Many-sorted logic + small changes

Bibliography.bib

@@ -844,12 +844,12 @@
 
 @techreport{BoolectorComp,
   title = {{Boolector at the SMT competition 2016}},
+  year={2016},
   series = {Technical Report 16/1},
   date = {June 2016},
   address =  {{FMV Reports Series, Institute for Formal Models and
     Verification, Johannes Kepler University, Altenbergerstr. 69, 4040
       Linz, Austria}},
-  year={2016}
 }
 
 @inproceedings{CMR97,
@@ -1098,4 +1098,27 @@ year = {2005},
                Design, 1993},
   pages     = {42--47},
   year      = {1993}
+}
+
+@inproceedings{Min93,
+  author    = {Shin{-}ichi Minato},
+  title     = {Zero-Suppressed BDDs for Set Manipulation in Combinatorial Problems},
+  booktitle = {{DAC}},
+  pages     = {272--277},
+  year      = {1993}
+}
+
+@inproceedings{BFGHMPS,
+  author    = {R. Iris Bahar and
+               Erica A. Frohm and
+               Charles M. Gaona and
+               Gary D. Hachtel and
+               Enrico Macii and
+               Abelardo Pardo and
+               Fabio Somenzi},
+  title     = {Algebraic decision diagrams and their applications},
+  booktitle = {Proceedings of the 1993 {IEEE/ACM} International Conference on Computer-Aided
+               Design, 1993, Santa Clara, California, USA, November 7-11, 1993},
+  pages     = {188--191},
+  year      = {1993}
 }
\ No newline at end of file

Chapters/Chapter02.tex

@@ -63,8 +63,8 @@ if $\varphi$ is satisfiable precisely if $\psi$ is satisfiable.
 \subsection{First-order logic and theories}
 
 A \emph{signature} $\Sigma$ consists of a set of \emph{function
-  symbols} $\Sigma^f$ and a set of \emph{predicate symbols} $\Sigma^p$
-and for each of the symbols a non-negative number called its
+  symbols} $\Sigma^f$, a set of \emph{predicate symbols} $\Sigma^p$
+and for each of these symbols a non-negative number called its
 \emph{arity}. Given a signature $\Sigma$, \emph{$\Sigma$-terms},
 \emph{$\Sigma$-atoms}, \emph{$\Sigma$-literals},
 \emph{$\Sigma$-clauses}, and \emph{$\Sigma$-formulas} are defined as
@@ -178,12 +178,45 @@ $\mathcal{T}$-\emph{solver}.
 % formula without existential quantifiers by introducing uninterpreted
 % functions; this process is known as \emph{Skolemization}.
 
+\subsection{Multi-sorted logic}
+
+For some theories, it can be convinient 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 many-sorted signature
+$\Sigma$ further contains a set of \emph{sort symbols}
+$\Sigma^S$. Arities in a many-sorted logic are also modified. Arity of
+a symbol determines not only a number of its arguments, but also their
+types -- each function symbol has assigned $(n+1)$-tuple of sort
+symbols for a non-negative integer $n$ and each predicate symbols has
+assigned a $n$-tuple of sort symbols for a non-negative integer
+$n$. Simultaneous inductive definition of terms of a sort
+$S \in \Sigma^S$ and definitions of atoms, literals, clauses and
+formulas are straightforward and can be found for example in Enderton.
+
+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
+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 arity $(S_1, \ldots, S_n, S_{n+1})$ assigns a
+function
+$f^\mathcal{A} \colon S_1^\mathcal{A} \times \ldots \times
+S_n^\mathcal{A} \rightarrow S_{n+1}^\mathcal{A}$, and to each
+predicate symbol $P \in \Sigma^p$ of arity $(S_1, \ldots, S_n)$
+assigns a relation
+$P^\mathcal{A} \subseteq S_1^\mathcal{A} \times \ldots \times
+S_n^\mathcal{A}$. Definitions of many-sorted satisfiability, models,
+entailment, and so on are also straightforward and can be found in
+Enderton.
+
 \section{Propositional satisfiability}
 
 A \emph{propositional satisfiability problem} (\sat) is, for a given
 formula $\varphi$ in \cnf, to decide whether it is satisfiable. The
 restriction to formulas in \cnf is without a loss of generality, as
-the Tseitin transformation can be used to transform every formula to
+the Tseytin transformation can be used to transform every formula to
 an equisatisfiable formula in \cnf with only linear increase of its
 size~\cite{Tse68}.
 
@@ -449,11 +482,11 @@ the search of the underlying \dpll solver has been named
 further improvements, we refer the reader to survey on the
 topic~\cite{Seb07}, or to the abstract description of the underlying
 transition system~\cite{NOT06}. The \dpllt approach is used in the
-majority of modern \smt solvers, including
+majority of modern \smt solvers, including solvers
 Barcelogic~\cite{Barcelogic}, CVC4~\cite{CVC4},
 MathSAT~\cite{MathSAT}, OpenSMT~\cite{OpenSMT},
 Simplify~\cite{Simplify}, Yices~\cite{Yices}, or
-Z3~\cite{Z3}.\marginpar{TODO: Přetok}
+Z3~\cite{Z3}.
 
 \subsection{Natural domain \smt}
 \label{ssec:natDomainSat}

Chapters/Chapter04.tex

@@ -19,9 +19,10 @@ could benefit from better refinement of the approximation 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 ??? and
-algebraic decision diagrams introduced by ??? and experimentally
-evaluate their effect on the performance of the \smt solver.
+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.
 
 \subsection{Hybrid approach to quantified bit-vectors}
 Although our results with the symbolic \smt solver for quantified