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msol_over_strings [2007/05/11 16:58] ghid.maatouk |
msol_over_strings [2007/05/27 12:52] vaibhav.rajan |
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====== MSOL over Strings ====== | ====== MSOL over Strings ====== | ||
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[\![F_1 \lor F_2]\!]e &=& [\![F_1]\!]e\ \lor\ [\![F_2]\!]e \\ \ | [\![F_1 \lor F_2]\!]e &=& [\![F_1]\!]e\ \lor\ [\![F_2]\!]e \\ \ | ||
[\![\lnot F]\!]e &=& \lnot ([\![F]\!]e) \\ \ | [\![\lnot F]\!]e &=& \lnot ([\![F]\!]e) \\ \ | ||
- | [\![\exists v. F]\!]e &=& \exists S. S\ \mbox{is finite}\ \land\ S \subseteq N_0. [\![F]\!](e[v \mapsto S]) | + | [\![\exists v. F]\!]e &=& \exists S. S\ \mbox{is finite}\ \land\ S \subseteq N_0 \land [\![F]\!](e[v \mapsto S]) |
\end{eqnarray*} | \end{eqnarray*} | ||
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* Intersection: $(A = B \cap C) = (A \subseteq B \land A \subseteq C) \land (\forall A_1. A_1 \subseteq B \land A_1 \subseteq C \rightarrow A_1 \subseteq A)$ | * Intersection: $(A = B \cap C) = (A \subseteq B \land A \subseteq C) \land (\forall A_1. A_1 \subseteq B \land A_1 \subseteq C \rightarrow A_1 \subseteq A)$ | ||
* Union: $(A = B \cup C) = (B \subseteq A \land C \subseteq A) \land (\forall A_1. B \subseteq A_1 \land C \subseteq A_1 \rightarrow A \subseteq A_1)$ | * Union: $(A = B \cup C) = (B \subseteq A \land C \subseteq A) \land (\forall A_1. B \subseteq A_1 \land C \subseteq A_1 \rightarrow A \subseteq A_1)$ | ||
- | * Set difference: $(A = B \setminus C) = (A \cup C = B \land A \cap C = \emptyset)$ | + | * Set difference: $(A = B \setminus C) = (A \cup (B \cap C) = B \land A \cap C = \emptyset)$ |
(or just use element-wise definitions with singletons) | (or just use element-wise definitions with singletons) | ||
* If $k$ is a fixed constant, properties $\mbox{card}(A) \geq k$, $\mbox{card}(A)\leq k$, $\mbox{card}(A)=k$ | * If $k$ is a fixed constant, properties $\mbox{card}(A) \geq k$, $\mbox{card}(A)\leq k$, $\mbox{card}(A)=k$ |