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The index of a language is the number of Myhill-Nerode classes that it has. It is also equal to the number of states in the minimal DFA for the language. What is an example of a language that has a different index from its reversal?

Raphael
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user3370201
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1 Answers1

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The $k$th letter equals $a$, for fixed $k$, assuming a two letter alphabet.

This generic answer is made precise in the comment by @AntonTrunov below.

Hendrik Jan
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    Not for arbitrary $k$ though. The number of states of the minimal DFA for the original language would be $k+2$ and for its reversal the number of states is $\frac{|\Sigma|^k - 1}{|\Sigma|-1} + 1$. So, for example, if $|\Sigma| = 2$ and $k = 2$, then the number of states for those DFAs equals $4$ (in both cases). – Anton Trunov Feb 16 '16 at 19:20
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    @AntonTrunov Thanks. That is very precise. – Hendrik Jan Feb 16 '16 at 20:35
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    Thank you. It would be very precise, if I added "assuming $|\Sigma|>1$" :) – Anton Trunov Feb 16 '16 at 20:47
  • @HendrikJan Add detail to you answer on the duplicate? – Raphael Feb 16 '16 at 23:58
  • @Raphael I followed your suggestion. Should we delete this answer? – Hendrik Jan Feb 17 '16 at 00:36
  • @HendrikJan Thanks! Your call. Does it do harm (eg. because it lacks the details)? If not, it can stay. – Raphael Feb 17 '16 at 07:10
  • I don't see where the denominator of $|\Sigma| - 1$ comes from, but then I also don't see the necessity of the term $|\Sigma|^k - 1$. If you encode all $k$ last seen characters, you need exactly the $|\Sigma|^k$ states you get with that expression for $|\Sigma| = 2$. If you only encode "position was $a$ (resp. $\bar a$)", you can always make do with $2^k$ states whenever $|\Sigma| \geq 2$. – G. Bach Feb 18 '16 at 13:01