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sav08:homework12 [2008/05/15 10:23]
vkuncak
sav08:homework12 [2015/04/21 17:30] (current)
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Write a regular expression in alphabet $\{x,y,z\} \to \{0,1\}$ denoting relation $z = x + y$ using [[:Regular expressions for automata with parallel inputs]]. ​ Try to make your regular expression as concise and understandable as possible. Write a regular expression in alphabet $\{x,y,z\} \to \{0,1\}$ denoting relation $z = x + y$ using [[:Regular expressions for automata with parallel inputs]]. ​ Try to make your regular expression as concise and understandable as possible.
+

===== Problem 2 ===== ===== Problem 2 =====

Describe the set of all binary relations $r^s_F$ definable through singleton sets Describe the set of all binary relations $r^s_F$ definable through singleton sets
-$+\begin{equation*} r^s_F = \{(p,q) \mid F(\{p\},​\{q\}) \} r^s_F = \{(p,q) \mid F(\{p\},​\{q\}) \} -$+\end{equation*}
where $F$ are formulas of WS1S.  How does this set of $r^s_F$ compare to the set of all binary relations definable in Presburger arithmetic ​ where $F$ are formulas of WS1S.  How does this set of $r^s_F$ compare to the set of all binary relations definable in Presburger arithmetic ​
-$+\begin{equation*} r^p_F = \{ (p,q) \mid G(p,q) \} r^p_F = \{ (p,q) \mid G(p,q) \} -$ +\end{equation*}
-where $G$ is a Presburger arithmetic formula. ​ Are the set of all $r^s_F$ and set of all $r^p_F$ equal, is one strict subset, or are they incomparable?​+where $G$ is a Presburger arithmetic formula. ​ Are the set of all $r^s_F$ and set of all $r^p_F$ equal, is one strict subset ​of the other, or are they incomparable?​

===== Optional Problem 3 ===== ===== Optional Problem 3 =====
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Extend the language of monadic second-order logic over strings with new predicate symbols and describe an algorithm that, given formulas $P(x,y)$ and $Q(y,z)$ in this extension (where $x$,$y$,$z$ are $n$-tuples of set variables) checks whether Extend the language of monadic second-order logic over strings with new predicate symbols and describe an algorithm that, given formulas $P(x,y)$ and $Q(y,z)$ in this extension (where $x$,$y$,$z$ are $n$-tuples of set variables) checks whether
-$+\begin{equation*} \forall x,y,z. (P(x,y) \rightarrow Q(y,z)) \forall x,y,z. (P(x,y) \rightarrow Q(y,z)) -$+\end{equation*}
holds, and, if it holds, finds an interpolant for $P(x,y)$ and $Q(y,z)$. holds, and, if it holds, finds an interpolant for $P(x,y)$ and $Q(y,z)$.

sav08/homework12.txt · Last modified: 2015/04/21 17:30 (external edit)

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