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......@@ -1237,4 +1237,104 @@ year = {2005},
Part {I}},
pages = {269--280},
year = {2015}
}
@incollection{BKOSSB07,
year={2007},
booktitle={Tools and Algorithms for the Construction and Analysis of Systems, {TACAS} 2007},
volume={4424},
series={LNCS},
title={{Deciding Bit-Vector Arithmetic with Abstraction}},
publisher={Springer},
author={Bryant, Randal E. and Kroening, Daniel and Ouaknine, Jo{\"{e}}l and Seshia, Sanjit A. and Strichman, Ofer and Brady, Bryan},
pages={358--372},
}
@book{Har09,
author = {Harrison, John},
title = {{Handbook of Practical Logic and Automated Reasoning}},
year = 2009,
edition = {1st},
publisher = {Cambridge University Press},
}
@TECHREPORT{BFT15,
author = {Clark Barrett and Pascal Fontaine and Cesare Tinelli},
title = {{The SMT-LIB Standard: Version 2.5}},
institution = {Department of Computer Science, The University of Iowa},
year = 2015,
note = {Available at {\tt www.SMT-LIB.org}}
}
@inproceedings{ZZG13,
author = {Yunhui Zheng and
Xiangyu Zhang and
Vijay Ganesh},
title = {Z3-str: a z3-based string solver for web application analysis},
booktitle = {Joint Meeting of the European Software Engineering Conference and
the {ACM} {SIGSOFT} Symposium on the Foundations of Software Engineering,
ESEC/FSE'13, Saint Petersburg, Russian Federation, August 18-26, 2013},
pages = {114--124},
year = {2013}
}
@inproceedings{RB16,
author = {Andrew Reynolds and
Jasmin Christian Blanchette},
title = {A Decision Procedure for (Co)datatypes in {SMT} Solvers},
booktitle = {Proceedings of the Twenty-Fifth International Joint Conference on
Artificial Intelligence, {IJCAI} 2016, New York, NY, USA, 9-15 July
2016},
pages = {4205--4209},
year = {2016}
}
@article{CGPD08,
author = {Cristian Cadar and
Vijay Ganesh and
Peter M. Pawlowski and
David L. Dill and
Dawson R. Engler},
title = {{EXE:} Automatically Generating Inputs of Death},
journal = {{ACM} Trans. Inf. Syst. Secur.},
volume = {12},
number = {2},
year = {2008}
}
@inproceedings{CDE08,
author = {Cristian Cadar and
Daniel Dunbar and
Dawson R. Engler},
title = {{KLEE:} Unassisted and Automatic Generation of High-Coverage Tests
for Complex Systems Programs},
booktitle = {8th {USENIX} Symposium on Operating Systems Design and Implementation,
{OSDI} 2008, December 8-10, 2008, San Diego, California, USA, Proceedings},
pages = {209--224},
year = {2008}
}
@article{CFM12,
author = {Lucas C. Cordeiro and
Bernd Fischer and
Jo{\~{a}}o Marques{-}Silva},
title = {SMT-Based Bounded Model Checking for Embedded {ANSI-C} Software},
journal = {{IEEE} Trans. Software Eng.},
volume = {38},
number = {4},
pages = {957--974},
year = {2012}
}
@inproceedings{KLW13,
author = {Daniel Kroening and Matt Lewis and Georg Weissenbacher},
title = {Under-Approximating Loops in {C} Programs for Fast Counterexample
Detection},
booktitle = {Computer Aided Verification - 25th International Conference, {CAV}
2013},
pages = {381--396},
year = {2013},
series = {LNCS},
volume = {8044},
publisher = {Springer}
}
\ No newline at end of file
CHANGES
View file @ b8836e93
Changes in version 4.2 (September 2015)
o Fixed issue between titlesec and tocloft regarding
part numbering in the ToC (classicthesis.sty).
part numbering in the ToC (classicthesis.sty).
Thanks to Enrico Gregorio and Henri Menke.
o Fixed bug with colored lines and graffiti
o Fixed bug with colored lines and graffiti
(classicthesis.sty). Thanks to Enrico Gregorio and
Stefano Bragaglia for the fix! (Also switched
graffiti from \slshape to \itshape.)
o Changed \vspace*{8ex} to \vspace{8ex} (Contents.tex)
o Moved \cleardoublepage from end of Contents.tex
o Changed \vspace*{8ex} to \vspace{8ex} (Contents.tex)
o Moved \cleardoublepage from end of Contents.tex
to appropriate position in ClassicThesis.tex
o \bflabel of acronym package removed, as it was
o \bflabel of acronym package removed, as it was
removed from acronym package and caused errors.
Now set \aclabelfont to \acsfont for consistency.
o Converted major files to UTF-8 (thanks to Jörn Hees)
(Bibliography.bib excluded for now as BibTeX cannot
handle UTF-8)
o Added support for BibLaTeX in
o Added support for BibLaTeX in
classicthesis-config.tex and Bibliography.tex
(thanks to Jörn Hees).
For LyX we will have a work-around for testing.
o Added an example for one's own publication list
(via BibLaTeX) in Publications.tex (renamed from
(via BibLaTeX) in Publications.tex (renamed from
Publication.tex)
o Replaced/updated scrpage2 with scrlayer-scrpage
in classicthesis.sty (thanks to Ivo Pletìkosić).
X Worked on Appendix case with floatperchapter
o Replaced/updated scrpage2 with scrlayer-scrpage
in classicthesis.sgty (thanks to Ivo Pletìkosić).
X Worked on Appendix case with floatperchapter
(works for captions but not for references so far)
Thanks to Claus Lahiri and Clemens Niederberger
Thanks to Claus Lahiri and Clemens Niederberger
for their help on this.
[REMOVED DUE TO OTHER PROBLEMS, SEE ISSUE #82]
[REMOVED DUE TO OTHER PROBLEMS, SEE ISSUE #82]
X Removed fixltx2e from classicthesis-config.tex and
have classicthesis.sty use or emulate the latest
have classicthesis.sty use or emulate the latest
LaTeX kernel via latexrelease (option: latest)
(thanks to Jörn Hees).
[REMOVED AS IT IS NOT AVAILABLE IN MANY
[REMOVED AS IT IS NOT AVAILABLE IN MANY
DISTRIBUTIONS, SEE ISSUE #107]
Changes in version 4.1 (August 2012)
o Working towards XeLaTeX support:
- "ifxetex" package for checking whether
XeTeX is in use (classicthesis.sty)
- Adjusted part, chapter, and tocEntry
- Adjusted part, chapter, and tocEntry
- Thanks to Joerg Weber for his ideas and code
- Option "pdfspacing" does not work for XeTeX
and is disabled when XeTeX is running
(maybe implement spacing with fontspec later)
o Load additional packages in order to
o Load additional packages in order to
reduce warnings (classicthesis-config.tex)
- "textcomp" to fix warnings with missing font
- "textcomp" to fix warnings with missing font
shapes
- "scrhack" to fix warnings when using KOMA
with listings package
- "scrhack" to fix warnings when using KOMA
with listings package
o Replaced \mathrm-command with \textrm-command
(Chapter03.tex)
Changes in version 4.0 (December 2011)
o Due to lots of issues and complaints,
classicthesis-preamble.sty is replaced by
o Due to lots of issues and complaints,
classicthesis-preamble.sty is replaced by
classicthesis-config.tex (which is cleaner,
nicer, better, etc. than before!)
o No per-chapter-numbering also for equations
o No per-chapter-numbering also for equations
(classicthesis.sty and Chapter03.tex)
o classicthesis.sty now also works without the
o classicthesis.sty now also works without the
KOMA-classes (scrbook, scrartcl, etc.):
- This affects the text body shape, where the
package "typearea" is always loaded in order
package "typearea" is always loaded in order
to use \areaset
- And affects the layout of footnotes where the
- And affects the layout of footnotes where the
package "footmisc" is loaded otherwise
(not 100\% identical look-and-feel but OK)
o Help for using spanish babel with classicthesis
o Help for using spanish babel with classicthesis
(classicthesis-config.tex)
o Note about pdfsync issues with graffiti
o Note about pdfsync issues with graffiti
(Chapter01.tex)
o Minor clean-up of comments and duplicate lines
o Minor clean-up of comments and duplicate lines
(classicthesis.sty)
o Adjusted the colophon text (Colophon.tex)
o Increased space for labels in the LoF, LoT, LoL
o Included version information on the title page and
in the drafting and final version information
in the drafting and final version information
(classicthesis.sty and Titlepage.tex)
Changes in version 3.0 (June 2011)
o There have been some major changes in order to port
o There have been some major changes in order to port
classicthesis to LyX easily. See below for details.
Special thanks go to Ivo Pletikosi\'c for all his
Special thanks go to Ivo Pletikosi\'c for all his
hard work he put into this project!
o Changes in ClassicThesis.tex:
- Redundant fleqn removed from documentclass options
- Redundant fleqn removed from documentclass options
(will be set by amsmath package in preamble)
- KOMA adjustments for documentclass:
- KOMA adjustments for documentclass:
fontsize=11pt,paper=14
- Moved language options into documentclass due to LyX
issues with babel
......@@ -108,18 +108,18 @@ o Changes to classicthesis-preamble.sty:
- Now contains all configuration information, such as
the title, your name etc.
- This package now also loads classicthesis.sty, change
the options classicthesis to be loaded with here
- Package inputenc now loaded with latin9 option
the options classicthesis to be loaded with here
- Package inputenc now loaded with latin9 option
- Package listings now also loaded here
- Now safer package loading regarding options by
using \PassOptionsToPackage command extensively
- Inserted new string for \mySubtitle
o Changes in Titlepage.tex and Titleback.tex because of new
o Changes in Titlepage.tex and Titleback.tex because of new
string \mySubtitle
o Changed a lot of the documentation, e.g., in Chapter01.tex
Changes in version 2.9 (January 2011)
o Fixed onlytext euler font for MinionPro font (in
o Fixed onlytext euler font for MinionPro font (in
classicthesis.sty)
o Option "listsseparated" deprecated, because it led to some
strange side-effects (in classicthesis.sty)
......@@ -127,7 +127,7 @@ o Changed default font size to 11pt (in ClassicThesis.tex) and
adjusted text block size accordingly (in classicthesis.sty)
o Fixed bug with backward compatibility regarding old commands
\myChapter and \myPart in classicthesis.sty
o New option "floatperchapter" for classicthesis.sty, removed
o New option "floatperchapter" for classicthesis.sty, removed
all code for that from classicthesis-ldpkg.sty
o Removed "nochapters" option from classicthesis-ldpkg.sty
o Removed redundant loading of listings package (in
......@@ -143,15 +143,15 @@ o Included backward compatibility regarding old commands
\myChapter and \myPart in classicthesis.sty
o Fixed drafting issues (draft information not shown at all)
caused by wrong code for listings and listsseparated options
in classicthesis.sty
in classicthesis.sty
Changes in version 2.7 (February 2010, cumulative release)
o Removed option for classicthesis-ldpkg.sty: nobackref
o Removed option for classicthesis-ldpkg.sty: nobackref
Introduced instead option "backref" for consistency
which works the other way (no backrefs by default)
Changes in version 2.6.2 (January 2010, released as beta)
o New option for classicthesis-ldpkg.sty: nobackref
o New option for classicthesis-ldpkg.sty: nobackref
(removes links to cited page in bibliography)
o Fixed some spacing issues for LoF, LoT, LoL with option
"listsseparated" (in classicthesis.sty)
......@@ -162,23 +162,23 @@ o Removed \myChapter and \myPart, you can now use the regular
(thanks to Hinrich Harms for this)
o Fixed used KOMA-options: numbers=noenddot, cleardoublepage=empty
(in ClassicThesis.tex)
o Included hack to use citations in float environments, will
o Included hack to use citations in float environments, will
be fixed with caption package version 3.2 (in classicthesis.sty)
o Included some font and text area fine-tuning code
o Included some font and text area fine-tuning code
(in ClassicThesis.tex)
o Some testing with Libertine font (classicthesis-book.tex)
o Fixed numbering of listings used in different chapters
(in ClassicThesis.tex, classicthesis.sty, classicthesis-ldpkg.sty)
o Spacing between listings in different chapters and after List
o Spacing between listings in different chapters and after List
of Listings header fixed (in classicthesis.sty)
o Increased \newnumberwidth to sizeof(999) which is better for
o Increased \newnumberwidth to sizeof(999) which is better for
documents with more than 99 pages (in classicthesis.sty)
o New option for classicthesis.sty: manychapters
Changes in version 2.6 (August 2009)
o Centered the title page with the addmargin environment
(thanks to Nicolas Repp for the hint)
o Fixed header for page two of Table of Contents (with 3
o Fixed header for page two of Table of Contents (with 3
or more pages) (in Contents.tex)
o Added numbering of \subsubsection (in Contents.tex)
o Added correct Table of Contents entries for \subsubsection
......@@ -202,11 +202,11 @@ o Cumulative release, some minor clean-up and bug-fixing
Changes in version 2.4.{1--3} (February and April 2008,
released as betas)
o Switched back to "obsolete" KOMA options due to
o Switched back to "obsolete" KOMA options due to
compatibility issues on some installations
o Increased robustness of \cauthor command in
o Increased robustness of \cauthor command in
classicthesis-ldpk.sty
o Spacing fix for \sectionmark (in classicthesis.sty)
o Spacing fix for \sectionmark (in classicthesis.sty)
o Made Booleans private (in classicthesis.sty)
o Acronyms header for long acronym lists (in Contents.tex)
o Moved "amsmath" package from classicthesis.sty to
......@@ -218,22 +218,22 @@ o Fixed issue with contents headings (in classicthesis.sty
o Thanks to Lorenzo
Changes in version 2.3 (November 2007)
o Fixed serious bug with \myChapter command (using plain
o Fixed serious bug with \myChapter command (using plain
LaTeX)
o Removed hypcap package from classicthesis-ldpkg.sty and
placeins package from classicthesis.sty. Put both into
ClassicThesis.tex for finer end-user control
o Removed obsolete KOMA options from ClassicThesis.tex
o Option "subfigure" deprecated, replaced by "subfig" (in
o Option "subfigure" deprecated, replaced by "subfig" (in
classicthesis.sty)
o Inserted some mechanisms to detect whether subfig(ure)
o Inserted some mechanisms to detect whether subfig(ure)
package was loaded or not (in classicthesis.sty)
o Inserted some mechanism to detect whether classicthesis
packackage was loaded or not (in classicthesis-ldpkg.sty)
o Moved "headlines" code block a bit up in
o Moved "headlines" code block a bit up in
classicthesis.sty (seemed to cause problems with new
KOMA-Script classes)
o Removed accidental "natbib" from classicthesis-ldpkg.sty
o Removed accidental "natbib" from classicthesis-ldpkg.sty
o Some fine-tuning in (Dirty)Titlepage.tex
o Thanks to Lorenzo and Denis! :-)
......@@ -242,20 +242,20 @@ o \myChapter and \myPart now take an optional argument
just as the regular \part and \chapter commands
o Did some minor changes to classicthesis-book.tex
so it's a better example
o Inserted some \cleardoublepage before \myPart
o Inserted some \cleardoublepage before \myPart
occurences to avoid pdfbookmark problems
o Introduced some general typographic fine-tuning
(linespread, penalties, etc.)
o Did some fine-tuning of the \graffito command
Changes in version 2.1 (July 2007)
o Removed duplicate xcolor package call from
o Removed duplicate xcolor package call from
classicthesis-ldpkg.sty and shifed some color
definitions around
o Removed re-definition of \ttdefault
Changes in version 2.0 (June 2007, cumulative release)
o Better setup of the list of acronyms
o Better setup of the list of acronyms
(in classicthesis-ldpkg.sty)
o Using KOMA option footinclude in ClassicThesis.tex
(text block sizes are adjusted to this)
......@@ -284,13 +284,13 @@ o Introduced new commands in classicthesis-ldpkg.sty to make
- \backrefnotcitedstring (ref is not cited)
- \backrefcitedsinglestring (ref is cited on one page)
- \backrefcitedmultistring (ref is cited on multiple pages)
o Removed ellipsis package from classicthesis-ldpkg.sty due to
o Removed ellipsis package from classicthesis-ldpkg.sty due to
some undesired results for some users
o Removed the following packages from classicthesis-ldpkg.sty as
o Removed the following packages from classicthesis-ldpkg.sty as
these seem to require changes to their options from most users:
inputenc, babel, natbib (now in ClassicThesis.tex)
o New option for classicthesis-ldpkg.sty: nochapters
Use the package with classes that do not have chapter divisions
Use the package with classes that do not have chapter divisions
Changes in version 1.5 (April 2007, not released separately)
o Fixed some bugs to use classicthesis.sty for article layout
......@@ -315,13 +315,13 @@ o Some minor bug fixes
Changes in version 1.4 (January 2007)
o Replaced \lineheight with \baselineskip in classicthesis.sty
o Removed obsolete "usenames" option from xcolor package (classicthesis.sty)
o Now classicthesis.sty processes the global a5paper option and adjusts
o Now classicthesis.sty processes the global a5paper option and adjusts
the layout accordingly (experimental!)
Changes in version 1.3.1 (August 2006)
o Urgent bug concerning ToC spacing etc. fixed
Changes in version 1.3 (August 2006)
Changes in version 1.3 (August 2006)
o New option for classicthesis.sty: parts
Turn this on if you use Part divisions. It fine-tunes the ToC and
adds support for Part division via \myPart{} (cf. \myChapter{})
......@@ -329,15 +329,15 @@ o New option for classicthesis.sty: parts
o New option for classicthesis.sty: eulermath
Uses the awesome Euler fonts for math (Palatino fonts are default).
o Added mparhack package to loadpackages.sty to get the graffiti right
o Added a hint to loadpackages.sty on how to typeset the List of
o Added a hint to loadpackages.sty on how to typeset the List of
Listings consistently with classicthesis
o Some minor fine-tuning of typographic details
o Collaboration with babel package improved (now easier to use
with other languages than English)
o Added appropriate textwidth values for 11pt Palatino and Minion
(See comments near \areaset in classicthesis.sty)
o Note: There is a problem with the case of math text in part-,
chapter-, and section titles and I have no clue how to fix this
o Note: There is a problem with the case of math text in part-,
chapter-, and section titles and I have no clue how to fix this
(either the case or the spacing breaks). So far, I chose the case. :-(
Changes in version 1.2 (June 2006)
......@@ -347,7 +347,7 @@ o New option for classicthesis.sty: nochapters
Allows to use the look-and-feel with classes that do not use chapters.
o New option for classicthesis.sty: beramono
Loads Bera Mono as typewriter font. (Default = no typewriter font.)
o Included list of files, see LISTOFFILES.
o Included list of files, see LISTOFFILES.
Changes in version 1.1 (May 2006)
o classicthesis.sty now stand-alone and usable w/o whole bundle!
......
Chapters/Chapter01.tex
View file @ b8836e93
......@@ -2,7 +2,7 @@
\chapter{Introduction}\label{ch:introduction}
% ************************************************
During the last decades, the area \emph{satisfiability modulo
During the last decades, the area of \emph{satisfiability modulo
theories} (\smt)~\cite{BSST09} has undergone steep development in
both theory and practice. Achieved advances of \smt solving opened new
research directions in program analysis and verification, where \smt
......@@ -15,13 +15,13 @@ also has the ability to provide its model. Modern \smt solvers support
wide range of different first-order theories -- for example theories
of integers, real numbers, floating-point numbers, arrays, strings,
inductively defined data types, bit-vectors, and various combinations
and sub-theories of these theories. From the software analysis and0
verification point of view, a particularly important of these theories
is the theory of bit-vectors, which can be used to describe properties
of computer programs, since programs usually use data-types of bounded
size instead of mathematical integers. Additionally, operations over
these bounded data-types naturally corespond to operations of the
bit-vector theory.
and sub-theories of these theories~\cite{BFT15, ZZG13, RB16}. From the
software analysis and verification point of view, a particularly
important of these theories is the theory of bit-vectors, which can be
used to describe properties of computer programs, since programs
usually use data-types of bounded size instead of mathematical
integers. Additionally, operations over these bounded data-types
naturally corespond to operations of the bit-vector theory.
The benefit of describing properties of programs by bit-vector
formulas is twofold. Formulas in the bit-vector theory allow to model
......@@ -32,26 +32,28 @@ even if the multiplication is allowed.
Therefore, quantifier-free bit-vector formulas are used in tools for
symbolic execution, bounded model checking, analysis of hardware
circuits, static analysis, or test generation. Most of the current
\smt solvers for the quantifier-free bit-vector formulas eagerly or
lazily translate the formula to the propositional logic
(\emph{bit-blasting}) and use an efficient \sat solver to decide its
satisfiability. Therefore, the efficiency of most of the bit-vector
\smt solvers is tightly connected to the efficiency of the \sat
solvers. Plenty of solvers for the quantifier-free bit-vector formulas
exist: for example Beaver~\cite{Beaver}, Boolector~\cite{Boolector},
CVC4~\cite{CVC4}, MathSAT5~\cite{MathSAT}, OpenSMT~\cite{OpenSMT},
Sonolar~\cite{Sonolar}, Spear~\cite{Spear}, STP~\cite{STP},
UCLID~\cite{LS04}, Yices~\cite{Yices}, and Z3~\cite{Z3}.
circuits, static analysis, or test generation~\cite{CGPD08, CDE08,
CFM12}. Most of the current \smt solvers for the quantifier-free
bit-vector formulas eagerly or lazily translate the formula to the
propositional logic (\emph{bit-blasting}) and use an efficient \sat
solver to decide its satisfiability. Therefore, the efficiency of most
of the bit-vector \smt solvers is tightly connected to the efficiency
of the \sat solvers. Plenty of solvers for the quantifier-free
bit-vector formulas exist: Beaver~\cite{Beaver},
Boolector~\cite{Boolector}, CVC4~\cite{CVC4}, MathSAT5~\cite{MathSAT},
OpenSMT~\cite{OpenSMT}, Sonolar~\cite{Sonolar}, Spear~\cite{Spear},
STP~\cite{STP}, UCLID~\cite{LS04}, Yices~\cite{Yices}, and
Z3~\cite{Z3}.
In some cases, quantifier-free formulas are not succinct enough and
using quantification is necessary keep size of the formula
reasonable. Bit-vector quantified formulas arise naturally for example
in applications that need to decide equality of two symbolically
represented states, or in applications that generate loop invariants,
ranking functions, or loop summaries. However, the current \smt
solvers' support of quantified bit-vector logic is much more modest --
only CVC4, Yices, and Z3 officially support quantifiers in bit-vector
represented states~\cite{BHB14}, or in applications that generate loop
invariants, ranking functions, or loop
summaries~\cite{WHD13,KLW13}. However, the current \smt solvers'
support of quantified bit-vector logic is much more modest -- only
CVC4, Yices, and Z3 officially support quantifiers in bit-vector
formulas. Recently, quantifiers have been also implemented in an
development version of Boolector~\cite{BoolectorComp}. All of these
\smt solvers solve quantified bit-vector formulas by some variant of
......
Chapters/Chapter02.tex
View file @ b8836e93
......@@ -760,16 +760,17 @@ its model $M$ and another call to the \QFBV solver is made to
determine whether $M$ is also a~model of
$\forall x_1, x_2, \dots, x_n \, (\psi)$. This is achieved by asking
the solver whether the formula $\neg \widehat{\psi}$ is satisfiable,
where $\widehat{\psi}$ is the formula $\psi$ with uninterpreted
constants replaced by their corresponding values in $M$. If
$\neg \widehat{\psi}$ is not satisfiable, then the structure $M$ is
indeed a model of the formula $\forall x_1, x_2, \dots, x_n\,(\psi)$,
therefore the entire formula is satisfiable and $M$ is its model. If
$\neg \widehat{\psi}$ is satisfiable, we get values $v_1, \dots, v_n$
such that $\neg\widehat{\psi}[v_1, \dots, v_n]$ holds. To rule out $M$
as a model of the input formula, the instance $\psi[v_1, \dots, v_n]$
of the quantified formula is added to the quantifier-free part,
i.e.~the formula $\varphi$ is modified to
where $\widehat{\psi}$ is the formula $\psi$ after substituting values
of all variables except for $x_1 \ldots x_n$ and uninterpreted
functions replaced by their values in $M$. If $\neg \widehat{\psi}$ is
not satisfiable, then the structure $M$ is indeed a model of the
formula $\forall x_1, x_2, \dots, x_n\,(\psi)$, therefore the entire
formula is satisfiable and $M$ is its model. If $\neg \widehat{\psi}$
is satisfiable, we get values $v_1, \dots, v_n$ such that
$\neg\widehat{\psi}[v_1, \dots, v_n]$ holds. To rule out $M$ as a
model of the input formula, the instance $\psi[v_1, \dots, v_n]$ of
the quantified formula is added to the quantifier-free part, i.e.~the
formula $\varphi$ is modified to
\[
\varphi' ~\equiv~ \varphi \wedge \psi[v_1, \dots, v_n],
\]
......
Chapters/Chapter03.tex
View file @ b8836e93
......@@ -57,9 +57,9 @@ the size of the formula. Dually, the existential quantification can be
distributed over disjunctions.
Moreover, universal quantification can distribute over disjunctions in
cases where the variable bound by the quantifier does not occur in one
cases where the variable bound by the quantifier does not occurg in one
of the disjuncts. This leads to the following rule and its dual
version, which is known as \emph{miniscoping}:
version, which is known as \emph{miniscoping}~\cite{Har09}:
\[
\forall x. \, (\varphi ~\vee~ \psi)~~\leadsto~~(\forall x . \,
\varphi) ~\vee~ \psi,
......@@ -139,8 +139,8 @@ unsatisfiability of an overapproximation implies unsatisfiability of
the input formula.
Q3B employs approximations similar to those used by the \smt solver
\uclid. In \uclid, the quantifier-free input formula is
underapproximated by representing each bit-vector variable by the
\uclid~\cite{BKOSSB07}. In \uclid, the quantifier-free input formula
is underapproximated by representing each bit-vector variable by the
smaller number of bit variables than its original bit-width. The
number of bits by which the bit-variable is represented is called the
\emph{effective bit-width}. For reducing the effective bit-width,
......
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