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\PaperNumber{60}
\Title{ Discrete Normalization and Standardization in
                        Deterministic Residual Structures}
\Authors{Z. Khasidashvili and J. Glauert         }
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\begin{Justification}

Deterministic residual structures were introduced by the authors in a
previous paper, and provide an abstract account of rewriting, along
similar lines to the Abstract Reduction Systems of Gonthier et al,
with in addition an axiomatised account  of residuals. Under mild
conditions, this paper develops a theory of neededness for these
reduction systems relative to a particular reduction $P: t
\rightarrow s$. They prove two main results:
\begin{enumerate}

\item a standardisation theorem for finite reductions---they also show
that this result does not extend to infinitary reductions;

\item a so-called discrete normalisation theorem.
\end{enumerate}

This paper is unfortunately extremely difficult to read; even the
introduction is not easy to decipher. The results are probably worth
publishing at some point, although it is difficult to assess the
statement of the discrete normalisation theorem without considerable
effort to understand the technical definitions. I recommend that the
paper is rejected at this stage.

\end{Justification}
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\begin{Recommendation}

An abstract account of reduction systems and residuals seems to be a
good idea, although I would have liked to have seen some examples to
motivate this level of abstraction and to illustate the differences
with other definitions of reduction systems.  It is not at all obvious
that the technical details need to be quite so complicated. Much more
intuition is required: for example, I'm sure that the introduction can
be written with much less notation.  Also, the paper should be
self-contained: in particular, more details regarding residuals should
be given.

Paul Mellies of Edinburgh University has some results in his thesis
which may relate to this work.

\end{Recommendation}
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