If the alphabet is $\{0,1\}$, then is the string '010' regular?

I think it is regular because DFA and regular languages are equivalent and this string has a DFA but at the same time it seems to contradict pumping lemma which implies not regular. Here is what I mean.

For pumping lemma first we have to take $w$ only in the language (here the language is just '010') So $w=$ '010'. I choose $k = 2$ then the new string $w=xy^kz$ has length more than 3 so it is definitely not '010', which means this language is not regular!

What am I missing?


1 Answer 1


Being regular is a property of languages, not of words.

However, it seems that by '010' you really mean the language consisting only of the single word '010', that is, $\{ 010 \}$. This language, just like any other finite language, is regular.

Where does your pumping lemma proof fail, then? Here is a complete statement of the pumping lemma.

If $L$ is a regular language then there exists a constant $n$ such that for all words $w \in L$ of length at least $n$ there is a decomposition $w = xyz$, where $|xy| \leq n$ and $y \neq \epsilon$, such that $xy^iz \in L$ for all $i \geq 0$.

Your argument is ignoring the constant $n$, which could be larger than the length of all words in $L$.


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