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Suppose that we have a grammar with the following rules:

$$S \to Aa\ |\ b\ |\ \varepsilon\\ A \to Ad\ |\ c$$

From looking at it I already know that $First(S) = \{b, \varepsilon, c\}$. My question is:

  • How to determine $First(A)$?

For instance, by looking at the rule $S \to Aa$ we know that $First(S)$ must include $First(A)$. $First(A)$ already includes $c$ (that's why $c \in First(S)$). but how to handle the rule $A \to Ad$.

Just by applying the algorithm for finding the $First$ set blindly, I will have that $First(A)$ "includes" $First(A)$, which confuses me.

As far as I understand in $A \to Ad\ |\ c$ there is a left recursion. By eliminating it we get the following rules

$A \to cA'\\ A' \to dA'$

Does this mean that $First(A)$ only contains $c$?

Does this mean that I can ignore a rule with left recursion and just look at the other (non-recursive) rules under that variable?

Thanks!

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  • $\begingroup$ Why do you find it confusing that $First(A)$ includes $First(A)$? :-) It seems tautological. However, it doesn't add any information. Since you (presumably) know that $A$ is not nullable, just accept the tautology and move on. $\endgroup$ – rici Dec 6 '18 at 21:36
  • $\begingroup$ (And to be clear, you can only ignore this left-recursive rule because $A$ is not nullable. If it were nullable, you would need to proceed with the rest of the RHS, as with any other nullable non-terminal.) $\endgroup$ – rici Dec 6 '18 at 21:37
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The grammar you have presented is left linear grammar. The production $A \rightarrow Ad$ in when transformed into $FM$, $A$ becomes state & an edge with $c$ terminal from $A$ to $A$ (self loop). So its clear for this particular grammar $First(A)$ is $c$.

Moreover, your approach is also correct or primarily correct.

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