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How to describe a $\operatorname{LL(1)}$ parsing algorithm for strings generated by a given grammar?

I have to design a parser for a specific grammar. Let $G$ be the grammar described as:

$$S \rightarrow aBC $$ $$B \rightarrow bB \mid cC $$ $$C \rightarrow c \mid S $$

My approach

1. Test if the grammar is $\operatorname{LL(1)}$

First I have to determine if the grammar is $\operatorname{LL(1)}$ or not. It seems that G is $\operatorname{LL(1)}$ because:

$\operatorname{FIRST}(aBC) \cap \operatorname{FIRST}(bB \mid cC) \cap \operatorname{FIRST}(c \mid S) = \{a\} \cap \{b,c\} \cap \{c,a\} = \emptyset $

2. Write a string generated by G

$s = abccaccc$

3. Design the algorithm for parsing strings

I'm getting stuck here

4. Test the algorithm for a given string

Lets test the algorithm for a given string $s = abccaccc$

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  • $\begingroup$ What conflict do you see? Did you compute first and follow for the rules? $\endgroup$ – Hendrik Jan Nov 6 '17 at 17:55
  • $\begingroup$ @HendrikJan First(B) = {b,c} ; First(C) = {c, a} . So First(B) $\cap$ First(C) = {c} then G is not LL(1). I don't have to compute follow because i don't see epsilon $\endgroup$ – Jack Nov 6 '17 at 18:00
  • $\begingroup$ @Jack: And in what context exactly might you have to choose between predicting a B and predicting a C? $\endgroup$ – rici Nov 6 '17 at 20:29
  • $\begingroup$ Instead you must distinguish the first of "$bB$" vs. the first of "$cC$" to see whether the two productions for $B$ are used in different situations! $\endgroup$ – Hendrik Jan Nov 6 '17 at 21:26
  • $\begingroup$ A language is a set of strings: it does not generate them, but a grammar does. Why jump on LL(1) parsing rather than some other technique: if it is a requirement of the problem, it should be stated in the question, not in the answer. $\endgroup$ – babou Nov 6 '17 at 22:23
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Here's a possibly useful quote from Aho, Lam et al ("the Dragon Book"). (Emphasis added.)

A grammar $G$ is $LL(1)$ if and only if whenever $a \to \alpha | \beta$ are two distinct productions of $G$, the following conditions hold:

  1. For no terminal $a$ do both $\alpha$ and $\beta$ derive strings beginning with $a$.

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  • $\begingroup$ thanks! But do you know how to design the algorithm? $\endgroup$ – Jack Nov 7 '17 at 12:40
  • $\begingroup$ See that book or some other textbook on compiler construction. Better still, use a parser generator (and you won't be restricted to LL(1) in most cases). Your question is to broad for this site, I think. $\endgroup$ – reinierpost Feb 5 '18 at 21:25

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