Strejda's comments

Chapters/Chapter02.tex

@@ -4,7 +4,7 @@
 
 \section{Preliminaries}
 
-This section introduces the notation which is used in the rest of this
+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 partly based on the same source, but the
@@ -72,16 +72,16 @@ contain free variables. For example, if we consider a set of atoms
 $\P = \{ x \geq y + 1, x = 4 \}$, where $x, y, 1$, and $4$ are
 constant symbols, $x \geq y + 1~\vee~\neg (x = 4)$ is a clause.
 
-A \emph{theory} $T$ is a set of first-order structures.  A formula $\varphi$
-is $T$-\emph{satisfiable} or $T$-\emph{consistent} if there is a
-structure $M \in T$ in which the formula $F$ holds in the standard
-first-order sense. The formula $\varphi$ is $T$-\emph{unsatisfiable} or
-$T$-\emph{inconsistent} otherwise. If the distinction is necessary, we
-call literals over a specific background theory $T$-literals. 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 $T$-literals over a
-$T$-\emph{solver}.
+A \emph{theory} $T$ is a set of first-order structures.  A formula
+$\varphi$ is $T$-\emph{satisfiable} or $T$-\emph{consistent} if there
+is a structure $M \in T$ in which the formula $F$ holds in the
+standard first-order sense. The formula $\varphi$ is
+$T$-\emph{unsatisfiable} or $T$-\emph{inconsistent} otherwise. If the
+distinction is necessary, we call literals over a specific background
+theory $T$-literals. 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
+$T$-literals a $T$-\emph{solver}.
 
 Using the same notation as in the propositional logic, a partial
 assignment is a set of literals and can be therefore seen as a
@@ -107,11 +107,11 @@ functions; this process is known as \emph{Skolemization}.
 
 \section{Propositional satisfiability}
 
-A \emph{propositional satisfiability problem} (\sat) is for a given
-formula $\varphi$ in \cnf decide whether it is satisfiable. The
+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
-Tseitin transformation can be used to transform every formula to a
-equisatisfiable formula in \cnf with only linear increase of its
+the Tseitin transformation can be used to transform every formula to
+an equisatisfiable formula in \cnf with only linear increase of its
 size~\cite{Tse68}.
 
 \subsection{Davis--Putnam--Logemann--Loveland algorithm}
@@ -145,7 +145,7 @@ search consists of decision and propagation steps. In decision steps,
 a variable and its new value are chosen and added to the current
 partial assignment. After each decision step, the \bcp is performed to
 set values of variables which are implied by the decision. The
-backtracking used in \dpll is \emph{chronological}, i.e. after the
+backtracking used in \dpll is \emph{chronological}, i.e. after a
 conflict the value of the last decided literal is changed.
 
 %It has been observed that \dpll based solver spends the
@@ -177,7 +177,7 @@ 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 a solution. There are multiple strategies of computing the
+contain any solution. 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
@@ -195,13 +195,14 @@ used VSIDS branching heuristic~\cite{LGPC16}.
 
 Another important part of modern \sat solvers are dynamic restarts,
 which allow restarting the search from scratch in hope that clauses
-learned during the conflict analyses will guide the search from regions
-of the search space that contain no solutions~\cite{GSK98}. Although
-most commonly used heuristic to decide when to restart the solver is
-based on the 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}
+learned during the conflict analyses will guide the search from
+regions of the search space that contain no
+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}
 
 In addition to the mentioned heuristics, an efficient implementation
 of \cdcl based \sat solver relies on lazy data structures used in the