Small changes

Bibliography.bib

@@ -956,7 +956,8 @@ year = {2005},
                   of Systems - 22nd International Conference, {TACAS}
                   2016, Held as Part of the ETAPS 2016, Eindhoven, The
                   Netherlands, April 2-8, 2016, Proceedings},
-  pages     = {946--949}
+  pages     = {946--949},
+  year = {2016}
 }
 
 @inproceedings{JS16,
@@ -1338,3 +1339,13 @@ pages={358--372},
   volume    = {8044},
   publisher = {Springer}
 }
+
+@inproceedings{Lei10,
+  author    = {K. Rustan M. Leino},
+  title     = {Dafny: An Automatic Program Verifier for Functional Correctness},
+  booktitle = {Logic for Programming, Artificial Intelligence, and Reasoning - 16th
+               International Conference, LPAR-16, Dakar, Senegal, April 25-May 1,
+               2010, Revised Selected Papers},
+  pages     = {348--370},
+  year      = {2010}
+}

Chapters/Chapter01.tex

@@ -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~\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.
+and fragments of formulas over 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 correspond 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
@@ -33,27 +33,27 @@ 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~\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},
+  Lei10, 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 supporting
+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
+using quantification is necessary to keep the size of the formula
+reasonable. Quantified bit-vector formulas arise naturally for example
 in applications that need to decide equality of two symbolically
 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
+support of quantified bit-vector logic is 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
@@ -61,8 +61,8 @@ quantifier-instantiation and using a solver for quantifier-free
 formulas as an oracle.
 
 In the last year, we have proposed a different approach. We have
-implemented a symbolic solver Q3B, which is based on binary decision
-diagrams, and have shown that it can not only compete with
+implemented a symbolic \smt solver Q3B, which is based on binary
+decision diagrams. We have shown that it can not only compete with
 state-of-the art \smt solvers, but outperforms them in many
 cases~\cite{JS16}. However, \bdds are not a silver bullet; the
 symbolic \smt solver fails on formulas containing non-trivial
@@ -71,14 +71,14 @@ study, I plan to develop a hybrid approach to solving quantified
 bit-vectors, which combines symbolic representation by \bdds with
 search techniques employed by existing state-of-the-art solvers. I
 also plan to continue developing the symbolic solver Q3B itself,
-namely improving its efficiency and adding support for features as
-uninterpreted functions and arrays, which are important in the software
-verification.
+namely improve its efficiency and add support for features as
+uninterpreted functions and arrays, which are important in the
+software verification.
 
 The thesis proposal is organized as follows. Chapter~\ref{ch:sota}
 summarizes the state of the art and is divided into six
 sections. Section~\ref{sec:prelim} introduces necessary background and
-notations from propositional logic and first-order
+notations from the propositional and first-order
 logic. Section~\ref{sec:sat} describes approaches to solving
 propositional satisfiability and Section~\ref{sec:smt} describes
 approaches to solving satisfiability modulo theories. Sections
@@ -86,7 +86,7 @@ approaches to solving satisfiability modulo theories. Sections
 and quantified formulas over the theory of bit-vectors,
 respectively. Finally, Section~\ref{sec:complexity} section describes
 results concerning the computational complexity of bit-vector
-logics. Chapter~\ref{ch:achieved} describes results, which we have
+logics. Chapter~\ref{ch:achieved} describes results that we have
 achieved during the first two years of my PhD
 study. Chapter~\ref{ch:aims} presents the aim of the thesis and states
 plans for the remaining part of my PhD study.

Chapters/Chapter02.tex

@@ -150,7 +150,7 @@ $\mathcal{T}$- prefixes.
 
 \subsection{Many-sorted logic}
 
-For some theories, it can be convinient to distinguish several types
+For some theories, it can be convenient to distinguish several types
 of objects instead of having only one universe. This can be achieved
 by using a \emph{many-sorted logic}. In contrast with a single-sorted
 signature, defined in the previous subsection, a \emph{many-sorted
@@ -168,7 +168,7 @@ Enderton~\cite{End01}.
 For a many-sorted signature $\Sigma$, a \emph{$\Sigma$-structure}
 $\mathcal{A}$ consists of an assignment $(\_)^\mathcal{A}$, which to
 each sort symbol $S \in \Sigma^S$ assigns a non-empty set
-$S^\mathcal{A}$, caled the \emph{domain of S in $\mathcal{A}$}, to
+$S^\mathcal{A}$, called the \emph{domain of S in $\mathcal{A}$}, to
 each variable $x$ of sort $S$ assigns an element
 $x^\mathcal{A} \in S^\mathcal{A}$, to each function symbol
 $f \in \Sigma^f$ of an arity $(S_1, \ldots, S_n, S_{n+1})$ assigns a
@@ -380,7 +380,7 @@ For a detailed description of these theories and implementation of the
 respective $\mathcal{T}$-solvers, we refer the reader for example to the book of
 Bradley and Manna~\cite{BM07}.
 
-In pracitce, a satisfiability of formulas using a combination of
+In practice, a satisfiability of formulas using a combination of
 theories is often needed. For example, a single formula may be using
 integers, arrays, and uninterpreted functions. In their seminal work,
 Nelson and Oppen have shown that satisfiability of combination of

Chapters/Chapter03.tex

@@ -57,12 +57,12 @@ 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 occurg in one
+cases where the variable bound by the quantifier does not occur in one
 of the disjuncts. This leads to the following rule and its dual
 version, which is known as \emph{miniscoping}~\cite{Har09}:
 \[
-  \forall x. \, (\varphi ~\vee~ \psi)~~\leadsto~~(\forall x . \,
-  \varphi) ~\vee~ \psi,
+  \forall x. \, (\varphi[x] ~\vee~ \psi)~~\leadsto~~(\forall x . \,
+  \varphi[x]) ~\vee~ \psi,
 \]
 where $x$ does not occur freely in $\psi$. Note that the scope of the
 quantifier is again reduced, which can lead to smaller intermediate
@@ -218,10 +218,15 @@ which uses quantified formulas to decide the equality of two symbolic
 states. On this set of benchmarks, our solver Q3B also has better
 performance than \smt solvers based on the quantifier instantiation
 and bit-blasting. Table \ref{tbl:results} shows the numbers of
-formulas solved by each solver and figure \ref{fig:quantilePlots}
+formulas solved by each solver and Figure \ref{fig:quantilePlots}
 presents the quantile plot of CPU times needed to solve these
 formulas.
 
+All experiments were performed on a Debian machine with two six-core
+Intel Xeon E5-2620 2.00GHz processors and 128 GB of RAM. Each
+benchmark run was limited to use 3 processor cores, 4 GB of RAM and 20
+minutes of CPU time (if not stated otherwise).
+
 \begin{table}
 \checkoddpage
 \edef\side{\ifoddpage l\else r\fi}
@@ -298,8 +303,9 @@ formulas.
   \end{minipage}}
 \end{table}
 
-Recently, these results were confirmed by the \smt competition 2016 --
-our solver is the winner of the quantified-bit vectors category. Table
+Recently, these results were confirmed by the \smt competition
+2016\footnote{\url{http://smtcomp.sourceforge.net/2016/}} -- our
+solver is the winner of the quantified-bit vectors category. Table
 \ref{tbl:smtcomp} shows official results for this category. The
 benchmarks are divided into two groups, with \emph{known} and
 \emph{unknown} status. A benchmark has a known status if at least two

Chapters/Chapter04.tex

@@ -5,13 +5,13 @@
 \section{Objectives and Expected Results}
 
 \subsection{Symbolic solver for quantified bit-vectors}
-I plan to continue developing the implemented symbolic \smt solver
-Q3B, which is aimed at solving quantified bit-vector formulas. In
-particular, I plan to add support for uninterpreted functions and the
-theory of arrays, which is highly desirable for applications in the
-program verification. I also want to implement extraction of an
-unsatisfiable core from the intermediate \bdds that were produced
-during the computation.
+Throughout the rest of my PhD study, I plan to continue developing the
+implemented symbolic \smt solver Q3B, which is aimed at solving
+quantified bit-vector formulas. In particular, I plan to add support
+for uninterpreted functions and the theory of arrays, which is highly
+desirable for applications in the program verification. I also want to
+implement extraction of an unsatisfiable core from the intermediate
+\bdds that were produced during the computation.
 
 Additionally, currently implemented approximations are very simple and
 could benefit from better refinement in the case that the current
@@ -61,7 +61,7 @@ correctness is not yet complete.
 Moreover, the simplifications of formulas with unconstrained variables
 can be further extended. For example, if a formula contains a term
 $k \times x$ in which the variable $x$ is unconstrained, this term can
-be replaced by a simpler term regargless the parity of the value
+be replaced by a simpler term regardless the parity of the value
 $k$. In particular, if $i$ is the largest number for which $2^i$
 divides the constant $k$ and the unconstrained variable $x$ has
 bit-width $n$, the term $k \times x$ can be rewritten to
@@ -87,18 +87,20 @@ those complexity classes. We are working on a proof which shows that
 BV2 is complete for the class of problems solvable by the
 \emph{alternating Turing machine} (\atm) with the exponential space
 and \emph{polynomial number of alternations} with respect to log-space
-reduction. This class is usually denoted as \AEXPTIMEp and is known to
-be in between \NEXPTIME and \EXPSPACE~\cite{HKVV15, Luc16}. However,
-whether any of the inclusions is proper is not known.
+reductions. This class is usually denoted as \AEXPTIMEp and is known
+to be in between \NEXPTIME and \EXPSPACE~\cite{HKVV15,
+  Luc16}. However, whether any of the inclusions is proper is not
+known.
 
 \section{Progression Schedule}
 The plan of my future study and research activities is following:
 
 \begin{description}[style=nextline,leftmargin=0.8cm]
-\item [now -- January 2017] Extending unconstrained variable
-  propagation to quantified formulas and to non-linear multiplication.
 \item [now -- January 2019] Improvements and maintaining of the
   developed symbolic \smt solver Q3B.
+\item [now -- January 2017] Extending unconstrained variable
+  propagation to quantified formulas and to non-linear multiplication
+  and division.
 \item [January 2017] Doctoral exam and defense of this thesis proposal.
 \item [February 2017 -- April 2017] Proving a precise complexity class
   of the quantified bit-vector formulas without uninterpreted functions