如何 L={ww^Rx| 其中 w, x 属于 {a,b}^* } 是常规语言？

Question

I have understood that L={wxw^r|w,x belongs to {a,b}^* } is regular because it turns out to be the pattern of starting and ending with same symbol but I am not getting the proper explanation that how to say L={ww^rx|w,x belongs to {a,b}*} is regular language using DFA design. Please help me in understanding this!

Answer 1

This is a trick question. The language L as you have specified is the language of the regular expression (a + b)* , that is, any string of a's and b's. The trick is that for any string y = s1.s2.s3...sk where si in {a, b} , we can write y = wxw^R where w is the empty string and x = y . Basically, the trick is that we can always choose w to be the empty string, and in that case we are left with L = {x | x in {a, b}^*} L = {x | x in {a, b}^*} , clearly regular. Another way of thinking about it is this: can you find any string of a's and b's that is not in L? Is it not in L even if you take w to be the empty string?