Quick/Simple Regex/Regular Language Clarification

Question

Hey all,

I feel like a moron posting such simple questions on here, but the knowledge base of this site is just amazing. Thanks for your understanding.

Concerning a question about finding the minimum pumping length (concerning the pumping lemma for regular languages) of a regular expression:

Regular Expression R = 1011 (over the alphabet {0,1})

Aren't the only matches to this string ε (empty string) and 1011?

EDIT-I've been staring at too many kleene stars. The empty string ε is not in this language.

Properties of regular languages state that if a language can be represented by a finite automaton (or regular expression), then it is regular, and certainly this one can be represented by both. (And the question obviously leads me to believe the language is regular)

But on the other hand, the pumping lemma (informally) states that all regular languages have a pumping length whereby all strings of at least this length can be split into xyz where y can be repeated without affecting the result. ε cannot be pumped by definition, and 1011 can't be pumped (I don't think, this is the question) because no other string matches it, so removing or adding instances of y will result in a string that is not accepted/matched.

Where is the error in my logic?

2nd EDIT-Is the answer for any p>4, the language can be pumped by setting x or y or z to be ϕ (the null set), which when concatenated with anything results in the null set?

Answer 1

The pumping lemma isn't too useful for finite languages. All finiate languages are regular - for example, in your case, you can set your "pumping length" to 4. In an empty sense, all words above that length can be pumped :)