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I came across the following problem while doing my Formal Language class assignment and hope someone can give me some hints:

  1. I have $\Sigma = \{0,1\}$ and $L=\{x0a \mid x \in \Sigma^*, a \in \Sigma\}$, namely $L$ contains all strings whose second-to-last character must be a $0$.
  2. I have found that there are four equivalence classes of $I_L$, namely $[\epsilon], [0], [00], [01]$, and I'm able to construct a DFA using this four equivalence classes to accept the language $L$.
  3. Then, I'm asked to show that there are at least three non-isomorphic NFAs of minimum size that all can accept $L$. I think the smallest NFA must have at least three states since if a NFA has only two states, this implies that there exists a single character that can be accepted by this NFA which is not true for the language we have.

Can someone provide some hints regarding how to approach this?

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Your argument that two states are not enough seems solid. The automaton needs a simple path of length two where only the last state is accepting.

Once we know this we can play with the automaton. Create three states: initial state $0$, seen letter $0$ as state $1$, and accept after $0a$ as state $2$. From this we can add extra edges to ensure that all and only strings ending in $0$ will reach the middle state $1$.

accepring 00 and 01

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  • $\begingroup$ Thank you! I think I see how this works. Just to make sure: non-isomorphic NFAs just mean that the NFAs do not have exactly the same transitions for each state, is this correct? $\endgroup$
    – x_Xbd
    Feb 23, 2023 at 16:12
  • $\begingroup$ Correct. This is called non-isomorphic (and not unequal) because the states can have different names. Just like with graphs (except here also edges are labelled and we distinguish initial and final states). $\endgroup$ Feb 23, 2023 at 21:35

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