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sav08:homework12 [2008/05/15 10:24]
vkuncak
sav08:homework12 [2015/04/21 17:30]
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-====== Homework 12, Due May 21 ====== 
- 
-===== Problem 1 ===== 
- 
-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 ===== 
- 
-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 of $r^s_F$ compare to the set of all binary relations definable in Presburger arithmetic ​ 
-\[ 
-   r^p_F = \{ (p,q) \mid G(p,q) \} 
-\] 
-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]]. 
- 
-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 
-\[ 
-    \forall x,y,z. (P(x,y) \rightarrow Q(y,z)) 
-\] 
-holds, and, if it holds, finds an interpolant for $P(x,y)$ and $Q(y,z)$.