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--- sav08:a_simple_sound_combination_method [2009/05/12 22:41]
vkuncak
+++ sav08:a_simple_sound_combination_method [2015/04/21 17:30] (current)
@@ Line 17: / Line 17: @@
 Improvement of precision: if $F$ is equivalent to disjunction $\vee_{j=1}^m D_j$ then to prove $F$ unsatisfiable, for each $1 \le j \le m$ check that $D_j$ is unsatisfiable by applying each of the provers.
-In particular, we may transform $F$ into [[Atomic Diagram Normal Form]].
+===== Defining Approximations =====
+We can define approximation for arbitrary first-order formulas.
+Start with $\alpha^1$, flip on negation.
+$\alpha^p(F_1 \land F_2) = \alpha^p(F_1) \land \alpha^p(F_2)$
+$\alpha^p(F_1 \lor F_2) = \alpha^p(F_1) \lor \alpha^p(F_2)$
+$\alpha^p(\lnot F) = \lnot \alpha^{1-p}(F)$
+$\alpha^p(\forall x.F) = \forall x. \alpha^p(F)$
+$\alpha^p(\exists x.F) = \exists x. \alpha^p(F)$
+$\alpha^p(A) = A$ for any supported atomic formula
+$\alpha^1(A) = true$ for unsupported atomic formula
+$\alpha^0(A) = false$ for unsupported atomic formula
+**Lemma:** $\alpha^0(F) \models F \models \alpha^1(F)$
+What does $\alpha^1$ do on a conjunction of flat literals?
 ===== Examples =====
@@ Line 24: / 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 $f(x) \neq f(1)$)
   * doing case analysis on equality of variables
 These are the key techniques that we use in methods that are complete for quantifier-free combinations of procedures that reason about disjoint function and predicate symbols.
-===== Defining Approximations =====
-Start with $\alpha^1$, flip on negation.
-$\alpha^p(F_1 \land F_2) = \alpha^p(F_1) \land \alpha^p(F_2)$
-$\alpha^p(F_1 \lor F_2) = \alpha^p(F_1) \lor \alpha^p(F_2)$
-$\alpha^p(\lnot F) = \lnot \alpha^{1-p}(F)$
-$\alpha^p(\forall x.F) = \forall x. \alpha^p(F)$
-$\alpha^p(\exists x.F) = \forall x. \alpha^p(F)$
-$\alpha^p(A) = A$ for any supported atomic formula
-$\alpha^1(A) = true$ for unsupported atomic formula
-$\alpha^0(A) = false$ for unsupported atomic formula
-**Lemma:** $\alpha^0(F) \models F \models \alpha^1(F)$
-What does $\alpha^1$ do on a conjunction of flat literals?
 ===== References =====
   * [[http://lara.epfl.ch/~kuncak/papers/ZeeETAL08FullFunctionalVerificationofLinkedDataStructures.html|Full Functional Verification of Linked Data Structures]]