Is {wxw^r} a regular language?

Question

Is $\{ WxW^{\mathrm{R}} \mid W,x\in\{0,1\}^+\}$ a regular language? If so, why?

The notation $W^{\mathrm{R}}$ means the reverse string of $W$?

If we consider the best answer in this solution, if the language is regular, then its FA should reject all strings not in the language. However, a string such as 0110100 would also be accepted by the FA, since it compares only the starting and end characters! Please explain.

@Shaull it doesn't explain why the string 0110100 would also be accepted by the FA!! — Somenath Sinha, Dec 13 '15 at 13:54
Ah, sorry, I misread your language. The string 0110100 should be accepted, since it's of the form $wxw^R$, where $w=0$ and $x=11010$. — Shaull, Dec 13 '15 at 14:01
@Shaull but the string 0110100 doesn't really fit the definition of the initial language where |w|>1 , does it? — Somenath Sinha, Dec 13 '15 at 14:05
First, where is it required that $|w|>1$? It's not in your definition. Also, if you require $|w|>1$, you can do the same trick by comparing the first two letters and the two last letters. This will work in the same way (but this time the string $0110100$ will of course not be accepted). — Shaull, Dec 13 '15 at 14:12
There's no requirement stated in the question that $|W|>1$. The requirement that $W$ matches $(0+1)^+$ just says that $|W|\ge1$. — David Richerby, Dec 13 '15 at 14:18
how can you consider w as only first character of the string and w^R as last character of the string? — TdBm, Nov 17 '17 at 10:16

Renato Sanhueza · Accepted Answer · 2015-12-15T16:00:20.743

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$W^R$ means the reverse string of $W$.
You are right about the DFA. If a DFA $A$ accepts the language $L$ then it rejects all the words that are not in $L$. But the word $\sigma = 0110100$ is in the language!. Consider $W=0$ and $x=11010$.
As G.Bach eloquently stated in his comment(a very interesting property of the language):

$$\{WxW^R : W,x\in \{0,1\}^+\} = \{WxW : W\in\{0,1\} , x\in \{0,1\}^+\}$$

¿Why?

Because there is little restriction to $x$. We can consider that $x$ is almost the whole word with the exception of the first and the last symbol(because in the original language $|W|>0$). The reverse string of any word with unitary size is the same word. So words like:

$$\sigma_1 = 01110000$$ $$\sigma_2 = 11111101$$

Are in the language. All that matter is the first and the last symbol. This language is very interesting because it appears to be irregular at the first sight.

edited Dec 15 '15 at 16:00

answered Dec 15 '15 at 15:55

Renato Sanhueza

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So this would be 0.+0|1.+1 with $.\in {0,1}$ :-) – Harald Dec 16 '15 at 12:05
$L = \mathscr{L}(,0 (0|1)^+ 0, |, 1 (0|1)^+ 1,)$ – Renato Sanhueza Dec 16 '15 at 14:13
So what happened is that a language L was described in a way that makes it look really complicated, but it’s a bluff and in reality it’s really simple. And therefore proving that it is regular also looks really difficult but turns out to be really simple, once you figure out someone’s playing a trick on you. – gnasher729 Oct 02 '20 at 21:02

score 1 · Answer 2 · answered Dec 16 '15 at 10:14

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Sure !! this is a Regular Language
Language : Start & End with same Symbol.

eg; W: 000111 W^R : 111000

WxW^R : 0 0011111100 0

First & Last symbol of language is W & W^R respectively & remaining is in x. You can check any input for W, the language hold the properties Start & End with same Symbol.

answered Dec 16 '15 at 10:14

S.N. Tiwari

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Welcome. While your answer is correct, I don't see what it adds. The observation that the language is just the set of strings that start and end with the same symbol has been made multiple times already, both in the existing answer and in the comments. We have plenty of unanswered questions and it would be much better to use your time answering some of those, rather than duplicating existing answers. – David Richerby Dec 16 '15 at 12:08

score 0 · Answer 3 · answered Jan 15 '19 at 09:16

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it's a tricky question but quite simpe one when you know the logic...

say w = 100, x= 101 => wr= 001

the complete string wxwr will be 100101001

now what we can do here is extend x in both direction so it consumes parts of w and wr leaving only the starting and ending symbol now x = 0010100 and w,wr = 1;

so simply the problem is reduced to string starting and ending with same symbol, now a DFA can be constructed.

so yes this is a regular language.

answered Jan 15 '19 at 09:16

Premjeet Prasad

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Thanks for taking the time to answer! Your answer is correct but we already have basically this answer twice already on this page so there's not really much benefit in posting it for a third time, just in slightly different words. – David Richerby Jan 15 '19 at 15:06

Is {wxw^r} a regular language?

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