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\markright{\footnotesize CS 451 Programming Paradigms, Spring 2001}

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\section*{Homework set 10: $\lambda$-calculus --- due Wednesday 7 March}

Total number of points available on this homework is 100. Full credit
is equivalent to 100 points.

\begin{enumerate}





\item (30 pts.)     (Turing) 
Let $A \triangleq \lambda xy. y (xxy)$. 
Let $\Theta \triangleq AA$.
Show that $\Theta$ is a fixed-point operator.





\item (70 pts.)      (Klop) 
Let:
\begin{quote}
$\pounds = \lambda abcdefghijklmnopqstuvwxyzr. \: r (thisisafixedpointcombinator)$ \\
$\$ = \pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds\pounds$
\end{quote}
Show that $\$$ is a fixed-point combinator.




\end{enumerate}


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