在递归下降解析器中计算语法产生的预测集的方法

Question

I understand first and follow but I am totally lost on the predict sets. can someone explain to me how to go about finding a predict set of a production in a grammar using the first and follow sets? I have not provided a grammar because this is for a homework assignment and I want to know how to do it not how to do it for this specific grammar.

Answer 1

Intuitively, the predict set for a production A → α [Note 1] is the set of terminal symbols which might be the next symbol to be read if that production is to be predicted. (That implies that the production's non-terminal ( A ) has already been predicted, and the parser must now decide which of the non-terminal's productions to predict.)

Obviously, that includes all the terminal symbols which might be the first symbol of the right-hand side. But what if the right-hand side might derive ε, the empty string? In that case, the next symbol in the input will be the first symbol which comes after the predicted non-terminal, A ; in other words, it will be a member of FOLLOW(A) . So the predict set contains the terminals which might start the right-hand side α , plus all the symbols in FOLLOW(A) if α could derive the empty string. [Note 2]

More formally, PREDICT(A → α) is:

• FIRST(α) if ε ∉ FIRST(α)
• (FIRST(α) ∪ FOLLOW(A)) - {ε} if ε ∈ FIRST(α)

Remember that we compute FIRST on a sentential form by "looking through" epsilons:

FIRST(aβ) is

• FIRST(a) if ε ∉ FIRST(a)

• (FIRST(a) - {ε}) ∪ FIRST(β) if ε ∈ FIRST(a)

Consequently, FIRST of a right hand side only include ε if every symbol in the right-hand side is nullable.

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Notes:

1. I use the common convention that capital letters ( A ...) refer to non-terminals, lower-case letters ( a ...) refer to grammar symbols (terminals or non-terminals) and Greek letters ( α ...) refer to possibly empty sequences of grammar symbols.

2. Aside from the first step when the start symbol is predicted, the current prediction always contains more than one symbol. So if A is the next non-terminal to expand and we see that it is nullable (ie, it could derive nothing), we don't really need to lookup FOLLOW(A) because we could just look at the predict stack and see what we've predicted will follow A . In some cases, this might allow us to avoid a conflict with one of the other alternatives for A .

   However, it is normal to use FOLLOW(A) , regardless. Always using FOLLOW(A) is usually referred to as the "Strong LL" (SLL) algorithm. Although it seems like computing the FIRST set of the known prediction stack is more powerful than using a precomputed FOLLOW set, it does not actually improve the power of LL parsing at all; every non-LL grammar can be converted to an SLL grammar.