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'atomic formula'
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| Title of object: |
atomic formula |
| Canonical Name: |
AtomicFormula |
| Type: |
Definition |
| Created on: |
2002-06-02 11:37:53 |
| Modified on: |
2007-11-23 13:13:33 |
| Classification: |
msc:03B10, msc:03C99 |
| Defines: |
literal, quantifier-free formula |
| Synonyms: |
atomic formula=quantifier free formula |
Preamble:
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Content:
Let $\Sigma$ be a signature and $T(\Sigma)$ the set of terms over $\Sigma$. The set $S$ of symbols for $T(\Sigma)$ is the disjoint union of $\Sigma$ and $V$, a countably infinite set whose elements are called \emph{variables}. Now, adjoin $S$ the set $\lbrace =, (, )\rbrace$, assumed to be disjoint from $S$. An \emph{atomic formula} $\varphi$ over $\Sigma$ is any one of the following:
\begin{enumerate}
\item either $(t_1=t_2)$, where $t_1$ and $t_2$ are terms in $T(\Sigma)$,
\item or $(R(t_1,...,t_n))$, where $R\in\Sigma$ is an $n$-ary relation symbol, and $t_i\in T(\Sigma)$.
\end{enumerate}
\textbf{Remarks}.
\begin{enumerate}
\item
Using atomic formulas, one can inductively build formulas using the logical connectives $\vee$, $\neg$, $\exists$ etc...
\item
A \emph{literal} is a formula that is either atomic or of the form $\neg \varphi$ where $\varphi$ is atomic.
\item
A \emph{qunatifier-free formula} is a formula that does not contain the symbols $\exists$ or $\forall$.
\item
If we identify a formula $\varphi$ with its double negation $\neg (\neg \varphi)$, then it can be shown that any quantifier-free formula can be identified with a formula that is in conjunctive normal form, that is, a finite conjunction of finite disjunctions of literals.
\end{enumerate} |
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