Making a Grammar LL(1)

Question

I have the following grammar:

S → a S b S | b S a S | ε

Since I'm trying to write a small compiler for it, I'd like to make it LL(1). I see that there seems to be a FIRST/FOLLOW conflict here, and I know I have to use substitution to resolve it, but I'm not exactly sure how to go about it. Here is my proposed grammar, but I'm not sure if it's correct:

S-> aSbT | epsilon

T-> bFaF| epsilon

F-> epsilon

Can someone help out?

Answer 1

In his original paper on LR parsing, Knuth gives the following grammar for this language, which he conjectures "is the briefest possible unambiguous grammar for this language:"

S → ε | aAbS | bBaS

A → ε | aAbA

B → ε | bBaB

Intuitively, this tries to break up any string of As and Bs into blocks that balance out completely. Some blocks start with a and end with b, while others start with b and end with a.

We can compute FIRST and FOLLOW sets as follows:

FIRST(S) = { ε, a, b }

FIRST(A) = { ε, a }

FIRST(B) = { ε, b }

FOLLOW(S) = { $ }

FOLLOW(A) = { b }

FOLLOW(B) = { a }

Based on this, we get the following LL(1) parse table:

   |   a   |   b   |   $   
 --+-------+-------+-------
 S | aAbS  | bBaS  |  e
 A | aAbA  |   e   |
 B |   e   | bBaB  |

And so this grammar is not only LR(1), but it's LL(1) as well.

Hope this helps!