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--- sav08:qe_from_conjunction_of_literals_suffices [2009/04/21 19:04]
vkuncak
+++ sav08:qe_from_conjunction_of_literals_suffices [2015/04/21 17:30] (current)
@@ Line 1: / Line 1: @@
 ====== Quantifier Elimination from Conjunction of Literals is Sufficient ======
-To show constructively that a theory has quantifier elimination, it suffices to show that we can eliminate an existential quantifier applied to a conjunction of [[wp>literal (mathematical logic)|literal]]s, that is, show that each formula of the form:
+((taken from [[wp>Quantifier Elimination]] to which it was originally contributed, so the Wikipedia sharing rights apply))To show constructively that a theory has quantifier elimination, it suffices to show that we can eliminate an existential quantifier applied to a conjunction of [[wp>literal (mathematical logic)|literal]]s, that is, show that each formula of the form:
-\[
+\begin{equation*}
    \exists x. \bigwedge_{i=1}^n L_i
-\]
+\end{equation*}
 where each $L_i$ is a literal, is equivalent to a quantifier-free formula. Indeed, suppose we know how to eliminate quantifiers from conjunctions of formulae, then if $F$ is a quantifier-free formula, we can write it in [[wp>disjunctive normal form]]
-\[
+\begin{equation*}
    \bigvee_{j=1}^m \bigwedge_{i=1}^n L_{ij}
-\]
+\end{equation*}
 and use the fact that
-\[
+\begin{equation*}
    \exists x. \bigvee_{j=1}^m \bigwedge_{i=1}^n L_{ij}
-\]
+\end{equation*}
 is equivalent to
-\[
+\begin{equation*}
    \bigvee_{j=1}^m \exists x. \bigwedge_{i=1}^n L_{ij}
-\]
+\end{equation*}
 Finally, to eliminate a universal quantifier
-\[
+\begin{equation*}
    \forall x. F
-\]
+\end{equation*}
 where $F$ is quantifier-free, we transform $\lnot F$ into disjunctive normal form, and use the fact that $\forall x. F$ is equivalent to $\lnot \exists x. \lnot F.$
-((taken from [[wp>Quantifier Elimination]] to which it was originally contributed, so the Wikipedia sharing rights apply))