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--- sav08:simple_qe_for_dense_linear_orders [2009/04/21 19:32]
vkuncak
+++ sav08:simple_qe_for_dense_linear_orders [2015/04/21 17:30] (current)
@@ Line 11: / Line 11: @@
 Formulas $T$ are the formulas that are closed formulas that are true in the structure $(\mathbb{Q},<)$ or rational numbers.
+**Example:** The 'successor' formula:
+\begin{equation*}
+   \forall x. \exists y.\ x < y \ \land\ (\forall z. (x < z \rightarrow z=y \lor y < z)
+\end{equation*}
+Is this formula true in dense linear orders? Is there a non-dense linear order where its truth value is different?
 ===== Normal form of Formulas =====
@@ Line 23: / Line 29: @@
 If we have three concrete values for $x,y,z$, what is the form of the strongest type of a formula that we could write about them in this language? //(atomic type formula)//
+Theorem: every quantifier-free formula in a language with only relational symbols is a disjunction of atomic type formulas
+  * if we know the set $T$ of axioms, we may be able to show that the atomic type formulas have a simple form
 ===== Quantifier Elimination Step =====
+Quantifier elimination from atomic type formulas:
+Quantifier elimination from more general formulas:
+**Example:** Use quantifier elimination to compute the truth value in dense linear orders for the example 'successor' formula above.
+===== References =====
-  * [[http://www4.informatik.tu-muenchen.de/~nipkow/pubs/lqe.pdf|Linear Quantifier Elimination]]
+  * [[http://www4.informatik.tu-muenchen.de/~nipkow/pubs/lqe.pdf|Linear Quantifier Elimination]] (Tobias Nipkow, IJCAR 2008)
   * [[http://citeseer.ist.psu.edu/loos93applying.html|Applying Linear Quantifier Elimination]]
   * [[http://citeseer.ist.psu.edu/71579.html|Parallel Fourier-Motzkin Elimination]]