LR(1) sets of items for $S\to CC$, $C\to cC\mid d$

Augmented grammar: $S'\to S$.

Items are written $[A\to\alpha\cdot\beta,\ a]$ where a is the lookahead terminal.

Closure hint: from $[S\to \cdot CC,\ $]$, the symbol after the dot is$C$and the lookahead context of$C\to\cdots$is$\mathrm{FIRST}(C$)={c,d}$, so inner$C$-items receive lookaheads${c,d}$.

Canonical LR(1) collection:

• $I_0$: $[S'\to\cdot S,$],\ [S\to\cdot CC,$],\ [C\to\cdot cC,\ c/d],\ [C\to\cdot d,\ c/d]$
• $I_1$ = goto$(I_0,S)$: $[S'\to S\cdot,$]$
• $I_2$ = goto$(I_0,C)$: $[S\to C\cdot C,$],\ [C\to\cdot cC,$],\ [C\to\cdot d,$]$
• $I_3$ = goto$(I_0,c)$: $[C\to c\cdot C,\ c/d],\ [C\to\cdot cC,\ c/d],\ [C\to\cdot d,\ c/d]$
• $I_4$ = goto$(I_0,d)$: $[C\to d\cdot,\ c/d]$
• $I_5$ = goto$(I_2,C)$: $[S\to CC\cdot,$]$
• $I_6$ = goto$(I_2,c)$: $[C\to c\cdot C,$],\ [C\to\cdot cC,$],\ [C\to\cdot d,$]$
• $I_7$ = goto$(I_2,d)$: $[C\to d\cdot,$]$
• $I_8$ = goto$(I_3,C)$: $[C\to cC\cdot,\ c/d]$
• $I_9$ = goto$(I_6,C)$: $[C\to cC\cdot,$]$

GOTO function:

• goto$(I_0,S)$= $I_1$; goto$(I_0,C)$=$I_2$; goto$(I_0,c)$=$I_3$; goto$(I_0,d)$=$I_4$
• goto$(I_2,C)$=$I_5$; goto$(I_2,c)$=$I_6$; goto$(I_2,d)$=$I_7$
• goto$(I_3,C)$=$I_8$; goto$(I_3,c)$=$I_3$; goto$(I_3,d)$=$I_4$
• goto$(I_6,C)$=$I_9$; goto$(I_6,c)$=$I_6$; goto$(I_6,d)$=$I_7$

So the canonical LR(1) machine has 10 states ($I_0$–$I_9$).

Reduce actions: $I_4$: $C\to d$ on $\{c,d\}$; $I_7$: $C\to d$ on $$$;$I_8$:$C\to cC$on${c,d}$;$I_9$:$C\to cC$on `$; $I_5$: $S\to CC$ on $`;$I_1$: accept on `$`.

Bonus — LALR(1): merging states with equal cores ($I_4\cup I_7$, $I_3\cup I_6$, $I_8\cup I_9$) gives the 7-state LALR(1) automaton, free of conflicts for this grammar.