Shortest word possible

Question

In a give context-free grammar language

$L = \{z^{3x} y^i w^j \ |\ x \ge 0 \land j \gt 2i \gt 0\}$

The shortest possible word that does not belong to the language would be $\lambda$.

But I'm having troubles to find the shortest work that does belong to the language.

x can be 0. But the minimum value for i is 1. Then the minimum value for j would be 2. Leaving $y^1c^w$ = $yw^2$ = yww.

Is my reasoning valid?

It's been a while since I studied this and I'm having troubles to find good resources to work this out.

Answer 1

The grammar requires a strict inequality $j > 2i > 0$, so there must be at least $3$ ws, and since $3^0 = 1$, there must be at least one z. The shortest word in L is hence zywww, corresponding to the smallest possible choices, $x = 0,\, i = 1,\, j = 2i+1 = 3$. The struck-out part refers to a mis-parsed grammar due to typesetting difficulties, the intended grammar $$\{ z^{3x}y^iw^j \mid x\geqslant 0 \land j > 2i > 0\}$$ does not require a $z$, hence the shortest word in $L$ is ywww, corresponding to $x = 0,\, i = 1,\, j = 2i+1 = 3$.