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sav08:quantifier_elimination_definition [2009/04/21 23:42] vkuncak |
sav08:quantifier_elimination_definition [2009/04/22 14:20] vkuncak |
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| ====== Definition of Quantifier Elimination ====== | ====== 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]]). We will write $\models F$ to denote $T \models F$ (which is equivalent to $F \in T$). | + | 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 | As a special case, we can have | ||
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| Then $\models F$ reduces to $\forall I \in {\cal I}. e_F(F)(I)$. | 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 $e_F(F)(I)$. | If we look at one interpretation, then ${\cal I}$ contains only that interpretation and the condition $\models F$ means $e_F(F)(I)$. | ||
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| + | **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, | **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, | ||