Multiple changes

  booktitle = {13th International Workshop on Satisfiability Modulo

                 Theories (SMT 2015)},

  year = 	 2015}

@inproceedings{FKB13,

  author    = {Andreas Fr{\"{o}}hlich and

               Gergely Kov{\'{a}}sznai and

               Armin Biere},

  title     = {More on the Complexity of Quantifier-Free Fixed-Size Bit-Vector Logics

               with Binary Encoding},

  booktitle = {Computer Science - Theory and Applications - 8th International Computer

               Science Symposium in Russia, {CSR} 2013, Ekaterinburg, Russia, June

               25-29, 2013. Proceedings},

  pages     = {378--390},

  year      = {2013}

}

@MISC{BST10,

  author =	 {Clark Barrett and Aaron Stump and Cesare Tinelli},

  title =	 {{The Satisfiability Modulo Theories Library (SMT-LIB)}},

  howpublished = {{\tt www.SMT-LIB.org}},

  year =	 2010,

}

@INCOLLECTION{BHB14,

  author = {Petr Bauch and Vojt\v{e}ch Havel and Ji\v{r}\'{\i} Barnat},

  title = {{LTL Model Checking of LLVM Bitcode with Symbolic Data}},

  booktitle = {{MEMICS} 2014},

  publisher = {Springer},

  year = 2014,

  volume = 8934,

  series = {LNCS},

  pages = {47--59},

}

@article{Bry91,

 author = {Bryant, Randal E.},

 title = {{On the Complexity of VLSI Implementations and Graph Representations of Boolean Functions with Application to Integer Multiplication}},

 journal = {IEEE Trans. Comput.},

 volume = {40},

 number = {2},

 year = {1991},

 issn = {0018-9340},

 pages = {205--213},

 publisher = {IEEE Computer Society},

}

@inproceedings{Rud93,

  author    = {Richard Rudell},

  title     = {Dynamic variable ordering for ordered binary decision diagrams},

  booktitle = {Proceedings of the 1993 {IEEE/ACM} International Conference on Computer-Aided

               Design, 1993},

  pages     = {42--47},

  year      = {1993}

}

 No newline at end of file

Notable examples of decidable first-order theories include

\begin{itemize}

\item the theory of \emph{equality and uninterpreted functions}

  $\teuf$, which contains all structures that interpret the predicate

  $=$ as a congruence with respect to all other functions,

\item the theory of \emph{equality and uninterpreted functions}, which

  contains all structures that interpret the predicate $=$ as a

  congruence with respect to all other functions,

\item the theory of \emph{linear integer arithmetic}, which consists

  of all structures from $\teuf$ that are isomorphic to the set of

  integers and interpret the function $+$ as addition and the

  predicate $\leq$ as the integer comparison;

\item the theory of \emph{linear real arithmetic}, which consists of

  all structures from $\teuf$ that are isomorphic to real numbers and

  interpret the function $+$ as addition, and the predicate $\leq$ as

  the real comparison;

  of all structures isomorphic to integers with the function $+$ and

  predicates $\leq$ and $=$,

\item the theory of \emph{linear real arithmetic}, which consists

  of all structures isomorphic to real numbers with the function $+$ and

  predicates $\leq$ and $=$,

\item the theory of \emph{real arithmetic}\marginpar{In contrast to

    real arithmetic, integer arithmetic with multiplication was shown

    to be undecidable by Gödel.}, which consists of all structures

  from $\teuf$ that are isomorphic to real numbers and interpret the

  function $+$ as addition, $\times$ as multiplication, and the

  predicate $\leq$ as the real comparison;

  isomorphic to real numbers with functions $+$ and $\times$ and

  predicates $\leq$ and $=$,

\item the theory of \emph{arrays}, which consists of all structures

  from $\teuf$ isomorphic to the set of arrays with a binary function

  $read(a, i)$ interpreted as a value in the index $i$ of the array

  $a$, and a ternary function $write(a, i, v)$ interpreted as an array

  $a$ modified to contain the value $v$ on the index $i$;

\item the theory of fixed-size bit-vectors.

  isomorphic to the set of arrays with a binary function $read(a, i)$

  interpreted as a value in the index $i$ of the array $a$, and a

  ternary function $write(a, i, v)$ interpreted as an array $a$

  modified to contain the value $v$ on the index $i$;

\item the theory of \emph{fixed-size bit-vectors}. \marginpar{TODO: describe}

\end{itemize}

For a detailed description of these theories and implementation of the

respective $T$-solvers, we refer the reader for example to the book of

to the equivalent propositional formula, which is subsequently solved

by a \sat solver. The transformation of a bit-vector formula to the

equivalent propositional formula is traditionally called

\emph{bit-blasting}~\cite{Kro08}. More lazy approach to the

\emph{bit-blasting}~\cite{Kro08}.  \marginpar{TODO: Nepřidat odstavec

  o propagating-complete \cnf encodings?} More lazy approach to the

bit-blasting is benefitial when the theory combination is

required. For example, solvers Z3 and Yices apply bit-blasting to all

operations except for the equality, which is handled by a specialized

reasoning and if the sub-solvers are not sufficient for deciding the

satisfiability of the formula, theory lemmas and propagated literals

generated by the sub-solvers are added to the formula and a lazy

\dpllt bit-blasting solver is

employed~\cite{HBJBT14}. \marginpar{TODO: Nepřidat odstavec o

  propagating-complete \cnf encodings?}

\dpllt bit-blasting solver is employed~\cite{HBJBT14}.

\subsection{Word-level techinques}

Although bit-blasting is highly efficient for most of practical

expanding all quantifiers with all possible bit-vector values of the

corresponding bit-width, this is rarely practical approach. Instead,

the formula is usually converted to a equisatisfiable formula by

Skolemization and then instances of the universal quantifiers are

lazily added to the formula until a model is found by a solver for

quantifier-free bit-vector formulas or the formula is found to be

unsatisfiable. There are multiple ways to choose quantifier instances

that are sufficient to decide the satisfiability of the formula. For

the bit-vector theory, the most widely used approach is the

\emph{model-based quantifier instantiation} approach~\cite{GM09},

supported by Z3, CVC4, and Yices, combined by heuristics as E-matching

or symbolic quantifier instantiation~\cite{WHD13,Dut15}. Additionally,

for dealing with quantifiers, CVC4 supports solving quantified

formulas by \emph{counter-example guided quantifier

  instantiation}~\cite{RDKT15}.

Skolemization and then instances of the universally quantified

formulas are lazily added to the formula until a model is found or the

formula is found to be unsatisfiable by a \QFBV solver. There are

multiple ways to choose quantifier instances that are sufficient to

decide the satisfiability of the formula. For the bit-vector theory,

the most widely used approach is the \emph{model-based quantifier

  instantiation} approach~\cite{GM09}, supported by Z3, CVC4, and

Yices, combined by heuristics as E-matching or symbolic quantifier

instantiation~\cite{WHD13,Dut15}. Additionally, for dealing with

quantifiers, CVC4 supports solving quantified formulas by

\emph{counter-example guided quantifier instantiation}~\cite{RDKT15}.

% and \emph{finite model finding}~\cite{RTG13}.

However, we describe only the model-based quantifier instantiation in

detail, as the counter-example guided quantifier considers all

\subsection{Model-based quantifier instantiation}

Given a closed formula with quantifiers, the first step is to convert

the formula to the negation normal form and apply Skolemization to

obtain equisatisfiable formula of the form

obtain equisatisfiable formula of the form \marginpar{TODO: V

  preliminaries vysvětlit notaci $\psi[x_1, \ldots, x_n]$}

\[

   \varphi ~\wedge~ \forall x_1, x_2, \dots, x_n\, (\psi[x_1, \dots, x_n]),

\]

number of distinct models $M$ of $\varphi$. However, in some cases

exponentially many such models have to be ruled out before the solver

is able to find a correct model or decide unsatisfiability of the

whole formula. To overcome this issue, state-of-the-art SMT solvers do

not use just instances of the form $\psi[v_1, \dots, v_n]$ with

concrete values, but employ heuristics such as

E-matching~\cite{DNS05,MB07} or symbolic quantifier

instantiation~\cite{WHD13} to choose instances with ground terms which

can potentially rule out more spurious models and thus significantly

reduce the number of iterations of the algorithm. In practice,

suitable ground terms substituted for quantified variables are

selected only from subterms of the input formula.

whole formula. To overcome this issue, state-of-the-art \smt solvers

do not use just instances of the form $\psi[v_1, \dots, v_n]$ with

concrete values, but employ mentioned heuristics such as E-matching or

symbolic quantifier instantiation to choose instances with ground

terms that can potentially rule out more spurious models and thus

significantly reduce the number of iterations of the algorithm. In

practice, suitable ground terms substituted for quantified variables

are selected only from subterms of the input formula.

Note that Skolemization introduces uninterpreted functions to the

quantified formulas of form other than

$\exists^*\forall^*\varphi$. This class of formulas is usually called

\emph{exists/forall}, or simply $\exists\forall$. Therefore, for the

model-based quantifier instantiation approach to be usable for

formulas not from the exists/forall class, the underlying

quantifier-free bit-vector solver has to support reasoning about

uninterpreted functions. In case of Z3, this is achieved by

\emph{template-based model finding} -- uninterpreted functions are

assigned templates and the quantifier-free \smt solver is used to find

parameters of these templates, which satisfy the formula. For example,

an uninterpreted function $f$ may be assigned a linear template

$f(x) = ax + b$ and finding a model of $f$ reduces to finding

parameters $a$ and $b$.

\section{Computational complexity}

formulas not from the exists/forall class, the underlying \QFBV solver

has to support reasoning about uninterpreted functions. In case of Z3,

this is achieved by \emph{template-based model finding} --

uninterpreted functions are assigned templates and the \QFBV solver is

used to find satisfying parameters of these templates. For example, an

uninterpreted function $f$ may be assigned a template for linear

functions $f(x) = ax + b$ and finding a satisfying function $f$

reduces to finding values $a$ and $b$.

\section{Computational complexity of bit-vector logics}

The propositional satisfiability problem is well known from the

computational complexity perspective, as it is the first problem that

was shown to be \NP-complete. The complexity of the satisfiability

the prefix UF, and the problems with unary and binary encoded

bit-widths are denoted by suffixes 1 and 2, respectively. For example,

QF\_UFBV2 is the decision problem for quantifier free formulas with

uninterpreted functions and binary encoded bit-withs.

uninterpreted functions and binary encoded bit-withs. The completeness

results for these classes are summarized in table

\ref{tbl:complexity}, and briefly explained in the rest of this section.

\begin{table}

\checkoddpage

\edef\side{\ifoddpage l\else r\fi}

\makebox[\textwidth][\side]{

\begin{minipage}[bt]{\fullwidth}

  \begin{center}

    \setlength{\tabcolsep}{0.6em}

    \begin{tabularx}{\textwidth}{l l l l X l r}

      \toprule

      & & \multicolumn{5}{c}{Quantifiers} \\

      \cmidrule(l){3-7}

      & & \multicolumn{2}{c}{No} & & \multicolumn{2}{c}{Yes} \\

      \cmidrule(l){3-4}

      \cmidrule(l){6-7}

      & & \multicolumn{2}{c}{Uninterpreted functions} & & \multicolumn{2}{c}{Uninterpreted functions} \\

      & & \multicolumn{1}{c}{No} & \multicolumn{1}{c}{Yes} & & \multicolumn{1}{c}{No} & \multicolumn{1}{c}{Yes} \\

      \midrule

      \multirow{2}{*}{Encoding~} & Unary & \NP & \NP & & \PSPACE & \NEXPTIME \\

       & Binary\hspace{2em} & \NEXPTIME & \NEXPTIME & & ? & \NNEXPTIME \\

      \bottomrule

    \end{tabularx}

  \end{center}

  \caption{Completeness results for various bit-vector logics and encodings~\cite{FKB13}.}

  \label{tbl:complexity}

\end{minipage}}

\end{table}

\paragraph{Unary encoded bit-widths} The bit-blasting yields a

polynomial time reduction from QF\_BV1 to \sat, showing that QF\_BV1

%\setcounter{page}{90}

% use \cleardoublepage here to avoid problems with pdfbookmark

\cleardoublepage

\newlength\fullmarginwidth

\fullmarginwidth=\marginparwidth

\advance\fullmarginwidth by \marginparsep

\newlength\fullwidth

\fullwidth=\textwidth

\advance\fullwidth by \fullmarginwidth

\include{Chapters/Chapter01}

\include{Chapters/Chapter02}

%\addtocontents{toc}{\protect\clearpage} % <--- just debug stuff, ignore

\thispagestyle{empty}

\hfill

%\hfill

\vfill

%\vfill

\noindent\myName: \textit{\myTitle,} \mySubtitle, %\myDegree, 

\textcopyright\ \myTime

%\noindent\myName: \textit{\myTitle,} \mySubtitle, %\myDegree,

%\textcopyright\ \myTime

%\bigskip

%