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--- sav08:homework11 [2008/05/08 18:54]
vkuncak
+++ sav08:homework11 [2015/04/21 17:30]
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-====== Homework 11, Due May 14 ======
-===== Problem 0 =====
-Prepare a 5-10 minute update on your project progress to present it on May 15 in exercises.
-===== Problem 1 =====
-Galois connection is defined by two monotonic functions $\alpha : C \to A$ and $\gamma : A \to C$ between [[:partial order]]s $\leq$ on $C$ and $\sqsubseteq$ on $A$, such that
-\begin{equation*}
-  \alpha(c) \sqsubseteq a\ \iff\ c \leq \gamma(a) \qquad (*)
-\end{equation*}
-for all $c$ and $a$ (intuitively, the condition means that $c$ is approximated by $a$).
-**Part a)** Show that the condition $(*)$ is equivalent to the conjunction of these two conditions:
-\begin{eqnarray*}
-  c &\leq& \gamma(\alpha(c)) \\
-  \alpha(\gamma(a)) &\sqsubseteq& a
-\end{eqnarray*}
-hold for all $c$ and $a$.
-**Part b)** Let $\alpha$ and $\gamma$ satisfy the condition of Galois connection.  Show that the following three conditions are equivalent:
-  * $\alpha(\gamma(a)) = a$ for all $a$
-  * $\alpha$ is a [[wk>surjective function]]
-  * $\gamma$ is an [[wk>injective function]]
-**Part c)** State the condition for $c=\gamma(\alpha(c))$ to hold for all $c$.  When $C$ is the set of sets of concrete states and $A$ is a domain of static analysis, is it more reasonable to expect that $c=\gamma(\alpha(c))$ or $\alpha(\gamma(a)) = a$ to be satisfied, and why?
-===== Problem 2 =====
-Suppose you are given a set of predicates ${\cap P} = \{P_0,P_1,\ldots,P_n\}$ in a decidable theory of first-order logic (for example, combination of uninterpreted function symbols with integer linear arithmetic) where $P_0$ is the predicate 'false'.
-**Part a)** Consider [[Conjunctions of Predicates]] as abstract interpretation domain.  Give example showing that it need not be the case that
-\[
-    a_1 \leq a_2 \leftrightarrow \gamma(a_1) \subseteq \gamma(a_2)
-\]
-**Part b)** Describe how to construct from $A$ a new, smaller, lattice $B$, where the above equivalence holds.  Is there an algorithm to compute $B$ and the partial order on $B$ using a decision procedure for the logic of predicates?
-**Part c)** Suppose that, for the same set of predicates, we use lattice $A$ and lattice $B$ to compute the fixpoints $g_A$ and $g_B$ of the function $F^{\#}$ from [[Abstract Interpretation Recipe]].  What can you say about
-  * the comparison of numbers of iterations needed to compute the fixpoint $g_A$ and $g_B$ (is one always less than the other, can they be equal, does it depend on set ${\cal P}$ of predicates)
-  * the precision of computed information, that is, comparison of sets $\gamma(g_A(p))$ and $\gamma(g_B(p))$ for an arbitrary program point $p \in V$.