Going to bed

Chapters/Chapter02.tex

@@ -8,11 +8,12 @@
 This section introduces the notation used in the rest of this
 chapter. The exposition of the propositional logic is mainly based on
 the work of Nieuwenhuis et al.~\cite{NOT06}. The exposition of the
-first-order logic is based on works of Enderton~\cite{End01} and of
-Barrett et al.~\cite{BSST09}. Note that instead of following the
-approach of Enderton, we follow the approach of Barrett et al. and
-define the theory as a set of first-order \emph{structures}, rather
-than a set of first-order sentences.
+first-order logic is based mainly on works of Enderton~\cite{End01}
+and of Barrett et al.~\cite{BSST09}. Note that instead of following
+the approach of Enderton to the first-order theories, we follow the
+approach of Barrett et al. and define the theory as a set of
+first-order \emph{structures}, rather than a set of first-order
+sentences.
 
 \subsection{Propositional formulas, assignments, and satisfaction}
 \label{prelim:prop}
@@ -21,7 +22,7 @@ Let $\P$ be a fixed finite set of propositional variables. For every
 variable $x \in \P$ there are two literals -- a \emph{positive
   literal} $x$ and a \emph{negative literal} $\overline{x}$. For a
 given literal $l$, we define $\neg l$ as $\overline{l}$ if $l = x$ for
-some variable $x$ and as $x$ if $l = \overline{x}$ for some variable
+a variable $x$ and as $x$ if $l = \overline{x}$ for a variable
 x. Literals $l$ and $\neg l$ are called \emph{complementary}. A
 \emph{clause} is a finite disjunction of literals. The empty clause is
 denoted by $\bot$. A formula in the \emph{conjunctive normal form}
@@ -65,7 +66,7 @@ if $\varphi$ is satisfiable precisely if $\psi$ is satisfiable.
 
 A \emph{signature} $\Sigma$ consists of a set of \emph{function
   symbols} $\Sigma^f$, a set of \emph{predicate symbols} $\Sigma^p$,
-and for each of these symbols a non-negative number called its
+and for each of these symbols of a non-negative number called its
 \emph{arity}. Given a signature $\Sigma$, \emph{$\Sigma$-terms},
 \emph{$\Sigma$-atoms}, \emph{$\Sigma$-literals},
 \emph{$\Sigma$-clauses}, \emph{$\Sigma$-formulas}, and
@@ -104,10 +105,10 @@ $\Sigma$-formula and $\Gamma$ is a set of $\Sigma$-formulas, we say
 that \emph{$\Gamma$ entails $\varphi$ in $\mathcal{T}$}, written
 $\Gamma \models_\mathcal{T} \varphi$, if every $\Sigma$-structure that
 satisfies all formulas in $\Gamma$ also satisfies $\varphi$. If
-$\emptyset \models_\mathcal{T} \varphi$ for a $\Sigma$-formula
-$\varphi$, the formula $\varphi$ is called \emph{$\mathcal{T}$-valid}
-or a \emph{theory lemma}. A set $\Gamma$ of $\Sigma$-formulas is
-called \emph{$\mathcal{T}$-inconsistent}, if
+$\emptyset \models_\mathcal{T} \varphi$, the formula $\varphi$ is
+called \emph{$\mathcal{T}$-valid} or a \emph{theory lemma}. A set
+$\Gamma$ of $\Sigma$-formulas is called
+\emph{$\mathcal{T}$-inconsistent}, if
 $\Gamma \models_{\mathcal{T}} \bot$.
 
 % If $\varphi$
@@ -141,8 +142,8 @@ simultaneous substitution of every free variable $x_i$ in $\varphi$ by
 the term $t_i$.
 
 In the rest of this chapter, we consider only theories for which the
-satisfiability of conjunctions of literals is decidable and we call
-any decision procedure for conjunctions of $\mathcal{T}$-literals a
+satisfiability of conjunctions of $\Sigma$-literals is decidable and
+we call any decision procedure for conjunctions of $\Sigma$-literals a
 $\mathcal{T}$-\emph{solver}. Moreover, if the signature or the theory
 is clear from the context, we drop the respective $\Sigma$- or
 $\mathcal{T}$- prefixes.
@@ -184,8 +185,8 @@ in Enderton~\cite{End01}.
 \section{Propositional satisfiability}
 \label{sec:sat}
 
-A \emph{propositional satisfiability problem} (\sat) is, for a given
-formula $\varphi$ in \cnf, to decide whether it is satisfiable. The
+A \emph{propositional satisfiability problem} (\sat) is to decide for
+a given formula $\varphi$ in \cnf, whether it is satisfiable. The
 restriction to formulas in \cnf is without a loss of generality, as
 the Tseytin transformation can be used to transform every formula to
 an equisatisfiable formula in \cnf with only linear increase of its
@@ -254,7 +255,7 @@ change the value of the last decision literal that occurs in the
 clause $C$. Note that this literal does not have to be the last
 decision literal in the search and therefore a bigger region of the
 search space can be ruled out from the search, as it is known not to
-contain any solution. There are multiple strategies of computing the
+contain any solutions. There are multiple strategies of computing the
 clausal reason for the conflict, but the majority of \cdcl based \sat
 solvers are using the \emph{first unique implication point} based
 learning scheme, which has been shown to produce small clauses in
@@ -310,14 +311,14 @@ using Reduced Ordered \bdds, which represent Boolean functions
 combinations of the represented Boolean functions using the recursive
 procedure \texttt{Apply}~\cite{Bry86}. The main idea of symbolic \sat
 solvers is to convert a \cnf formula to the corresponding \robdd and
-check the root of the resulting \bdd. If the resulting \bdd has the
-root 0, the formula is unsatisfiable, and it is satisfiable
-otherwise. However, in order to keep the size of the \bdd small, it is
-necessary to existentially quantify the variables during the
-computation. This technique is known as \emph{early
-  quantification}~\cite{HKB96}. A simplified symbolic \sat algorithm
-can be found for example in the survey of \sat solving by Darwiche and
-Pipatsriswat~\cite{DP09}.
+check the root of the resulting \bdd. If the resulting \bdd is
+equivalent to the \bdd for the constant $0$ function, the formula is
+unsatisfiable, and it is satisfiable otherwise. However, in order to
+keep the size of the \bdd small, it is necessary to existentially
+quantify the variables during the computation. This technique is known
+as \emph{early quantification}~\cite{HKB96}. A simplified symbolic
+\sat algorithm can be found for example in the survey of \sat solving
+by Darwiche and Pipatsriswat~\cite{DP09}.
 
 Look-ahead based algorithm, in contrast to the \cdcl, are employing
 expensive heuristics to guide the \dpll search to a satisfying
@@ -370,11 +371,10 @@ Notable examples of decidable first-order theories include
   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}, 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}.
+  structure contains set of all bit-vectors as an universe, bit-wise
+  and arithmetic functions on those bit-vectors, and ordering
+  relations 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
@@ -395,7 +395,7 @@ $\mathcal{T}$-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
+combination in which one of the theories does not have to be stably
 infinite have been proposed~\cite{TZ05, RRZ05, JB10}.
 
 \subsection{DPLL modulo theories}
@@ -405,14 +405,14 @@ Most of the \smt approaches can be classified as \emph{eager} or
 is directly translated to an equivalent propositional formula and an
 off-the-shelf \sat solver is used to decide satisfiability of this
 formula. The eager \smt approach is implemented for example in the
-\smt solver \uclid, which supports the combination of the theory of
-uninterpreted functions and linear integer arithmetic~\cite{LS04}.
+\smt solver \uclid for the combination of the theory of uninterpreted
+functions and linear integer arithmetic~\cite{LS04}.
 
 On the other hand, the lazy \smt approach uses a \sat solver to reason
 about the Boolean structure of the formula and a specialized
 $\mathcal{T}$-solver to reason about conjunctions of
-$\mathcal{T}$-literals. There are two variants of the lazy approach to
-the \smt solving -- \emph{online} and \emph{offline}~\cite{FJOS03}.
+$\Sigma$-literals. There are two variants of the lazy approach to the
+\smt solving -- \emph{online} and \emph{offline}~\cite{FJOS03}.
 
 \paragraph{Offline approach}
 In the offline approach, the input formula $\varphi$ is first
@@ -423,8 +423,8 @@ $\varphi^p$. If $\varphi^p$ is unsatisfiable, the input formula
 $\varphi$ is also unsatisfiable. On the other hand, if $\varphi^p$ is
 satisfiable, the \sat solver returns its propositional model $M^p$,
 and the $\mathcal{T}$-solver can be used to determine
-$\mathcal{T}$-satisfiability of the corresponding set of literals
-$M$. If the $\mathcal{T}$-solver decides $M$ to be
+$\mathcal{T}$-satisfiability of the corresponding set of
+$\Sigma$-literals $M$. If the $\mathcal{T}$-solver decides $M$ to be
 $\mathcal{T}$-satisfiable, the formula $\varphi$ is also
 $\mathcal{T}$-satisfiable. If the propositional model is not
 $\mathcal{T}$-satisfiable, the corresponding theory lemma $\neg M^p$
@@ -468,11 +468,11 @@ Simplify~\cite{Simplify}, Yices~\cite{Yices}, and Z3~\cite{Z3}.
 \subsection{Natural domain \smt}
 \label{ssec:natDomainSat}
 Although the separation of the propositional and theory reasoning in
-the \dpllt approach allows the solver to be modular, it can be also
-restricting in some cases. In particular, \dpllt-based solvers can not
-directly reason about values of first-order variables, but have to
-rely on the $\mathcal{T}$-solver guiding the search over Boolean
-valuations. While there are some techniques like \emph{splitting on
+\dpllt allows the solver to be modular, it can be also restricting in
+some cases. In particular, \dpllt-based solvers can not directly
+reason about values of first-order variables, but have to rely on the
+$\mathcal{T}$-solver guiding the search over Boolean valuations. While
+there are some techniques like \emph{splitting on
   demand}~\cite{BNOT06} , which allow the $\mathcal{T}$-solver to add
 new atoms to the formula and delegate the case splitting to the
 underlying \sat solver, their performance depends on the internal
@@ -514,7 +514,7 @@ computed by the conflict analysis.
 In thus specified framework, an abstract domain can be replaced by an
 arbitrary domain that satisfies a certain set of requirements and this
 yields an abstract conflict-driven clause learning \acdcl algorithm
-using this particular domain. As an example, Brain et al. have
+using this particular domain. For example, Brain et al. have
 implemented the \fpacdcl solver for the floating point arithmetic
 using an abstract interval domain and have shown that on a certain
 class of formulas it significantly outperforms state-of the art \smt
@@ -556,14 +556,15 @@ core. In the second approach each clause can be represented by a fresh
 selector variable and after a top level conflict, clauses whose
 selector variable appears in the final conflict clause are returned as
 an unsatisfiable core. Third approach consists in combination of an
-\smt solver with an external propositional core~\cite{CGS07}.
+\smt solver with an external propositional core
+extractor~\cite{CGS07}.
 
 A (Craig) interpolant for a pair of formulas $F$ and $G$ such that
 $F \wedge G$ is unsatisfiable is a formula $I$ such that
 $F \models_T I$, $G \wedge I$ is unsatisfiable, and all uninterpreted
 symbols in $I$ are contained in both $F$ and $G$. Following the work
-of Pudlák~\cite{Pud97} or and McMillan~\cite{McM05}, an interpolant
-for the pair $(F, G)$ over the theory $\mathcal{T}$ can be computed from a
+of Pudlák~\cite{Pud97} and McMillan~\cite{McM05}, an interpolant for
+the pair $(F, G)$ over the theory $\mathcal{T}$ can be computed from a
 resolution proof of unsatisfiability of $F \wedge G$ by using a
 combination of a propositional interpolating algorithm and a theory
 specific interpolation procedure, which computes interpolants of
@@ -578,20 +579,18 @@ and uninterpreted functions~\cite{McM11}.
 The \emph{theory of fixed sized bit-vectors (\BV)} is a many-sorted
 first-order theory with infinitely many sorts $\sort{n}$ corresponding
 to bit-vectors of length $n$. The only predicate symbols in the \BV
-theory are $=$, $\leq_u$, and $\leq_s$, interpreted as equality,
-unsigned inequality of binary-encoded natural numbers, and signed
-inequality of integers in $2$'s complement representation,
-respectively. Function symbols in the theory are $+$, $\times$,
-$\div$, $\&$, $\mid$, $\oplus$, $\ll$, $\gg$, $\cdot$,
-$\extract{n}{p}$, interpreted as addition, multiplication, unsigned
-division, bit-wise and, bit-wise or, bit-wise exclusive or,
-left-shift, right-shift, concatenation, and extraction of $n$ bits
-starting from the position $p$, respectively. For the detailed
-description of the \BV theory syntax and semantics, see for example
-Hadarean's PhD thesis~\cite{Had15}. This section focuses on the
-problem of satisfiability of the quantifier-free fragment of the \BV
-theory, denoted \QFBV. The the full \BV logic is treated in the next
-section.
+theory are $\leq_u$ and $\leq_s$, interpreted as inequality of
+binary-encoded unsigned and signed integers, respectively. Function
+symbols in the theory are $+$, $\times$, $\div$, $\&$, $\mid$,
+$\oplus$, $\ll$, $\gg$, $\cdot$, $\extract{n}{p}$, interpreted as
+addition, multiplication, unsigned division, bit-wise and, bit-wise
+or, bit-wise exclusive or, left-shift, right-shift, concatenation, and
+extraction of $n$ bits starting from the position $p$,
+respectively. For the detailed description of the \BV theory syntax
+and semantics, see for example Hadarean's PhD
+thesis~\cite{Had15}. This section focuses on the problem of
+satisfiability of the quantifier-free fragment of the \BV theory,
+denoted \QFBV. The the full \BV logic is treated in the next section.
 
 Current state-of-the-art \smt solvers for the \QFBV logic rely on
 rewriting techniques used to simplify the formula during the
@@ -599,8 +598,7 @@ preprocessing, and eager or lazy translation of the bit-vector formula
 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}.  \marginpar{TODO: Nepřidat odstavec
-  o propagating-complete \cnf encodings?} More lazy approach to the
+\emph{bit-blasting}~\cite{Kro08}. More lazy approach to the
 bit-blasting is beneficial 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
@@ -620,13 +618,13 @@ Although bit-blasting is highly efficient for most of practical
 problems, it can exhaust memory of the solver if the input formula
 contains complex arithmetic or variables with large bit-width. Several
 techniques that avoid the bit-blasting and work directly on the level
-of individual bit-vectors (\emph{word-level}) have been proposed to
+of individual bit-vectors (\emph{word level}) have been proposed to
 alleviate this problem.
 
 Some useful fragments of the bit-vector theory can be solved by
 specialized algorithms for deciding satisfiability. For example,
 conjunctions of literals in the \emph{core theory of bit-vectors}
-consisting of equality, concatenation and extraction, is decidable in
+consisting of equality, concatenation and extraction, are decidable in
 polynomial time by reduction to the theory of equality~\cite{CMR97,
   BS09}. Additionally, Babić and Musuvathi have described algorithms
 for solving linear and non-linear modular arithmetic, which can be
@@ -709,13 +707,14 @@ industrial benchmarks and especially in benchmarks produced during a
 verification of programs in a single static assignment form, such as
 LLVM bit-code.
 
-With a slight abuse of notation which identifies interpreted function
-symbols with their intended interpretations, a simple definition of
-invertibility for binary function symbols, which can be easily
-generalized, is as follows. A binary function $f$ can be inverted with
-respect to its first argument if for every two values $a_i, a_o$ there
-exists a value $b$ such that $f(b,a_i) = a_o$. For example, as the
-addition is invertible with respect to its first argument, the formula
+With a slight abuse of notation which identifies function symbols with
+their intended interpretations, a simple definition of invertibility
+for binary function symbols is as follows. A binary function $f$ can
+be inverted with respect to its first argument if for every two values
+$a_i, a_o$ there exists a value $b$ such that $f(b,a_i) = a_o$. This
+definition can be easily generalized to functions of different
+arities. For example, as the addition is invertible with respect to
+its first argument, the formula
 $\varphi \equiv x + (y * y - 30 * z * y) = 0$ can be transformed to an
 equisatisfiable formula $v = 0$, where $v$ is a fresh variable,
 because $x$ is unconstrained in $\varphi$. Note that in contrast to
@@ -736,7 +735,7 @@ 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}~\cite{GM09}, implemented in Z3, CVC4, and Yices,
-combined by heuristics as \emph{E-matching} or \emph{symbolic
+combined with heuristics as \emph{E-matching} or \emph{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}.
@@ -767,10 +766,10 @@ $\neg \widehat{\psi}$ is not satisfiable, then the structure $M$ is
 indeed a model of the formula $\forall x_1, x_2, \dots, x_n\,(\psi)$,
 therefore the entire formula is satisfiable and $M$ is its model. If
 $\neg \widehat{\psi}$ is satisfiable, we get values $v_1, \dots, v_n$
-such that $\neg\widehat{\psi}[v_1, \dots, v_n]$ holds.  To rule out
-$M$ as a model as the formula $\varphi$, the instance
-$\psi[v_1, \dots, v_n]$ of the quantified formula is added to the
-quantifier-free part, i.e.~the formula $\varphi$ is modified to
+such that $\neg\widehat{\psi}[v_1, \dots, v_n]$ holds. To rule out $M$
+as a model of the input formula, the instance $\psi[v_1, \dots, v_n]$
+of the quantified formula is added to the quantifier-free part,
+i.e.~the formula $\varphi$ is modified to
 \[
    \varphi' ~\equiv~ \varphi \wedge \psi[v_1, \dots, v_n],
 \]
@@ -833,7 +832,7 @@ studied and was shown to range from \NP-complete to
 of the bit-vector satisfiability problem differ in allowing
 uninterpreted functions, allowing quantifiers, and in encoding of the
 bit-widths (unary vs. binary). In the following, we follow the
-notation of Kováznai et al~\cite{KFB16} -- decision problems for
+notation of Kováznai et al.~\cite{KFB16} -- decision problems for
 quantifier-free fragments are denoted by the prefix QF\_, the
 combination with the theory of uninterpreted functions is denoted by
 the prefix UF, and the problems with unary and binary encoded
@@ -874,7 +873,7 @@ results for these classes are summarized in table
 polynomial time reduction from QF\_BV1 to \sat, showing that QF\_BV1
 is in NP. A similar reduction from BV1 to \qbf can show that BV is in
 \PSPACE. For lower bounds, \NP-hardness of QF\_BV1 follows from a
-simple reduction from \sat, by encoding each propositional variable as
+simple reduction from \sat by encoding each propositional variable as
 a bit-vector of bit-width 1. Similarly, BV1 can be shown to be
 \PSPACE-hard by considering every bit-vector as a quantified variable
 of bit-width 1.

Chapters/Chapter03.tex

@@ -19,8 +19,8 @@ The benefit of a \bdd-based solver over the quantifier
 instantiation-based one is the ability of representing all models of a
 universally quantified formula even in cases in which finding a
 suitable set of quantifier instantiations would be infeasible. The
-following formula, in which all variables have bit-with 32 bits, shows
-this difference:
+following formula, in which all variables have bit-width 32 bits,
+shows this difference:
 \[
   \varphi \equiv a=16 \cdot b \,+\, 16 \cdot c~\wedge~\forall (x :
   [32]) \, (a \not = 16 \cdot x).
@@ -29,14 +29,15 @@ this difference:
 The \bdd-based solver can quickly compute the \bdd for the quantified
 subformula $\forall (x : [32]) \, (a \not = 16 \cdot x)$, which shows
 that at least one of the last 4 bits of the variable $a$ has to be
-$0$, i.e. the value of $a$ is not divisible by $16$. After conjoining
+$1$, i.e. the value of $a$ is not divisible by $16$. After conjoining
 this \bdd to the \bdd for $a=16 \cdot b \,+\, 16 \cdot c$, the formula
-is decided unsatisfiable, as the resulting \bdd has root $0$. On the
-other hand, if one considers only quantifier instantiations by the
-subterm of the input formula, exponentially many quantifier instances
-have to be added to the formula to show its unsatisfiability, as there
-is no subterm of $\varphi$ that can be instantiated as $x$ to yield an
-unsatisfiable quantifier-free formula.
+is decided unsatisfiable, as the resulting \bdd represents the
+constant $0$ function. On the other hand, if one considers only
+quantifier instantiations by the subterm of the input formula,
+exponentially many quantifier instances have to be added to the
+formula to show its unsatisfiability, as there is no subterm of
+$\varphi$ that can be instantiated as $x$ to yield an unsatisfiable
+quantifier-free formula.
 
 \subsection{Simplifications}
 
@@ -57,7 +58,7 @@ distributed over disjunctions.
 
 Moreover, universal quantification can distribute over disjunctions in
 cases where the variable bound by the quantifier does not occur in one
-of the conjuncts. This leads to the following rule and its dual
+of the disjuncts. This leads to the following rule and its dual
 version, which is known as \emph{miniscoping}:
 \[
   \forall x. \, (\varphi ~\vee~ \psi)~~\leadsto~~(\forall x . \,
@@ -91,9 +92,9 @@ where $t$ is an arbitrary term that does not contain $x$.
 Ordering of \bdd variables is crucial for efficiency. The size of a
 \bdd can differ exponentially when choosing a different
 ordering. Therefore, besides well known methods of dynamic variable
-reordering during the computation, Q3B precomputes
-initial variable ordering, which is based on the dependencies among
-the variables in the formula.
+reordering during the computation, Q3B computes an initial variable
+ordering, which is based on the syntactic dependencies among the
+variables in the formula.
 
 We have implemented and compared three different initial orderings,
 which we denote $\leq_1, \leq_2, \leq_3$.
@@ -297,7 +298,7 @@ formulas.
   \end{minipage}}
 \end{table}
 
-Recently, this results was confirmed by the \smt competition 2016 --
+Recently, these results were confirmed by the \smt competition 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
@@ -309,7 +310,7 @@ them in the shortest time. Moreover, Q3B can solve more benchmarks
 with unknown status than any of the other solvers.
 
 \section{Other areas of research}
-Outside the field of \smt solving, my research also consists in
+Outside the field of \smt solving, my research interests also include
 software verification. In this field, I have co-authored the following
 paper on the tool Symbiotic, which combines instrumentation, slicing
 and, symbolic execution to allow verification of a real-world
@@ -324,7 +325,7 @@ code~\cite{CJSSV16}.
     Formulas Using Binary Decision Diagrams.}" In: Theory and
   Applications of Satisfiability Testing -- SAT 2016 -- 19th
   International Conference, Bordeaux, France, July 5-8, 2016,
-  Proceedings. 2016, pp. 267-283
+  Proceedings. 2016, pp. 267-283~\cite{JS16}
   \smallskip
 
   I am the main author of the text of the paper. I have also
@@ -335,7 +336,7 @@ code~\cite{CJSSV16}.
   Algorithms for the Construction and Analysis of Systems -- 22nd
   International Conference, TACAS 2016, Held as Part of the ETAPS
   2016, Eindhoven, The Netherlands, April 2-8, 2016, Proceedings,
-  pp. 946-949
+  pp. 946-949~\cite{CJSSV16}
 
   \smallskip
 

Chapters/Chapter04.tex

@@ -39,14 +39,9 @@ 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 value
-of a variable is decided by during the \mcbv computation, it 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, an implementation of a solver using the
-hybrid approach and its evaluation on the representative set of
-benchmarks is expected. The hybrid approach to quantified bit-vectors
-is the main aim of my PhD study.
+of a variable is decided during the \mcbv computation, it can be used
+to partially instantiate a quantified part of the formula, for which
+the \bdd will be computed.
 
 \subsection{Unconstrained variable propagation for quantified
   bit-vectors}
@@ -70,7 +65,7 @@ be replaced by a simpler term regargless 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
-$extract_0^{n-i}(x) \cdot 0^i$.. This approach can be also extended to
+$extract_0^{n-i}(x) \cdot 0^i$. This approach can be also extended to
 the multiplication of two variables from which one is unconstrained. I
 plan to investigate further extensions to the terms containing
 division and remainder operations and to publish a paper concerning

classicthesis-config.tex

@@ -26,7 +26,7 @@
 % 1. Configure classicthesis for your needs here, e.g., remove "drafting" below
 % in order to deactivate the time-stamp on the pages
 % ****************************************************************************************************
-\PassOptionsToPackage{eulerchapternumbers,listings,drafting,dottedtoc,
+\PassOptionsToPackage{eulerchapternumbers,listings,dottedtoc,
 					 %floatperchapter,%linedheaders,%
 					 subfig,beramono,eulermath,minionprospacing}{classicthesis}
 % ********************************************************************