# Recursion

Consider unbounded H-tone spread to penult.

$\begin{array}[t]{lcl} \acute{\sigma}\sigma\sigma & \mapsto & \acute{\sigma}\acute{\sigma}\sigma \\ \sigma\acute{\sigma}\sigma\sigma\sigma\sigma & \mapsto & \sigma\acute{\sigma}\acute{\sigma}\acute{\sigma}\acute{\sigma}\sigma \\ \sigma\sigma\acute{\sigma}\sigma\sigma\sigma & \mapsto & \sigma\sigma\acute{\sigma}\acute{\sigma}\acute{\sigma}\sigma \\ \sigma\acute{\sigma}\sigma\sigma\sigma\sigma\sigma & \mapsto & \sigma\acute{\sigma}\acute{\sigma}\acute{\sigma}\acute{\sigma}\acute{\sigma}\acute{\sigma}\sigma \\ \mathrm{etc.} \end{array}$

This is an output-oriented process. That is, whether a syllable is H-toned depends on the output tonal value of the preceding syllable.

Let $$\acute{\sigma}_i(x)$$ represent the input contrast between H and non-H syllables, and $$\acute{\sigma}_o(x)$$ representing the same in the output, we can do this explicitly in BMRS using a recursive definition of $$\acute{\sigma}_o(x)$$.

The below is a first approximation, although it is a bit wrong.

$\acute{\sigma}_o(x):= \mathtt{if}~\acute{\sigma}_o(p(x))~\mathtt{then}~\top~\mathtt{else}~\acute{\sigma}_i(x)$

This works as shown below:

$\begin{array}[t]{rccccccc} \mathrm{in:} & \sigma & \acute{\sigma} & \sigma & \sigma & \sigma & \sigma \\ \acute{\sigma}_i(x) & \bot & \top & \bot & \bot & \bot & \bot \\ \acute{\sigma}_o(x) & \bot & \top & \top & \top & \top & ^*\top \\ \mathrm{out:} & \sigma & \acute{\sigma} & \acute{\sigma} & \acute{\sigma} & \acute{\sigma} & \acute{\sigma} \\ \end{array}$

The H tone spreads leftward all the way to the final syllable, as indicated by $$^*\top$$. So we preempt this in the case that $$x$$ is the final syllable:

$\acute{\sigma}_o(x):= \begin{array}[t]{l} \mathtt{if}~\mathrm{final}(x)~\mathtt{then}~\bot~\mathtt{else}\\ \mathtt{if}~\acute{\sigma}_o(p(x))~\mathtt{then}~\top~\mathtt{else}\\ \acute{\sigma}_i(x) \end{array}$

In this way, $$\mathrm{final}(x)$$ acts like a higher-ranking constraint, blocking the spread of the H tone only in the final syllable:

$\begin{array}[t]{rccccccc} \mathrm{in:} & \sigma & \acute{\sigma} & \sigma & \sigma & \sigma & \sigma \\ \acute{\sigma}_i(x) & \bot & \top & \bot & \bot & \bot & \bot \\ \acute{\sigma}_o(x) & \bot & \top & \top & \top & \top & \bot \\ \mathrm{out:} & \sigma & \acute{\sigma} & \acute{\sigma} & \acute{\sigma} & \acute{\sigma} & \sigma \\ \end{array}$