Predictive (LL(1)) parsing table

Grammar:
$$S \to AaAb \mid BbBa\\ A \to \epsilon\\ B \to \epsilon$$

Step 1 — FIRST and FOLLOW

$\mathrm{FIRST}(A) = \{\epsilon\}$, $\mathrm{FIRST}(B) = \{\epsilon\}$.

For $S\to AAAb$ : because $A\to\epsilon$, the first of the right-hand string is a. For $S\to BbBa$: the first is b. Hence:
$$\mathrm{FIRST}(S) = \{a, b\}$$

FOLLOW:

• $S$ is start → $\mathrm{FOLLOW}(S) = {$ }$.
• From $S\to AaAb$: the first $A$ is followed by a, the second $A$ by b.
  $$\mathrm{FOLLOW}(A) = \{a, b\}$$
• From $S\to BbBa$: first $B$ followed by b, second $B$ by a.
  $$\mathrm{FOLLOW}(B) = \{a, b\}$$

Step 2 — build table $M[X][\text{terminal}]$

For each production $X\to\alpha$, put $X\to\alpha$ into $M[X][t]$ for every $t\in \mathrm{FIRST}(\alpha)$; if $\alpha \Rightarrow \epsilon$, also put it into $M[X][t]$ for every $t\in\mathrm{FOLLOW}(X)$.

• For $S \to AaAb$: $\mathrm{FIRST}(AaAb)=\{a\}$ → entry $M[S][a] = S\to AaAb$.
• For $S \to BbBa$: $\mathrm{FIRST}=\{b\}$ → $M[S][b] = S\to BbBa$.
• For $A\to\epsilon$: place under $\mathrm{FOLLOW}(A)=\{a,b\}$ → $M[A][a]=A\to\epsilon$, $M[A][b]=A\to\epsilon$.
• For $B\to\epsilon$: $\mathrm{FOLLOW}(B)=\{a,b\}$ → $M[B][a]=B\to\epsilon$, $M[B][b]=B\to\epsilon$.

Predictive parsing table (blank cells = error, no production):

Nonterminal	a	b	$
$S$	$S\to AaAb$	$S\to BbBa$	—
$A$	$A\to\epsilon$	$A\to\epsilon$	—
$B$	$B\to\epsilon$	$B\to\epsilon$	—

Verification: Every cell has at most one entry, so the grammar is LL(1) — no conflicts. This is the classic example of an LL(1) grammar with $A\to\epsilon,\ B\to\epsilon$ that resolves the choice using lookahead.