Question

I have three grammars:

A -> aB | b | CBB

B -> aB | ba | aBb

C -> aaA | b | caB

I need to “determine whether [they] are LL grammars by performing the pairwise disjoint test, showing the first sets of each RHS of each nonterminal.

This is what I have so far…

A -> aB | b | CBB

first(aB) = a

first(b) = b

first(CBB) = aaA = a

This is the one I’m having trouble with. Did I do CBB correctly? If so I would say that they intersect & the rule fails the test. (right?)

B -> aB | ba | aBb

first(aB) = a

first(ba) = b

first(aBb) = a

They are intersected & thus the rule fails the test.

C -> aaA | b | caB

first(aaA) = a

first(b) = b

first(caB) = c

They are not intersected & thus the rule passes

Answer 1

The point of the test is to see if, looking at the first terminal, you can tell which rule to use (a requirement for LL). Its pretty obvious for B that there are 2 rules that could apply for the terminal a; its also pretty obvious the each rule for C starts with a different terminal. And you can see that the possible first terminals for C (and hence CBB) overlaps for the other rules for A.

Bottom line: looks good (although, if you had stopped at a single terminal for CBB and happened to choose c, you would have come to the wrong conclusion).