Multiple changes

Bibliography.bib

@@ -1133,3 +1133,11 @@ year = {2005},
   pages     = {245--257},
   year      = {1979}
 }
+
+@article{Luc16,
+  author    = {Martin L{\"{u}}ck},
+  title     = {Complete Problems of Propositional Logic for the Exponential Hierarchy},
+  journal   = {CoRR},
+  volume    = {abs/1602.03050},
+  year      = {2016}
+}
\ No newline at end of file

Chapters/Chapter04.tex

@@ -4,66 +4,77 @@
 
 \section{Objectives and Expected Results}
 
-\subsection{Unconstrained variable propagation for quantified
-  bit-vectors}
-Simplifications using unconstrained variables can be extended to
-quantified formulas. However, in the quantified setting, constraints
-can be induced also by the order of the quantified variables. We have
-hypothesis of the necessary condition for the quantified variable to
-be unconstrained and we have implemented an proof-of-concept
-simplification procedure using unconstrained variables for the
-quantified bit-vector formulas. Although the initial experimental
-results, conducted on the formulas from the semi-symbolic model
-checker \SymDivine look promissing, the formal proof of the correctnes
-is not yet complete.
-
-Furthermore, we suggest simplifications even for term in the form
-$k \times x$ with odd values of $x$. If $x$ has bit-width $n$, $i$ is
-the largest number such that $2^i$ which divides the constant $k$ and
-the value $x$ is unconstrained, the term $k \times x$ can be rewritten
-to $extract_0^{n-i}(x) \cdot 0^i$. This approach can possibly be
-extended to the multiplication of two variables from one is
-unconstrained and further generalized. We plan to prove the
-correctness of these rules and develop a formal framework to classify
-such rewrite rules.
-
 \subsection{Symbolic solver for quantified bit-vectors}
-I plan to continue developing the implemented symbolic \smt solver for
-quantified bit-vecors Q3B. Besides implementing the proposed
-simplifiactions for quantified formulas containing unconstrained
-variables, I plan to add support of uninterpreted functions and theory
-of arrays, which are highly desirable for the usage in program
-verification. I also want to implement a support for extracting
-unsatisfiable cores from the intermediate \bdds, which were produced
+I will continue developing the implemented symbolic \smt solver Q3B,
+which is aimed for 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 an extraction of an
+unsatisfiable core from the intermediate \bdds that were produced
 during the computation of the solver.
 
-Additionally, approximations implemented are right now very simple and
-could benefit from better refinement of the approximation in the case
-that the current approximation is too coarse.
+Additionally, currently implemented approximations are very simple and
+could benefit from better refinement in the case that the current
+approximation is too coarse.
 
-Moreover, we want to implement other variants of decision diagrams
-such as zero-suppressed decision diagrams introduced by
-Minato~\cite{Min93} and algebraic decision diagrams introduced by
-Bahar et al.~\cite{BFGHMPS} and experimentally evaluate their effect
-on the performance of the \smt solver.
+Moreover, I want to implement other variants of decision diagrams such
+as zero-suppressed decision diagrams introduced by Minato~\cite{Min93}
+and algebraic decision diagrams introduced by Bahar et
+al.~\cite{BFGHMPS} and experimentally evaluate their effect on the
+performance of the symbolic \smt solver for the quantified
+bit-vectors.
 
 \subsection{Hybrid approach to quantified bit-vectors}
-Although our results with the symbolic \smt solver for quantified
-bit-vectors look promissing, standard \smt solvers still perform
-better on simple queries and on queries containing
-multiplication. Therefore, I want to develop a hybrid approach to \smt
-solving of quantified bit-vector formulas, which combines strengths of
-both of these approaches. For example, a part of the quantified
-formula without multiplication can be converted to the \bdd, which can
-be used to guide the model search in the model-based quantifier
-instantiation. One possible way of achieving this is adding \bdd based
-representation of sets of assignments to the \mcbv solver developed by
-Zeljić et al. The \bdd representation can be added to the current
-over-approximations by bit-patterns and arithmetic intervals.
+Although results of the symbolic \smt solver for quantified
+bit-vectors look promising, standard \smt solvers still perform
+better on queries containing multiplication and complex
+arithmetic. Therefore, I plan to develop a hybrid approach to \smt
+solving of quantified bit-vector formulas that combines strengths of
+both of these approaches. For example, parts of the quantified
+formula, which do not contain multiplication, can be converted to the
+\bdd that can be used to guide the model search in the model-based
+quantifier instantiation. One possible way of achieving this is adding
+support for \bdds to the solver \mcbv developed by Zeljić et al. The
+\bdd representation can be added to the currently used
+over-approximations by bit-patterns and arithmetic
+intervals. Moreover, a tighter cooperation is possible -- if a \mcsat
+solver decides a value, this can be used to partially instantiate a
+quantified part of the formula, for which the \bdd will be computed.
 
-As the part of my PhD study, also an implementation of a proposed
+As the part of my PhD study, an implementation of a solver using the
 hybrid approach and its evaluation on the representative set of
-benchmark is expected.
+benchmark is expected. The hybrid approach to quantified bit-vectors
+is the main aim of my PhD study.
+
+\subsection{Unconstrained variable propagation for quantified
+  bit-vectors}
+Simplifications using unconstrained variables can be extended to
+quantified formulas. However, in the quantified setting, constraints
+among variables can be introduced also by the order in which the
+variables are quantified. We have hypothesis that describes the
+necessary condition for the quantified variable to be considered as
+unconstrained. Based on this hypothesis and a partial proof of its
+validity, we have implemented an proof-of-concept simplification
+procedure that can simplify quantified formulas which contain
+unconstrained variables. Although the initial experimental results,
+conducted on the formulas from the semi-symbolic model checker
+\SymDivine, look promising, the formal proof of the 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
+$k \times x$ with an odd values of $k$ and the variable $x$ is
+unconstrained, this term can be replaced by a simpler term. In
+particular, if $i$ is the largest number for which $2^i$ divides the
+constant $k$ and the variable $x$ has bit-width $n$, the term
+$k \times x$ can be rewritten to $extract_0^{n-i}(x) \cdot 0^i$. In
+turn, this approach can be also extended to the multiplication of two
+variables from one is unconstrained. I plan to investigate further
+extensions of to the terms containing division and remainder
+operations and publish all described extension to the unconstrained
+variable propagation. I will also experimentally evaluate the effect
+of such simplifications on our solver Q3B and on state-of the art
+solvers such as Boolector, CVC4, and Z3.
 
 \subsection{Complexity of BV2}
 As was explained in section \ref{sec:complexity}, the precise
@@ -72,36 +83,38 @@ bit-widths and without uninterpreted functions is not known. It is
 known to be in \EXPSPACE and to be \NEXPTIME-hard. However, a class
 for which the problem is complete is not known.
 
-Acording to our investigation, it is probably complete for neither of
+According to our investigation, it is probably complete for neither of
 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 \EXPSPACE and \NEXPTIME. However, whether any of the
-inclusions is proper is not known.
+inclusions is proper is not known. The \AEXPTIMEp-hardness of the BV2
+can be shown by reducing the problem of satisfiability of
+\emph{quantified second order boolean formulas}, which was recently
+proven to be \AEXPTIMEp-hard by Lück~\cite{Luc16}.
 
 Expected result is a paper published at an international conference or
 in a journal.
 
-\newpage
 \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 mantaining of the
+\item [now -- January 2019] Improvements and maintaining of the
   developed symbolic \smt solver Q3B.
-\item [January 2017] Doctoral exam and defence of this thesis proposal.
-\item [January 2017 -- May 2017] Proving a precise complexity class of
-  the quantfied bit-vector formulas without uninterpreted functions
-  and with binary encoded bit-widths.
-\item [May 2017 -- January 2018] Development of a hybrid approach to
+\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
+  and with binary encoded bit-widths. Preparing a paper on this topic.
+\item [May 2017 -- December 2017] Development of a hybrid approach to
   \smt solving of quantified bit-vector formulas combining symbolic
   representation of formulas with \dpllt or \mcsat.
 \item [January 2018 -- August 2018] Implementing a hybrid \smt solver
-\item [August 2018 -- January 2019] Text of the Ph.D. thesis.
+\item [September 2018 -- January 2019] Text of the PhD thesis.
 \item [January 2019] The final version of the thesis.
 \end{description}
 

FrontBackmatter/Titlepage.tex

@@ -9,28 +9,31 @@
 
         \hfill
 
-        \vfill
+        \bigskip \bigskip
 
         \begingroup
-            \color{Maroon}\spacedallcaps{\myTitle} \\ \bigskip
+            \Large{\color{Maroon}\spacedallcaps{\myTitle}} \\ \bigskip
         \endgroup
 
         \spacedlowsmallcaps{\myName}
 
         \vfill
 
-        \includegraphics[width=6cm]{gfx/fi_logo} \\ \bigskip
+        \includegraphics[width=5cm]{gfx/fi_logo}
+
+        \vfill
 
-        \mySubtitle \\ \medskip
+        \mySubtitle \\
         %\myDegree \\
         %\myDepartment \\
-        %\myFaculty \\
-        %\myUni \\ \bigskip
+        \smallskip
 
         \myTime
 
         \vfill
 
-    \end{center}
+      \end{center}
+      \noindent \myFaculty, \myUni \\
+      \noindent Supervisor: \myProf
   \end{addmargin}
 \end{titlepage}

Includes/notation.tex

@@ -18,6 +18,7 @@
 \newcommand{\sls}{\textsc{sls}\xspace}
 \newcommand{\mcbv}{\textsc{mcbv}\xspace}
 \newcommand{\nnf}{\textsc{nnf}\xspace}
+\newcommand{\qsbf}{\textsc{qsbf}\xspace}
 
 \newcommand{\true}{\ensuremath{\texttt{true}}}
 \newcommand{\false}{\ensuremath{\texttt{false}}}

classicthesis-config.tex

@@ -49,7 +49,7 @@
 \newcommand{\myProf}{doc. RNDr. Jan Strejček, Ph.D.\xspace}
 %\newcommand{\myOtherProf}{Put name here\xspace}
 %\newcommand{\mySupervisor}{Put name here\xspace}
-\newcommand{\myFaculty}{Faculty of informatics\xspace}
+\newcommand{\myFaculty}{Faculty of Informatics\xspace}
 %\newcommand{\myDepartment}{Put data here\xspace}
 \newcommand{\myUni}{Masaryk University\xspace}
 \newcommand{\myLocation}{Brno\xspace}
@@ -185,13 +185,17 @@
 % ********************************************************************
 % Using PDFLaTeX
 % ********************************************************************
+%\usepackage{tikz}
+%\usetikzlibrary{backgrounds,calc}
+%\usepackage{graphicx,lipsum}
+
 \PassOptionsToPackage{pdftex,hyperfootnotes=false,pdfpagelabels}{hyperref}
     \usepackage{hyperref}  % backref linktocpage pagebackref
 \pdfcompresslevel=9
 \pdfadjustspacing=1
 \PassOptionsToPackage{pdftex}{graphicx}
     \usepackage{graphicx}
-
+%    \usepackage{eso-pic}
 
 % ********************************************************************
 % Hyperreferences