = \vec x + \sum_i k_i \, \vec \delta_i
+\wedge
+\sum_i k_i = k > 0
+\,\}
+\end{equation}
+and then the approximation computed in \eqref{eq:transitive:approx}
+is essentially the same as that of \shortciteN{Beletska2009}.
+If some of $\Delta_i$s are not singleton sets or if
+some of $\vec \delta_i$s are parametric, then we need
+to resort to further approximations.
+
+To ease both the exposition and the implementation, we will for
+the remainder of this section work with extended offsets
+$\Delta_i' = \Delta_i \times \{\, 1 \,\}$.
+That is, each offset is extended with an extra coordinate that is
+set equal to one. The paths constructed by summing such extended
+offsets have the length encoded as the difference of their
+final coordinates. The path $P'$ can then be decomposed into
+paths $P_i'$, one for each $\Delta_i$,
+\begin{equation}
+\label{eq:transitive:decompose}
+P' = \left(
+(P_m' \cup \identity) \circ \cdots \circ
+(P_2' \cup \identity) \circ
+(P_1' \cup \identity)
+\right) \cap
+\{\,
+\vec x' \to \vec y' \mid y_{d+1} - x_{d+1} = k > 0
+\,\}
+,
+\end{equation}
+with
+$$
+P_i' = \vec s \mapsto \{\, \vec x' \to \vec y' \mid
+\exists k \in \Z_{\ge 1} :
+\vec y' = \vec x' + k \, \Delta_i'(\vec s)
+\,\}
+.
+$$
+Note that each $P_i'$ contains paths of length at least one.
+We therefore need to take the union with the identity relation
+when composing the $P_i'$s to allow for paths that do not contain
+any offsets from one or more $\Delta_i'$.
+The path that consists of only identity relations is removed
+by imposing the constraint $y_{d+1} - x_{d+1} > 0$.
+
+Let us now consider how to compute an overapproximation of $P_i'$.
+Those that correspond to singleton $\Delta_i$s are grouped together
+and handled as in \eqref{eq:transitive:singleton}.
+Note that this is just an optimization. The procedure described
+below would produce results that are at least as accurate.
+For simplicity, we will assume that no constraint in $\Delta_i'$
+involves any existentially quantified variables. Dropping such
+constraints results in a larger $\Delta_i'$. An overapproximation
+of the paths corresponding to this larger set will also be an
+overapproximation of the paths corresponding to the original set.
+Without existentially quantified variables, we can classify
+the constraints of $\Delta_i'$ as follows
+\begin{enumerate}
+\item non-parametric constraints
+\begin{equation}
+\label{eq:transitive:non-parametric}
+A_1 \vec x + \vec c_1 \geq \vec 0
+\end{equation}
+\item purely parametric constraints
+\begin{equation}
+\label{eq:transitive:parametric}
+B_2 \vec s + \vec c_2 \geq \vec 0
+\end{equation}
+\item negative mixed constraints
+\begin{equation}
+\label{eq:transitive:mixed}
+A_3 \vec x + B_3 \vec s + \vec c_3 \geq \vec 0
+\end{equation}
+such that for each row $j$ and for all $\vec s$,
+$$
+\Delta_i'(\vec s) \cap
+\{\, \vec x' \to \vec y' \mid B_{3,j} \vec s + c_{3,j} > 0 \,\}
+= \emptyset
+$$
+\item positive mixed constraints
+$$
+A_4 \vec x + B_4 \vec s + \vec c_4 \geq \vec 0
+$$
+such that for each row $j$, there is at least one $\vec s$ such that
+$$
+\Delta_i'(\vec s) \cap
+\{\, \vec x' \to \vec y' \mid B_{4,j} \vec s + c_{4,j} > 0 \,\}
+\ne \emptyset
+$$
+\end{enumerate}
+We will use the following approximation $Q_i$ for $P_i'$:
+\begin{equation}
+\label{eq:transitive:Q}
+\begin{aligned}
+Q_i = \vec s \mapsto
+\{\,
+\vec x' \to \vec y'
+\mid {} & \exists k \in \Z_{\ge 1}, \vec f \in \Z^d :
+\vec y' = \vec x' + (\vec f, k)
+\wedge {}
+\\
+&
+A_1 \vec f + k \vec c_1 \geq \vec 0
+\wedge
+B_2 \vec s + \vec c_2 \geq \vec 0
+\wedge
+A_3 \vec f + B_3 \vec s + \vec c_3 \geq \vec 0
+\,\}
+.
+\end{aligned}
+\end{equation}
+To prove that $Q_i$ is indeed an overapproximation of $P_i'$,
+we need to show that for every $\vec s \in \Z^n$, for every
+$k \in \Z_{\ge 1}$ and for every $\vec f \in k \, \Delta_i(\vec s)$
+we have that
+$(\vec f, k)$ satisfies the constraints in \eqref{eq:transitive:Q}.
+If $\Delta_i(\vec s)$ is non-empty, then $\vec s$ must satisfy
+the constraints in \eqref{eq:transitive:parametric}.
+Each element $(\vec f, k) \in k \, \Delta_i'(\vec s)$ is a sum
+of $k$ elements $(\vec f_j, 1)$ in $\Delta_i'(\vec s)$.
+Each of these elements satisfies the constraints in
+\eqref{eq:transitive:non-parametric}, i.e.,
+$$
+\left[
+\begin{matrix}
+A_1 & \vec c_1
+\end{matrix}
+\right]
+\left[
+\begin{matrix}
+\vec f_j \\ 1
+\end{matrix}
+\right]
+\ge \vec 0
+.
+$$
+The sum of these elements therefore satisfies the same set of inequalities,
+i.e., $A_1 \vec f + k \vec c_1 \geq \vec 0$.
+Finally, the constraints in \eqref{eq:transitive:mixed} are such
+that for any $\vec s$ in the parameter domain of $\Delta$,
+we have $-\vec r(\vec s) \coloneqq B_3 \vec s + \vec c_3 \le \vec 0$,
+i.e., $A_3 \vec f_j \ge \vec r(\vec s) \ge \vec 0$
+and therefore also $A_3 \vec f \ge \vec r(\vec s)$.
+Note that if there are no mixed constraints and if the
+rational relaxation of $\Delta_i(\vec s)$, i.e.,
+$\{\, \vec x \in \Q^d \mid A_1 \vec x + \vec c_1 \ge \vec 0\,\}$,
+has integer vertices, then the approximation is exact, i.e.,
+$Q_i = P_i'$. In this case, the vertices of $\Delta'_i(\vec s)$
+generate the rational cone
+$\{\, \vec x' \in \Q^{d+1} \mid \left[
+\begin{matrix}
+A_1 & \vec c_1
+\end{matrix}
+\right] \vec x' \,\}$ and therefore $\Delta'_i(\vec s)$ is
+a Hilbert basis of this cone \shortcite[Theorem~16.4]{Schrijver1986}.
+
+The accurateness of the above approach
+can be improved by also classifying the existentially
+quantified variables into variables that are uniquely
+determined by the parameters, variables that are independent
+of the parameters and others. The first set can be treated
+as parameters and the second as variables. Constraints involving
+the other existentially quantified variables should continue to
+be removed.
+
+\subsection{Checking Exactness}
+
+The approximation $T$ for the transitive closure $R^+$ can be obtained
+by projecting out the parameter $k$ from the approximation $K$
+\eqref{eq:transitive:approx} of the power $R^k$.
+Since $K$ is an overapproximation of $R^k$, $T$ will also be an
+overapproximation of $R^+$.
+To check whether the results are exact, we need to consider two
+cases depending on whether $R$ is {\em cyclic}, where $R$ is defined
+to be cyclic if $R^+$ maps any element to itself, i.e.,
+$R^+ \cap \identity \ne \emptyset$.
+If $R$ is acyclic, then the inductive definition of
+\eqref{eq:transitive:inductive} is equivalent to its completion,
+i.e.,
+$$
+R^+ = R \cup \left(R \circ R^+\right)
+$$
+is a defining property.
+Since $T$ is known to be an overapproximation, we only need to check
+whether
+$$
+T \subseteq R \cup \left(R \circ T\right)
+.
+$$
+This is essentially Theorem~5 of \shortciteN{Kelly1996closure}.
+The only difference is that they only consider lexicographically
+forward relations, a special case of acyclic relation.
+
+If, on the other hand, $R$ is cyclic, then we have to resort
+to checking whether the approximation $K$ of the power is exact.
+Note that $T$ may be exact even if $K$ is not exact, so the check
+is sound, but incomplete.
+To check exactness of the power, we simply need to check
+\eqref{eq:transitive:power}. Since again $K$ is known
+to be an overapproximation, we only need to check whether
+$$
+\begin{aligned}
+K'|_{y_{d+1} - x_{d+1} = 1} & \subseteq R'
+\\
+K'|_{y_{d+1} - x_{d+1} \ge 2} & \subseteq R' \circ K'|_{y_{d+1} - x_{d+1} \ge 1}
+,
+\end{aligned}
+$$
+where $R' = \{\, \vec x' \to \vec y' \mid \vec x \to \vec y \in R
+\wedge y_{d+1} - x_{d+1} = 1\,\}$, i.e., $R$ extended with path
+lengths equal to 1.
+
+All that remains is to explain how to check the cyclicity of $R$.
+Note that the exactness on the power is always sound, even
+in the acyclic case, so we only need to be careful that we find
+all cyclic cases. Now, if $R$ is cyclic, i.e.,
+$R^+ \cap \identity \ne \emptyset$, then, since $T$ is
+an overapproximation of $R^+$, also
+$T \cap \identity \ne \emptyset$. This in turn means
+that $\Delta \, K'$ contains a point whose first $d$ coordinates
+are zero and whose final coordinate is positive.
+In the implementation we currently perform this test on $P'$ instead of $K'$.
+Note that if $R^+$ is acyclic and $T$ is not, then the approximation
+is clearly not exact and the approximation of the power $K$
+will not be exact either.
+
+\subsection{Decomposing $R$ into strongly connected components}
+
+If the input relation $R$ is a union of several basic relations
+that can be partially ordered
+then the accuracy of the approximation may be improved by computing
+an approximation of each strongly connected components separately.
+For example, if $R = R_1 \cup R_2$ and $R_1 \circ R_2 = \emptyset$,
+then we know that any path that passes through $R_2$ cannot later
+pass through $R_1$, i.e.,
+$$
+R^+ = R_1^+ \cup R_2^+ \cup \left(R_2^+ \circ R_1^+\right)
+.
+$$
+We can therefore compute (approximations of) transitive closures
+of $R_1$ and $R_2$ separately.
+Note, however, that the condition $R_1 \circ R_2 = \emptyset$
+is actually too strong.
+If $R_1 \circ R_2$ is a subset of $R_2 \circ R_1$
+then we can reorder the segments
+in any path that moves through both $R_1$ and $R_2$ to
+first move through $R_1$ and then through $R_2$.
+
+This idea can be generalized to relations that are unions
+of more than two basic relations by constructing the
+strongly connected components in the graph with as vertices
+the basic relations and an edge between two basic relations
+$R_i$ and $R_j$ if $R_i$ needs to follow $R_j$ in some paths.
+That is, there is an edge from $R_i$ to $R_j$ iff
+\begin{equation}
+\label{eq:transitive:edge}
+R_i \circ R_j
+\not\subseteq
+R_j \circ R_i
+.
+\end{equation}
+The components can be obtained from the graph by applying
+Tarjan's algorithm \shortcite{Tarjan1972}.
+
+In practice, we compute the (extended) powers $K_i'$ of each component
+separately and then compose them as in \eqref{eq:transitive:decompose}.
+Note, however, that in this case the order in which we apply them is
+important and should correspond to a topological ordering of the
+strongly connected components. Simply applying Tarjan's
+algorithm will produce topologically sorted strongly connected components.
+The graph on which Tarjan's algorithm is applied is constructed on-the-fly.
+That is, whenever the algorithm checks if there is an edge between
+two vertices, we evaluate \eqref{eq:transitive:edge}.
+The exactness check is performed on each component separately.
+If the approximation turns out to be inexact for any of the components,
+then the entire result is marked inexact and the exactness check
+is skipped on the components that still need to be handled.
+
+\begin{figure}
+\begin{center}
+\begin{tikzpicture}[x=0.5cm,y=0.5cm,>=stealth,shorten >=1pt]
+\foreach \x in {1,...,10}{
+ \foreach \y in {1,...,10}{
+ \draw[->] (\x,\y) -- (\x,\y+1);
+ }
+}
+\foreach \x in {1,...,20}{
+ \foreach \y in {5,...,15}{
+ \draw[->] (\x,\y) -- (\x+1,\y);
+ }
+}
+\end{tikzpicture}
+\end{center}
+\caption{The relation from \autoref{ex:closure4}}
+\label{f:closure4}
+\end{figure}
+\begin{example}
+\label{ex:closure4}
+Consider the relation in example {\tt closure4} that comes with
+the Omega calculator~\shortcite{Omega_calc}, $R = R_1 \cup R_2$,
+with
+$$
+\begin{aligned}
+R_1 & = \{\, (x,y) \to (x,y+1) \mid 1 \le x,y \le 10 \,\}
+\\
+R_2 & = \{\, (x,y) \to (x+1,y) \mid 1 \le x \le 20 \wedge 5 \le y \le 15 \,\}
+.
+\end{aligned}
+$$
+This relation is shown graphically in \autoref{f:closure4}.
+We have
+$$
+\begin{aligned}
+R_1 \circ R_2 &=
+\{\, (x,y) \to (x+1,y+1) \mid 1 \le x \le 9 \wedge 5 \le y \le 10 \,\}
+\\
+R_2 \circ R_1 &=
+\{\, (x,y) \to (x+1,y+1) \mid 1 \le x \le 10 \wedge 4 \le y \le 10 \,\}
+.
+\end{aligned}
+$$
+Clearly, $R_1 \circ R_2 \subseteq R_2 \circ R_1$ and so
+$$
+\left(
+R_1 \cup R_2
+\right)^+
+=
+R_2^+ \circ R_1^+
+.
+$$
+\end{example}
+
+\begin{figure}
+\newcounter{n}
+\newcounter{t1}
+\newcounter{t2}
+\newcounter{t3}
+\newcounter{t4}
+\begin{center}
+\begin{tikzpicture}[>=stealth,shorten >=1pt]
+\setcounter{n}{7}
+\foreach \i in {1,...,\value{n}}{
+ \foreach \j in {1,...,\value{n}}{
+ \setcounter{t1}{2 * \j - 4 - \i + 1}
+ \setcounter{t2}{\value{n} - 3 - \i + 1}
+ \setcounter{t3}{2 * \i - 1 - \j + 1}
+ \setcounter{t4}{\value{n} - \j + 1}
+ \ifnum\value{t1}>0\ifnum\value{t2}>0
+ \ifnum\value{t3}>0\ifnum\value{t4}>0
+ \draw[thick,->] (\i,\j) to[out=20] (\i+3,\j);
+ \fi\fi\fi\fi
+ \setcounter{t1}{2 * \j - 1 - \i + 1}
+ \setcounter{t2}{\value{n} - \i + 1}
+ \setcounter{t3}{2 * \i - 4 - \j + 1}
+ \setcounter{t4}{\value{n} - 3 - \j + 1}
+ \ifnum\value{t1}>0\ifnum\value{t2}>0
+ \ifnum\value{t3}>0\ifnum\value{t4}>0
+ \draw[thick,->] (\i,\j) to[in=-20,out=20] (\i,\j+3);
+ \fi\fi\fi\fi
+ \setcounter{t1}{2 * \j - 1 - \i + 1}
+ \setcounter{t2}{\value{n} - 1 - \i + 1}
+ \setcounter{t3}{2 * \i - 1 - \j + 1}
+ \setcounter{t4}{\value{n} - 1 - \j + 1}
+ \ifnum\value{t1}>0\ifnum\value{t2}>0
+ \ifnum\value{t3}>0\ifnum\value{t4}>0
+ \draw[thick,->] (\i,\j) to (\i+1,\j+1);
+ \fi\fi\fi\fi
+ }
+}
+\end{tikzpicture}
+\end{center}
+\caption{The relation from \autoref{ex:decomposition}}
+\label{f:decomposition}
+\end{figure}
+\begin{example}
+\label{ex:decomposition}
+Consider the relation on the right of \shortciteN[Figure~2]{Beletska2009},
+reproduced in \autoref{f:decomposition}.
+The relation can be described as $R = R_1 \cup R_2 \cup R_3$,
+with
+$$
+\begin{aligned}
+R_1 &= n \mapsto \{\, (i,j) \to (i+3,j) \mid
+i \le 2 j - 4 \wedge
+i \le n - 3 \wedge
+j \le 2 i - 1 \wedge
+j \le n \,\}
+\\
+R_2 &= n \mapsto \{\, (i,j) \to (i,j+3) \mid
+i \le 2 j - 1 \wedge
+i \le n \wedge
+j \le 2 i - 4 \wedge
+j \le n - 3 \,\}
+\\
+R_3 &= n \mapsto \{\, (i,j) \to (i+1,j+1) \mid
+i \le 2 j - 1 \wedge
+i \le n - 1 \wedge
+j \le 2 i - 1 \wedge
+j \le n - 1\,\}
+.
+\end{aligned}
+$$
+The figure shows this relation for $n = 7$.
+Both
+$R_3 \circ R_1 \subseteq R_1 \circ R_3$
+and
+$R_3 \circ R_2 \subseteq R_2 \circ R_3$,
+which the reader can verify using the {\tt iscc} calculator:
+\begin{verbatim}
+R1 := [n] -> { [i,j] -> [i+3,j] : i <= 2 j - 4 and i <= n - 3 and
+ j <= 2 i - 1 and j <= n };
+R2 := [n] -> { [i,j] -> [i,j+3] : i <= 2 j - 1 and i <= n and
+ j <= 2 i - 4 and j <= n - 3 };
+R3 := [n] -> { [i,j] -> [i+1,j+1] : i <= 2 j - 1 and i <= n - 1 and
+ j <= 2 i - 1 and j <= n - 1 };
+(R1 . R3) - (R3 . R1);
+(R2 . R3) - (R3 . R2);
+\end{verbatim}
+$R_3$ can therefore be moved forward in any path.
+For the other two basic relations, we have both
+$R_2 \circ R_1 \not\subseteq R_1 \circ R_2$
+and
+$R_1 \circ R_2 \not\subseteq R_2 \circ R_1$
+and so $R_1$ and $R_2$ form a strongly connected component.
+By computing the power of $R_3$ and $R_1 \cup R_2$ separately
+and composing the results, the power of $R$ can be computed exactly
+using \eqref{eq:transitive:singleton}.
+As explained by \shortciteN{Beletska2009}, applying the same formula
+to $R$ directly, without a decomposition, would result in
+an overapproximation of the power.
+\end{example}
--- /dev/null
+@inproceedings{Kelly1996closure,
+ author = {Wayne Kelly and
+ William Pugh and
+ Evan Rosser and
+ Tatiana Shpeisman},
+ title = {Transitive Closure of Infinite Graphs and Its Applications},
+ pages = {126-140},
+ editor = {Chua-Huang Huang and
+ P. Sadayappan and
+ Utpal Banerjee and
+ David Gelernter and
+ Alexandru Nicolau and
+ David A. Padua},
+ booktitle = {Languages and Compilers for Parallel Computing, 8th International
+ Workshop, LCPC'95, Columbus, Ohio, USA, August 10-12, 1995,
+ Proceedings},
+ booktitle = {LCPC},
+ publisher = {Springer},
+ series = {Lecture Notes in Computer Science},
+ volume = {1033},
+ year = {1996},
+ isbn = {3-540-60765-X},
+}
+
+@inproceedings{Beletska2009,
+ author = {Beletska, Anna and Barthou, Denis and Bielecki, Wlodzimierz and Cohen, Albert},
+ title = {Computing the Transitive Closure of a Union of Affine Integer Tuple Relations},
+ booktitle = {COCOA '09: Proceedings of the 3rd International Conference on Combinatorial Optimization and Applications},
+ year = {2009},
+ isbn = {978-3-642-02025-4},
+ pages = {98--109},
+ location = {Huangshan, China},
+ doi = {10.1007/978-3-642-02026-1_9},
+ publisher = {Springer-Verlag},
+ address = {Berlin, Heidelberg},
+}
+
+@book{Schrijver1986,
+ author = "Schrijver, Alexander",
+ title = "Theory of Linear and Integer Programming",
+ publisher = "John Wiley \& Sons",
+ year = 1986
+}
+
+@article{Tarjan1972,
+ author = {Tarjan, Robert},
+ journal = {SIAM Journal on Computing},
+ number = {2},
+ pages = {146--160},
+ publisher = {SIAM},
+ title = {Depth-First Search and Linear Graph Algorithms},
+ volume = {1},
+ year = {1972}
+}
+
+@TechReport{ Omega_calc,
+ author = "Wayne Kelly and Vadim Maslov and William Pugh and Evan Rosser and Tatiana Shpeisman and Dave Wonnacott",
+ title = "The {Omega} Calculator and Library",
+ month = nov,
+ institution = "University of Maryland",
+ year = 1996
+}
-\documentclass{article}
+\documentclass{report}
+\usepackage[plainpages=false,pdfpagelabels,breaklinks,pagebackref]{hyperref}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{txfonts}
+\usepackage{chicago}
+\usepackage{aliascnt}
+\usepackage{tikz}
+\usepackage{calc}
+
+\def\vec#1{\mathchoice{\mbox{\boldmath$\displaystyle\bf#1$}}
+{\mbox{\boldmath$\textstyle\bf#1$}}
+{\mbox{\boldmath$\scriptstyle\bf#1$}}
+{\mbox{\boldmath$\scriptscriptstyle\bf#1$}}}
+
+\newtheorem{theorem}{Theorem}
+\newaliascnt{example}{theorem}
+\newtheorem{example}[example]{Example}
+\newaliascnt{def}{theorem}
+\newtheorem{definition}[def]{Definition}
+\aliascntresetthe{example}
+\aliascntresetthe{def}
+\numberwithin{theorem}{section}
+\numberwithin{def}{section}
+\numberwithin{example}{section}
+
+\newcommand{\exampleautorefname}{Example}
+
+\def\Z{\mathbb{Z}}
+\def\Q{\mathbb{Q}}
+
+\def\pdom{\mathop{\rm pdom}\nolimits}
+\def\domain{\mathop{\rm dom}\nolimits}
+\def\range{\mathop{\rm ran}\nolimits}
+\def\identity{\mathop{\rm Id}\nolimits}
\begin{document}
\maketitle
\tableofcontents
+\chapter{User Manual}
+
\input{user}
+\chapter{Implementation Details}
+
+\input{implementation}
+
+\bibliography{isl}
+\bibliographystyle{chicago}
+
\end{document}
*
* R_1 \circ R_2
*
- * is non-empty and that moreover, it is non-empty on the set
- * of elements that do not get mapped to the same set of elements
- * by both "R_1 \circ R_2" and "R_2 \circ R_1".
- * For elements that do get mapped to the same elements by these
- * two compositions, R_1 and R_2 are commutative, so if these
- * elements are the only ones for which R_1 \circ R_2 is non-empty,
- * then you may just as well apply R_1 first.
+ * is a subset of
+ *
+ * R_2 \circ R_1
+ *
+ * If so, then there is no reason for R_1 to immediately follow R_2
+ * in any path.
*/
static int basic_map_follows(__isl_keep isl_basic_map *bmap1,
__isl_keep isl_basic_map *bmap2)
{
struct isl_map *map12 = NULL;
struct isl_map *map21 = NULL;
- struct isl_map *d = NULL;
- struct isl_set *dom = NULL;
- int empty;
+ int subset;
map21 = isl_map_from_basic_map(
isl_basic_map_apply_range(
isl_basic_map_copy(bmap2),
isl_basic_map_copy(bmap1)));
- empty = isl_map_is_empty(map21);
- if (empty < 0)
+ subset = isl_map_is_empty(map21);
+ if (subset < 0)
goto error;
- if (empty) {
+ if (subset) {
isl_map_free(map21);
return 0;
}
isl_basic_map_apply_range(
isl_basic_map_copy(bmap1),
isl_basic_map_copy(bmap2)));
- d = isl_map_subtract(isl_map_copy(map12), isl_map_copy(map21));
- d = isl_map_union(d,
- isl_map_subtract(isl_map_copy(map21), isl_map_copy(map12)));
- dom = isl_map_domain(d);
- map21 = isl_map_intersect_domain(map21, dom);
- empty = isl_map_is_empty(map21);
+ subset = isl_map_is_subset(map21, map12);
isl_map_free(map12);
isl_map_free(map21);
- return empty < 0 ? -1 : !empty;
+ return subset < 0 ? -1 : !subset;
error:
isl_map_free(map21);
return -1;
projects / platform / upstream / isl.git / commitdiff