Question 41

In an $LR(0)$ parser, a state with a reduce item $X \rightarrow \alpha \cdot$ places a reduce action in every terminal column of the ACTION table (or we count each conflicting (state, terminal) entry or state-level shift-reduce conflict):

1. Initial state $I_0$: Closure of $[S' \rightarrow \cdot S]$ contains $[S \rightarrow \cdot ACB]$, $[A \rightarrow \cdot aA]$, and the reduce item $[A \rightarrow \cdot]$ (from $\epsilon$-production), with shift on $a$ ($I_a = \text{GOTO}(I_0, a)$ contains identical closure items $[A \rightarrow a \cdot A]$, $[A \rightarrow \cdot aA]$, $[A \rightarrow \cdot]$).
2. State $I_A = \text{GOTO}(I_0, A)$: Contains $[S \rightarrow A \cdot CB]$, $[C \rightarrow \cdot cC]$, and the reduce item $[C \rightarrow \cdot]$, with shift on $c$.
3. State $I_c = \text{GOTO}(I_A, c)$: Contains $[C \rightarrow c \cdot C]$, $[C \rightarrow \cdot cC]$, and the reduce item $[C \rightarrow \cdot]$, with shift on $c$.
4. State $I_b = \text{GOTO}(I_{AC}, b)$: Contains $[B \rightarrow b \cdot B]$, $[B \rightarrow b \cdot]$, $[B \rightarrow \cdot bB]$, $[B \rightarrow \cdot b]$, which has reduce item $[B \rightarrow b \cdot]$ and shift on $b$.

These 5 states ($I_0$, $I_a$, $I_A$, $I_c$, and $I_b$) each contain both a shift item on a terminal ($a, a, c, c, b$ respectively) and a completed reduce item ($A \rightarrow \cdot$ in $I_0, I_a$; $C \rightarrow \cdot$ in $I_A, I_c$; $B \rightarrow b \cdot$ in $I_b$), producing exactly 5 shift-reduce conflicts in the $LR(0)$ ACTION table.