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\handout{11}{3}{Homework Assignment 5 \\
Due Thursday, March 28 2013 at 3:30pm}
The answers to the homework assignment should be your own individual work.
Please hand in a hard copy of your solutions in class on the due date.
\begin{enumerate}
\item (20 points)
For each of the following claims, state whether it is true or false.
Give a clear and concise explanation to justify your answer.
\begin{enumerate}
\item Consider a first order-theory $T$ with signature $\Sigma$ and axioms $A$.
Let $F_A$ denote the conjunction of every axiom in $A$, and let $\phi$ be a formula over $\Sigma$. If $\phi$ is satisfiable modulo $T$, then $F_A \rightarrow \phi$ is also satisfiable in standard first-order logic.
\item Consider a first order-theory $T$ with signature $\Sigma$ and axioms $A$.
Let $F_A$ denote the conjunction of every axiom in $A$, and let $\phi$ be a formula over $\Sigma$. If $F_A \rightarrow \phi$ is satisfiable in standard first-order logic, then $\phi$ is satisfiable modulo $T$.
\item Consider a first-order theory $T$ with signature $\{ \}$ and no axioms. The set of sentences that belong to this theory is $\emptyset$.
\item Consider a first order theory $T$ with signature $\{p\}$ and the following axioms:
\[
\begin{array}{rl}
{\rm Axiom \ 1:} & \forall x. \exists y. p(x,y) \\
{\rm Axiom \ 2:} & (\exists x. \exists y. p(x,y)) \rightarrow (\exists x. \forall y. p(x,y)) \\
{\rm Axiom \ 3:} & \forall x. \neg p(x, x)
\end{array}
\]
This theory is complete.
\end{enumerate}
\item (10 points) Decide whether the formula below is valid. If it is valid or unsatisfiable, use the semantic argument method to prove your claim. If the formula is satisfiable but not valid, provide both a falsifying and satisfying structure.
\[
\ard{\awt{a}{i}{e}}{j} = e \ \rightarrow \ i=j \lor a[j] = e
\]
\item (10 points) Apply the congruence closure algorithm to decide the satisfiability of the following $T_=$ formula:
\[
f(g(x)) = g(f(x)) \land f(g(f(y))) = x \land f(y) = x \land g(f(x)) \neq x
\]
You solution should provide detail at the level of the examples in slides 24-31 of lecture 11.
\item (17 points) Solve the following linear program using Simplex:
\[
\begin{array}{ll}
{\rm Maximize:} & x_1 + 3x_2 \\
{\rm Subject \ to:} & \\
& \
\begin{array}{rll}
-x_1 + x_2 & \leq & -1 \\
-2x_1 - 2x_2 & \leq & -6 \\
-x_1 + 4x_2 & \leq & 2 \\
x_1,x_2 & \geq & 0
\end{array}
\end{array}
\]
Show the initial slack form representation and the auxiliary linear program needed to obtain a feasible basic solution. Also, show each step of the Simplex algorithm after performing a pivot operation. Your answer should provide detail at the level of the examples done in class (e.g., slides 37-40 of lecture 12). Use Bland's rule for pivot selection.
\item (8 points) Recall that the Omega Test uses Fourier-Motzkin variable elimination to compute the real shadow.
\begin{enumerate}
\item Use the Fourier-Motzkin technique to eliminate variable $x_1$ from the following system:
\[
\begin{array}{ccc}
-5x_1 - 5x_2 & \leq & -11 \\
2x_1 - 2x_2& \leq & 1 \\
-x_1 + 3x_2 & \leq & 3
\end{array}
\]
\item Using your answer to part (a), what does the Omega test conclude about the existence of an integer solution to this system based on the real shadow?
\end{enumerate}
\end{enumerate}
\end{document}