Fix todos

Bibliography.bib

@@ -150,13 +150,10 @@
   year      = {2015}
 }
 
-@inproceedings{BF15restarts,
+@misc{BF15restarts,
   author    = {Armin Biere and
                Andreas Fr{\"{o}}hlich},
   title     = {Evaluating {CDCL} Restart Schemes},
-  booktitle = {Workshop on Pragmatics of SAT -- {POS'15}, Austin, TX,
-                  USA, September 24-27, 2015, Proceedings},
-  pages     = {405--422},
   year      = {2015}
 }
 
@@ -1140,4 +1137,38 @@ year = {2005},
   journal   = {CoRR},
   volume    = {abs/1602.03050},
   year      = {2016}
+}
+
+@article{TZ05,
+  author    = {Cesare Tinelli and
+               Calogero G. Zarba},
+  title     = {Combining Nonstably Infinite Theories},
+  journal   = {J. Autom. Reasoning},
+  volume    = {34},
+  number    = {3},
+  pages     = {209--238},
+  year      = {2005}
+}
+
+@inproceedings{JB10,
+  author    = {Dejan Jovanovic and
+               Clark Barrett},
+  title     = {Polite Theories Revisited},
+  booktitle = {Logic for Programming, Artificial Intelligence, and Reasoning - 17th
+               International Conference, LPAR-17, Yogyakarta, Indonesia, October
+               10-15, 2010. Proceedings},
+  pages     = {402--416},
+  year      = {2010}
+}
+
+@inproceedings{RRZ05,
+  author    = {Silvio Ranise and
+               Christophe Ringeissen and
+               Calogero G. Zarba},
+  title     = {Combining Data Structures with Nonstably Infinite Theories Using Many-Sorted
+               Logic},
+  booktitle = {Frontiers of Combining Systems, 5th International Workshop, FroCoS
+               2005, Vienna, Austria, September 19-21, 2005, Proceedings},
+  pages     = {48--64},
+  year      = {2005}
 }
\ No newline at end of file

Chapters/Chapter02.tex

@@ -122,10 +122,11 @@ For two signatures $\Sigma$ and $\Omega$, we call $\Omega$ a
 $\Omega^p \subseteq \Sigma^p$. Given a $\Sigma$-structure
 $\mathcal{A}$ and a signature $\Omega$ that is a sub-signature of
 $\Sigma$, the \emph{reduct} of $\mathcal{A}$ to a sub-signature
-$\Omega$ is an $\Omega$-structure $\mathcal{A}'$ that coincides with
-$\mathcal{A}$ on all symbols from $\Omega$. For a $\Sigma_1$-theory
-$\mathcal{T}_1$ and a $\Sigma_2$-theory $\mathcal{T}_2$, their
-\emph{combination} $\mathcal{T}_1 + \mathcal{T}_2$ is the largest
+$\Omega$ is an $\Omega$-structure $\mathcal{A}'$ that has the same
+universe as $\mathcal{A}$ and coincides with $\mathcal{A}$ on all
+symbols from $\Omega$. For a $\Sigma_1$-theory $\mathcal{T}_1$ and a
+$\Sigma_2$-theory $\mathcal{T}_2$, their \emph{combination}
+$\mathcal{T}_1 + \mathcal{T}_2$ is the largest
 $(\Sigma_1 \cup \Sigma_2)$-theory that contains all
 $(\Sigma_1 \cup \Sigma_2)$-structures $\mathcal{A}$ for which the
 reduct of $\mathcal{A}$ to $\Sigma_1$ is in $\mathcal{T}_1$ and the
@@ -317,8 +318,7 @@ solutions~\cite{GSK98}. Although most commonly used heuristic to
 decide when to restart the solver is based on the \emph{Luby
   sequence}~\cite{LSZ93}, the recent survey has shown that it is
 outperformed by a heuristic based on the concept of exponential moving
-averages~\cite{BF15restarts}.\marginpar{spravit citaci, BF15a nevyšlo
-  v proceedings}
+averages~\cite{BF15restarts}.
 
 In addition to the mentioned heuristics, an efficient implementation
 of \cdcl based \sat solver relies on lazy data structures used in the
@@ -409,7 +409,12 @@ Notable examples of decidable first-order theories include
   interpreted as a value on the index $i$ of the array $a$, and a
   ternary function $\arwrite(a, i, v)$ 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}
+\item the theory of \emph{fixed-size bit-vectors}, in which the
+  structure contains all bit-vectors as an universe, ordering
+  relations according to the corresponding integers, and interpreted
+  functions are bit-wise operations on the bit-vectors and modular
+  arithmetic operations on the corresponding integers. This theory is
+  in detail described in section \ref{sec:qfbv}.
 \end{itemize}
 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
@@ -426,10 +431,12 @@ variables~\cite{NO79}. A theory $\mathcal{T}$ is \emph{stably
   infinite} if every $\mathcal{T}$-satisfiable formula has a
 $\mathcal{T}$-model whose universe is infinite. For a theory over a
 many-sorted logic, the theory is \emph{stably infinite} if every
-$\mathcal{T}$-satisfiable formula has a model, whose every sort
-has an infinite domain. Although almost all practically used theories
-are stably infinite, this is not true for inherently finite theories
-like the theory of bit-vectors.
+$\mathcal{T}$-satisfiable formula has a model, whose every sort has an
+infinite domain. Although almost all practically used theories are
+stably infinite, this is not true for inherently finite theories like
+the theory of bit-vectors. Therefore, variants of the theory
+combinations in which one of the theories does not have to be stably
+infinite have been proposed~\cite{TZ05, RRZ05, JB10}.
 
 \subsection{DPLL modulo theories}
 
@@ -608,6 +615,7 @@ arithmetic~\cite{CHN12}, linear integer arithmetic~\cite{KLR10, GLS11,
 and uninterpreted functions~\cite{McM11}.
 
 \section{Satisfiability of quantifier-free bit-vector formulas}
+\label{sec:qfbv}
 
 The \emph{theory of fixed sized bit-vectors (\BV)} is a many-sorted
 first-order theory with infinitely many sorts $\sort{n}$ corresponding