# Question about the definition of strings that GNFA's accept

In Sipser's "Introduction to the Theory of Computation" it states "A GNFA accepts a string $w$ in $Σ^{*}$ if $w = w_1 w_2 · · · w_k$ , where each $w_i$ is in $Σ^{*}$".

However, wouldn't a more precise definition be "A GNFA accepts a string $w$ in $Σ^{*}$ if $w = w_1 w_2 · · · w_k$ , where each $w_i$ is in $Σ$" ?

• You only quote part of the definition. Please give the full one; it may be relevant. – Raphael Aug 13 '16 at 9:00

If a GNFA $A$ reconizes $\Sigma^{*}$, then $\epsilon \in \Sigma^{*}$ must be accepted. However $\epsilon \not\in \Sigma$.
• WIth $k=0$ the proposed alternate definition admits $\varepsilon$ as well. – Raphael Aug 13 '16 at 9:01