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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)
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====== Quantifier Elimination from Conjunction of Literals is Sufficient ====== ====== 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:
-\[+\begin{equation*}
​\exists x. \bigwedge_{i=1}^n L_i    ​\exists x. \bigwedge_{i=1}^n L_i
-\]+\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]] ​ 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]] ​
-\[+\begin{equation*}
​\bigvee_{j=1}^m \bigwedge_{i=1}^n L_{ij}    ​\bigvee_{j=1}^m \bigwedge_{i=1}^n L_{ij}
-\]+\end{equation*}
and use the fact that  and use the fact that
-\[+\begin{equation*}
​\exists x. \bigvee_{j=1}^m \bigwedge_{i=1}^n L_{ij}    ​\exists x. \bigvee_{j=1}^m \bigwedge_{i=1}^n L_{ij}
-\]+\end{equation*}
is equivalent to  is equivalent to
-\[+\begin{equation*}
​\bigvee_{j=1}^m \exists x. \bigwedge_{i=1}^n L_{ij}    ​\bigvee_{j=1}^m \exists x. \bigwedge_{i=1}^n L_{ij}
-\]+\end{equation*}

Finally, to eliminate a universal quantifier ​ Finally, to eliminate a universal quantifier ​
-\[+\begin{equation*}
​\forall x. F    ​\forall x. F
-\]+\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.\$ 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))