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 sav08:homework12 [2008/05/15 10:08]vkuncak sav08:homework12 [2015/04/21 17:30] (current) Both sides previous revision Previous revision 2008/05/15 10:24 vkuncak 2008/05/15 10:23 vkuncak 2008/05/15 10:08 vkuncak 2008/05/15 10:06 vkuncak 2008/05/15 10:05 vkuncak 2008/05/14 20:08 vkuncak 2008/05/14 20:03 vkuncak 2008/05/14 19:59 vkuncak 2008/05/14 19:59 vkuncak 2008/05/14 19:58 vkuncak created Next revision Previous revision 2008/05/15 10:24 vkuncak 2008/05/15 10:23 vkuncak 2008/05/15 10:08 vkuncak 2008/05/15 10:06 vkuncak 2008/05/15 10:05 vkuncak 2008/05/14 20:08 vkuncak 2008/05/14 20:03 vkuncak 2008/05/14 19:59 vkuncak 2008/05/14 19:59 vkuncak 2008/05/14 19:58 vkuncak created Line 4: Line 4: 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 + \begin{equation*} + 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 ​ + \begin{equation*} + 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 of the other, or are they incomparable?​ + + ===== Optional Problem 3 ===== In [[Quantifier elimination definition]] we noted that if the validity of first-order formulas in some theory is decidable, then we can extend the language of formulas so that we have quantifier elimination. ​ Previously we also observed that quantifier elimination implies the existence of interpolants (see the definition of [[interpolation for propositional logic]] as well as the [[Calculus of Computation Textbook]]). ​ We next apply these observations to weak monadic second-order logic over strings described in [[Lecture23]]. In [[Quantifier elimination definition]] we noted that if the validity of first-order formulas in some theory is decidable, then we can extend the language of formulas so that we have quantifier elimination. ​ Previously we also observed that quantifier elimination implies the existence of interpolants (see the definition of [[interpolation for propositional logic]] as well as the [[Calculus of Computation Textbook]]). ​ We next apply these observations to weak monadic second-order logic over strings described in [[Lecture23]]. 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 $Q(y,z)$. - - ===== Optional Problem 3 ===== - - Describe the set of all binary relations $r^s_F$ definable through singleton sets - $- r^s_F = \{(p,q) \mid F(\{p\},​\{q\}) \} -$ - where $F$ are formulas of WS1S.  How does this set compare to the set of all binary relations definable in Presburger arithmetic?