# What is the regular expression for the following language?

What is the regular expression for the following language?
$$L = \{acbc: a,b,c \in \{0,1\}^+ \}$$ maybe we can say $$L = ((0 + 1)^+ 0 (0 + 1)^+ 0) + ((0 + 1)^+ 1 (0 + 1)^+ 1)$$ Is it true??

• Hint: it's something of a "trick" question. See if you can work out how to make almost any string be in the language. – David Richerby Nov 8 '18 at 13:42
• Hint: you can assume $|c|=1$ (why?). – Yuval Filmus Nov 8 '18 at 17:22
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• Try to prove my hint. – Yuval Filmus Nov 9 '18 at 4:38
• That isn't a hint- it doesn't satisfy the specification of the language. It could be placed as an additional constraint on the language, but then no proof would be. required. And unless this constraint is applied, the language is not regular, and therefore cannot be expressed as an RE – Brishna Batool May 23 '19 at 6:48

I claim that $$L$$ is equal to the set of all strings of the form $$a\gamma b\gamma$$, where $$a,b \in \{0,1\}^+$$ and $$\gamma \in \{0,1\}$$. Indeed, a string of this form clearly belongs to $$L$$. Conversely, if $$acbc \in L$$ and $$c=d\gamma$$, then $$acbc = ad\gamma bd\gamma$$ belongs to the new set. The new set is clearly described by the regular expression $$(0+1)^+0(0+1)^+0 + (0+1)^+1(0+1)^+1.$$