START $->$ sym START

START $->$ sym

START $->$ sibilant IH$\_$Z

START $->$ voiced$\_$nonsibilant Z

START $->$ voiceless$\_$nonsibilant S

S $->$ sibilant IH$\_$Z

S $->$ voiced$\_$nonsibilant Z

S $->$ voiceless$\_$nonsibilant S$1$

IH$\_$Z $->\text{'}+\text{'}\text{'}$S$\text{'}$

Z $->\text{'}+\text{'}\text{'}$S$\text{'}$

S$1->\text{'}+\text{'}\text{'}$S$\text{'}$

sibilant $->\text{'}$CH$\text{'}$

sibilant $->\text{'}$JH$\text{'}$

sibilant $->\text{'}$S$\text{'}$

$...$

voiced$\_$nonsibilant $->\text{'}$AA$\text{'}$

voiced$\_$nonsibilant $->\text{'}$AE$\text{'}$

$...$

voiced$\_$nonsibilant $->\text{'}$B$\text{'}$

voiced$\_$nonsibilant $->\text{'}$D$\text{'}$

$...$

voiceless$\_$nonsibilant $->\text{'}$P$\text{'}$

$...$

sym $->\text{'}$AA$\text{'}$

$...$

sym $->\text{'}$Z$\text{'}$

If we wanted to implement this grammar using our python format, we would have a lot of rules with the form

preterminal $->$ 'terminal$1\text{'}$

preterminal $->$ 'terminal$2\text{'}$

It turns out there is a convenient way to write these rules:

preterminal $->$ 'terminal$1\text{'}|$ 'terminal$2\text{'}$

For example, we can define all our sibilants like this:

sibilant $->\text{'}$CH$\text{'}|\text{'}$JH$\text{'}|\text{'}$S$\text{'}|\text{'}$SH$\text{'}|\text{'}$Z$\text{'}|\text{'}$ZH$\text{'}$

When parsing a string like

D IH SH $+$ S

the symbols $"+$ S$"$ will form a phrase under the non$-$terminal IH$\_$Z$,$ and that's how we would know that the chosen allomorph is IH Z$.$ The tree would look like this:

$($S $($sym D$)($S $($sym IH$)($S $($sibilant SH$)($IH$\_$Z $+$ S$))))$

First, try to understand the example above $($you don't need to implement it$),$ then attempt to write the grammar to solve the problem below.

Aspiration Revisited

Consider the aspiration rule from HW $2$ part $2$: Voiceless stops are aspirated when they occur immediately before a stressed vowel, and there is no $[$s$]$ immediately preceding the voiceless stop.

Write a context$-$free grammar that will parse the same input strings from HW $2$ part $2,$ so that each consonant that should be aspirated has the pre$-$terminal "aspirated" as its parent. Implement the grammar as a CFG in the format used in HW $4.$ As a starting point, consider the following grammar:

g $=$ cfg$("""$

S $->$ sym S

S $->$ sym

sym $->\text{'}$AA$\text{'}|\text{'}$AE$\text{'}|\text{'}$AH$\text{'}|\text{'}$AO$\text{'}|\text{'}$AW$\text{'}|\text{'}$AY$\text{'}|\text{'}$B$\text{'}|\text{'}$CH$\text{'}|\text{'}$D$\text{'}|\text{'}$DH$\text{'}|\text{'}$EH$\text{'}|\text{'}$ER$\text{'}|\text{'}$EY$\text{'}|\text{'}$F$\text{'}|\text{'}$G$\text{'}|\text{'}$HH$\text{'}|\text{'}$IH$\text{'}|\text{'}$IY$\text{'}|\text{'}$JH$\text{'}|\text{'}$K$\text{'}|\text{'}$L$\text{'}|\text{'}$M$\text{'}|\text{'}$N$\text{'}|\text{'}$NG$\text{'}|\text{'}$OW$\text{'}|\text{'}$OY$\text{'}|\text{'}$P$\text{'}|\text{'}$R$\text{'}|\text{'}$S$\text{'}|\text{'}$SH$\text{'}|\text{'}$T$\text{'}|\text{'}$TH$\text{'}|\text{'}$UH$\text{'}|\text{'}$UW$\text{'}|\text{'}$V$\text{'}|\text{'}$W$\text{'}|\text{'}$Y$\text{'}|\text{'}$Z$\text{'}|\text{'}$ZH$\text{'}|\text{'}$AA$1\text{'}|\text{'}$AE$1\text{'}|\text{'}$AH$1\text{'}|\text{'}$AO$1\text{'}|\text{'}$AW$1\text{'}|\text{'}$AY$1\text{'}|\text{'}$EH$1\text{'}|\text{'}$ER$1\text{'}|\text{'}$EY$1\text{'}|\text{'}$IH$1\text{'}|\text{'}$IY$1\text{'}|\text{'}$OW$1\text{'}|\text{'}$OY$1\text{'}|\text{'}$UH$1\text{'}|\text{'}$UW$1\text{'}$

$""")$

In the last line of the grammar, we used a shorthand notation to create lexical rules for all the symbols we will need. In this grammar, every symbol has the pre$-$terminal symbol sym$.$ Modify it to model aspiration according to the description above, and enter your resulting grammar in your submission.

If it helps, strings may be ambiguous according to your grammar. However, only aspirated consonants should ever have the "aspirated" non$-$terminal. If a consonant should be aspirated, the string may have multiple trees, as long as the aspirated consonant has the pre$-$terminal "aspirated" in at least one.