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sav08:homework11 [2008/05/08 18:54]
vkuncak
sav08:homework11 [2015/04/21 17:30] (current)
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===== Problem 2 ===== ===== 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'​.  ​+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, ​quantifier-free formulas in the 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  **Part a)** Consider [[Conjunctions of Predicates]] as abstract interpretation domain. ​ Give example showing that it need not be the case that
-$+\begin{equation*} a_1 \leq a_2 \leftrightarrow \gamma(a_1) \subseteq \gamma(a_2) a_1 \leq a_2 \leftrightarrow \gamma(a_1) \subseteq \gamma(a_2) -$+\end{equation*}

**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 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?