# The problem

## Background in Iny

Iny (Ribeiro, 2002 2012) has regressive ATR harmony in which both input and output ATR values of triggers are important. The surface vowel inventory is as follows.

[+ATR]a ã ə̃ (ə) ɔ ɛ ʊ ɨ ɪ
[-ATR]ə o õ e u i̘ i ĩ

The following forms illustrate harmony occuring.

/r-ɛ-rɔ=r-e/[rerore]
/b-∅-r-krɔ=kre/[bikrokre]
/r-ɛ-hãɗɛ=r-e/[rɛhãɗere]

The following forms harmonize as well, but in a different way. What is the difference? Why?

/brɔrɛ-dĩ/[broreni]
/wa-θɛ-rit͡ʃɔrɛ/[waθerit͡ʃɔrɛ]
/kɔɗʊ-dĩ/[kɔɗuni]
/r-ɛ-hI=r-e[rɛhire]

The following definitions and chart may help you sort things out. Using these analytical categories, think about how high, mid, and low vowels behave with respect to ATR harmony.

• Triggers: for some vowel $$V$$ at index $$x$$, $$V_x$$, if underlying [+ATR] on $$V_x$$ results in $$V_{x-1}$$ becoming [+ATR], then $$V_x$$ triggers [+ATR] harmony.
• Undergoes: for some vowel $$V$$ at index $$y$$, $$V_y$$, if [+ATR] (underlying or derived) on $$V_{y+1}$$ results in $$V_y$$ becoming [+ATR], then $$V_y$$ undergoes [+ATR] harmony.
• Propagates: for some vowel $$V$$ at index $$V_z$$, if [+ATR] (underlying or derived) on $$V_{z+1}$$ results in $$V_z$$ and $$V_{z-1}$$ becoming [+ATR], then $$V_z$$ propagates [+ATR].

## Analysis

There are two natural classes we will want to refer to. Let's define those first.

• $$[+\mathrm{ATR},+\mathrm{hi}]_i(x)=$$
• $$[+\mathrm{ATR},-\mathrm{hi},-\mathrm{lo},-\mathrm{nas}]_o(x)=$$

Now we can define a BMRS for Iny. Assume the following output features. Hint: How are you going to capture the directionality?

1. $$[\mathrm{high}]_o(x)=$$
2. $$[\mathrm{low}]_o(x)=$$
3. $$[\mathrm{nasal}]_o(x)=$$
4. $$[\mathrm{ATR}]_o(x)=$$