Can somebody provide an example of two equivalent (recognizing the same language) minimal non-deterministic automata (NFA) which are not isomorphic?


2 Answers 2


See the paper (postscript)

Arnold, Dicky, Nivat. A note about minimal non-deterministic automata

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    $\begingroup$ Thank you for sharing! Their paper includes a nice example. :) $\endgroup$ Commented Jun 11, 2015 at 5:10

Along a different line: the set $L_6$ of strings of the form $a^n$, where $n$ is not a multiple of 6 has two very different minimal NFAs. Two minimal NFAs for $L_6$.

One of them is basically the minimal DFA, the other guesses whether it is not a multiple of 2 or not a multiple of 3.

  • $\begingroup$ But isn't the later automaton not minimal? One can discard the state a and set b and d as inital states (and moving the accepted states to c, e and f). So a minimal NFA has actually 5 states $\endgroup$
    – Andrei Kh
    Commented Feb 15, 2023 at 14:02
  • $\begingroup$ That depends on the actual definition of NFA. You are right: I silently assumed that NFAs have one initial state. That's at least a frequent definition, see, e.g., the Wikipedia article on NFAs. $\endgroup$
    – Thomas S
    Commented Feb 16, 2023 at 15:24

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