Differences

This shows you the differences between two versions of the page.

--- sav08:quantifier_elimination_definition [2009/04/22 14:20]
vkuncak
+++ sav08:quantifier_elimination_definition [2015/04/21 17:30]
@@ Line 1: / Line 1: @@
-====== Definition of Quantifier Elimination ======
-In this section, we will consider some language ${\cal L}$ and some set $T$ of formulas such that $Conseq(T)=T$ (see [[sav08:First-Order Logic Semantics]]).
-As a special case, we can have
-\[
-    T = \{ F \mid \forall I \in {\cal I}. e_F(F)(I) \}
-\]
-where ${\cal I}$ is a set of interpretations of interest.
-Then $\models F$ reduces to $\forall I \in {\cal I}. e_F(F)(I)$.
-If we look at one interpretation, then ${\cal I}$ contains only that interpretation and the condition $\models F$ means $e_F(F)(I)$.
-**Shorthand:** We will generally fix $T$ and write $\models F$ as a shorthand for $T \models F$.
-**Example:** Let $M = ({\cal Z},+)$ be the structure of integers with addition. If we let ${\cal I} = \{ M \}$ then $T$ in the definition above is the set of all formulas involving only $+$ that are true about integers. In this example,
-  * $T$ contains e.g. formulas
-    * $(\forall x. x = x)$
-    * $(\forall x.\forall y. x+y=y+x)$
-    * $(\forall x.\forall y. \forall z. (x+y=z \rightarrow z+x = x+x+y)$
-    * $(\exists y. \forall x. x + y = x)$
-  * $T$ does not contain e.g. formula $(\forall x. x + x = x)$
-**Definition:** We say that $T$ //admits quantifier elimination// iff for every FOL formula $F$ in ${\cal L}$ there exists some formula $qe(F)$ such that
-  * $\models (F \leftrightarrow qe(F))$
-  * $qe(F)$ has no quantifiers
-  * $FV(qe(F)) \subseteq FV(F)$
-(In words: for every formula there exists an equivalent quantifier-free formula with same free variables.)
-**Definition:** We say that $T$ admits //effective quantifier elimination// if there is an //effectively computable// function $qe$ with the property in the previous definition.
-**Definition:** A //ground formula// is a formula $F$ such that
-  * $F$ contains no quantifiers and
-  * $FV(F)=\emptyset$.
-Example: in language ${\cal L} = \{0,1,+\}$ where $0,1$ are constants and $+$ is a binary operation, an example of a ground formula is
-\[
-    \lnot (0 = 1) \rightarrow \lnot (0 + 1 = 1 + 1)
-\]
-**Lemma:** Suppose that
-  * $T$ has effective quantifier elimination;
-  * there is an algorithm for deciding $F \in T$ when $F$ is a ground formula.
-Then there is an algorithm for deciding, given a //sentence// $F$, whether $F \in T$.
-**Proof:**
-++++|
-Take $F$ a sentence.  Note $FV(F)=\emptyset$.  Compute $qe(F)$. Note that also $FV(qe(F)) \subseteq FV(F) = \emptyset$.  So $qe(F)$ is a ground formula and there is an algorithm to check $\models qe(F)$.  But $\models F$ is equivalent to $\models qe(F)$.
-++++