Trivium

December 2008

12dec2008

Remember the nice outline calculus I presented recently? I did a fatal mistake there: The outline normal form is not confluent, here is a counterexample:

\begin{align}
(((x))(y)) &\xrightarrow{()()} ((x)y)\\
(((x))(y)) &\xrightarrow{(())} ((x)(y)) \xrightarrow{()()} (x\ y)
\end{align}

Thus, order of rule application matters.

Which kind of destroys all usefulness… cause then, fully-parenthesized form is not a valid transformation.

I discovered this issue when I wanted to find a nicer way to write the x.0y items, e.g. the preface to a chapter:

(Chapter 1
((Preface)
Section 1
(...))
(Section 2 (...))
)

It is misleading that the Preface is in the same parenthesis as Section 1, therefore I applied a variant of FPF and wrote:

Chapter 1
((Preface))
(Section 1)
((...))
(Section 2)
((...))

And then it’s easy to see.

It’s always sad to see a nice thing disappear (especially if you found it). I wonder how I could fix this? (Removing the (())-rule would be an option, but make everything much harder to deal with.)

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