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So in my attempt to convert the following DFA into a regular expression, I ended up with ((ba)*(ab(ab)*)*(aa(ba)*a)*)*. The exercise I'm following wants me to convert the DFA into a regular expression then describe its language in plain English, however, I can't seem to come up with anything for the converted expression. Any hints on how to go from here?

DFA in question

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    $\begingroup$ It's not a simple task. One thing I notice is that you can't have two B's in a row. Anything else? Also, your regular expression seems wrong because aaaba matches the DFA but not the regex $\endgroup$
    – user253751
    Aug 26, 2022 at 22:48
  • $\begingroup$ Yeah, I also noticed the part about no bb's, but I'm not sure about any other pattern. I tried redoing the regex but I'm still quite lost ^^" $\endgroup$
    – apollonia
    Aug 27, 2022 at 5:32

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To accept a string, you must start in $q_0$ and end in $q_0$ or in $q_3$.
A path from $q_0$ to $q_0$ can make a cycle through $q_5$ (so you can have $(ba)^*$) or through $q_1,q_2,q_3,q_4$ (and so $a(ba)^*a(ba)^*a$), and then you can put these together to obtain $((ba)^*+a(ba)^*a(ba)^*a)^*$.
A path from $q_0$ to $q_3$ is a path from $q_0$ to $q_0$ and then a path that do not pass twice through $q_0$ from $q_0$ to $q_3$, i.e. $ab(ab)^*$.

Now, we can put all together: $$((ba)^*+a(ba)^*a(ba)^*a)^*(\varepsilon+(ab)^+)$$

A description of this language in plain English could be something like this: strings of the type $((ba)^*+a(ba)^*a(ba)^*a)^*$ or $((ba)^*+a(ba)^*a(ba)^*a)^*(ab)^+$. The first type if formed by strings starting either with $a$ or $b$, with no consecutive $b$s, such that the number $a$s exceeds the number of $b$s by a multiple of 3 (0 included), and ending with $a$; the second type if formed by strings of the first type concatenated with $(ab)^+$, so with a nonempty string starting with $a$, ending with $b$ and such that no two consecutive letters are equal.

Here is the simplest description: strings without consecutive $b$s such that the number $a$s exceeds the number of $b$s by a multiple of 3 (0 included).

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