Question

FIRST(A)  = { b, epsilon  }
FIRST(S) = { b, epsilon  }

FOLLOW(S)  = { a, $ }
FOLLOW(A) = { a, b, $ }

What is the Arithmetic Expressions for this First & Follow set?

Answer 1

FIRST(X) = the terminals which can appear first when trying parse the non-terminal X. If it can match an empty string, epsilon is also included.

FOLLOW(X) = the terminals which can appear immediately after the non-terminal X. This is a union of the FIRST-sets of all symbols appearing after X in any parsing rule.

Read more: LL parser

The clues given are:

1. FIRST(A), FIRST(S) ⇒ All of the derivations of A and S respectively, must either begin with the terminal b, or be zero-length.

   S → b … | ε
   A → b … | ε

2. FOLLOW(S) ⇒ There must be some construction where S is followed by the terminal a, or a non-terminal which can begin with a. (Neither A nor S qualify).

   S → b S a | ε
   A → b … | ε

3. FOLLOW(A) ⇒ There must be some construction where A is followed by each of the terminals a and b, or some non-terminal which can begin with those.

   S → b S a | ε
   A → b A b | b A a | ε

4. FOLLOW(A) ⇒ Assuming S is the start-symbol, A must appear at the end of some branch of S, possibly followed by other nullable non-terminals.

   S → b S a | A | ε
   A → b A b | b A a | ε

   (NB. Adding A to S did not break the constraint on FIRST(S))

We can make the grammar a little smaller:

S → b S a | A | ε
A → b A b | ε

We can no longer generate strings like “bbbabb“, but it does not violate the constraints.