Polishing

Chapters/Chapter02.tex

@@ -85,7 +85,7 @@ and to each predicate symbol $P \in \Sigma^p$ of arity $n$ assigns a
 relation $P^\mathcal{A} \subseteq A^n$. Given a $\Sigma$-structure
 $\mathcal{A}$, there exists a unique homomorphic extension of
 $(\_)^\mathcal{A}$ to $\Sigma$-terms, which to each $\Sigma$-term $t$
-assigns an element $t^\mathcal{A} \in A$. In turn, using standard
+assigns an element $t^\mathcal{A} \in A$. In turn, by using standard
 definitions, the assignment $(\_)^\mathcal{A}$ can be further extended
 to assign a truth value $\varphi^\mathcal{A} \in \{ \true, \false \}$
 to each $\Sigma$-formula $\varphi$. A $\Sigma$-structure $\mathcal{A}$
@@ -159,9 +159,9 @@ signature, defined in the previous subsection, a \emph{many-sorted
 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
+predicate symbol 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
+of sorts $S \in \Sigma^S$ and definitions of atoms, literals, clauses
 and formulas are straightforward and can be found for example in
 Enderton~\cite{End01}.
 
@@ -204,7 +204,7 @@ the other hand, if after elimination of all variables no clauses
 remain, the formula is satisfiable. The main problem of \dppr is its
 space complexity, since the number of the clauses may grow
 exponentially even for simple formulas. To alleviate this problem, the
-refinement of \dppr algorithm was introduced in 1962 by Davis,
+refinement of the \dppr algorithm was introduced in 1962 by Davis,
 Logemann and Loveland~\cite{DPLL62}.
 
 The Davis--Putnam--Logemann--Loveland \quotegraffito{If you don't know