### Polishing

parent cc7d22bb
 ... ... @@ -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 ... ...
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