this is not regular.
wxw. Let W=01 X= 100
wxw= 01 100 01.Since there is no mention about the lentgh of X,we try to match elements as many as we can to make it regular.In worst case , even if you go till last but one, LHS has 0 and RHS has 1. W cant take two different values 0,1.
If it had been WXW^R IT is regular.
It is regular.
Lets see the strings in L = { a, b, aa, ab, ba, bb, aaa, ..... } (When w is ϵ, wxw generates all these strings and hence we don't need to consider any other case for w) = Σ* - {ϵ} As we have no restriction on the length of x, we can leave 1st and last symbol as it is and consider the rest part as x.
eg : w=bab x=bb
wxw = babbbbab; we may assume x = abbbba so we can read it as : bxb which is finite.
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