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--- sav08:a_simple_sound_combination_method [2009/05/13 11:25]
vkuncak
+++ sav08:a_simple_sound_combination_method [2015/04/21 17:30] (current)
@@ Line 49: / Line 49: @@
 **Example** (from [[Calculus of Computation Textbook]], page 210, Example 10.1): Let $F$ be
-\[
+\begin{equation*}
 \le x \land x \le 2 \land f(x) \neq f(1) \land f(x) \neq f(2)
-\]
+\end{equation*}
 This formula is unsatisfiable.  However,
-\[
+\begin{equation*}
    \alpha_U(F) = true
-\]
+\end{equation*}
-\[
+\begin{equation*}
    \alpha_A(F) = 1 \le x \land x \le 2
-\]
+\end{equation*}
 and both approximations are satisfiable in resulting theories.
 Instead of doing approximation directly, let us transform the original formula into formula $F'$ where function application and arithmetic are separated:
-\[
+\begin{equation*}
 \le x \land x \le 2 \land y_1=1 \land f(x) \neq f(y_1) \land y_2=2 \land f(x) \neq f(y_2)
-\]
+\end{equation*}
 Variables $x$ and $y_1$ both appear in the formula, so let us convert $F'$ into disjunction of $(F' \land x=y_1) \lor (F' \land x \neq y_1)$.  We then check satisfiability for each of the disjuncts.  Consider, for example, $F' \land x = y_1$, which is formula $F''$
-\[
+\begin{equation*}
 \le x \land x \le 2 \land y_1=1 \land f(x) \neq f(y_1) \land y_2=2 \land f(x) \neq f(y_2) \land x = y_1
-\]
+\end{equation*}
 We have
-\[
+\begin{equation*}
    \alpha_U(F'') = f(x) \neq f(y_1) \land f(x) \neq f(y_2) \land x=y_1
-\]
+\end{equation*}
 which is unsatisfiable.  Then, considering formula $F' \land x=y_1$, we can do further case analysis on equality of two variables, say $x=y_2$.  For $F' \land x \neq y_1 \land x = y_2$ we similarly obtain unsatisfiability of $\alpha_U$-approximation.  The remaining case is $F' \land x \neq y_1 \land x \neq y_2$.  For this formula, denoted $F'''$, we have
-\[
+\begin{equation*}
    \alpha_A(F''') = 1 \le x \land x \le 2 \land y_1=1 \land y_2=2 \land x \neq y_1 \land x \neq y_2
-\]
+\end{equation*}
 which is unsatisfiable.  Therefore, by transforming formula into disjunction of formulas, we were able to prove unsatisfiability.  Two things helped precision
   * separating literals into literals understood by individual theories (unlike mixed literals $f(x) \neq f(1)$)