\documentclass{rtaloop}
\rtalabel{left-linear}

\begin{document}
\begin{solvedproblem}{Vincent van Oostrom}{}{June 1993}

\begin{abstract}
Is the union of two left-linear, confluent combinatory reduction systems over
the same alphabet, where the rules of the first system do not overlap the
rules of the second, confluent?
\end{abstract}

Let $R$ and $S$ be two left-linear, confluent combinatory reduction systems
with the same alphabet.  Suppose the rules of $R$ do not overlap the rules of
$S$.  Is $R \cup S$ confluent?  This is true for the restricted case when $R$
is a term-rewriting system (an easy generalization of a result by F. M\"uller
\cite{Mull:92}), or if neither system has critical pairs.  (The
restriction to 
the same alphabet is essential, since confluence is in general not preserved
under the addition of function symbols, not even for left-linear systems.)

\begin{remark}
The answer is yes (Thm. 3.1 of \cite{Oost:Raam:94}).
\end{remark}

\end{solvedproblem}
\end{document}
