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A word doubled

Is L = { ww : w ∈ {0,1}* } regular?

you attack · engine defends

Engine · move 1

The engine claims L = { ww : w ∈ {0,1}* } is regular and announces a pumping length of n = 6. Choose a string in the language with at least 6 symbols.

You · move 2

w ∈ L, |w| ≥ 6