\documentclass{rtaloop}
\rtalabel{infinite-parallel}


\begin{document}
\begin{solvedproblem}{Jan Willem Klop}{}{December 1991}

\begin{abstract}
  Prove a \emph{Parallel Moves Lemma} for reductions of infinite length.
\end{abstract}

For reductions of transfinite length, a version of the Parallel Moves Lemma
can be proved if one considers only ``strongly converging'' infinite reductions
in the sense of \cite{KKSV91:rta}. However, if one wants to consider
converging reductions, as in \cite{DKP91:tcs}, then it is not difficult to
construct a counterexample, not to the infinite Parallel Moves Lemma itself,
but to the method of proof (cf. \cite{KKSV90:cwi}). An infinite Parallel Moves
Lemma might involve a different notion of ``descendant''.

\begin{remark}
  \cite{simonsen04ipl} shows that it is not possible to obtain a Parallel
  Moves Lemma for (Cauchy-)convergent infinite reductions which relies on a
  notion of \emph{residual} maintaining some of the basic properties of
  residuals known from the finite case. His counterexamples, however, are
  somewhat particular in that the right-hand sides of the rewrite rule are not
  normalized. The question remains whether it is possible to salvage a
  Parallel Moves Lemma for (Cauchy-)convergent reductions for restricted
  classes of rewrite systems.
\end{remark}

\end{solvedproblem}
\end{document}