# Construct a Turing Machine for language L = {ww^{r} | w ∈ {0, 1}}

Prerequisite – Turing Machine

The language L = {ww^{r} | w ∈ {0, 1}} represents a kind of language where you use only 2 character, i.e., 0 and 1. The first part of language can be any string of 0 and 1. The second part is the reverse of the first part. Combining both these parts out string will be formed. Any such string which falls in this category will be accepted by this language. The beginning and end of string is marked by $ sign.

For example, if first part w = 1 1 0 0 1 then second part w^{r} = 1 0 0 1 1. It is clearly visible that w^{r} is the reverse of w, so the string 1 1 0 0 1 1 0 0 1 1 is a part of given language.

**Examples –**

Input : 0 0 1 1 1 1 0 0 Output : Accepted Input : 1 0 1 0 0 1 0 1 Output : Accepted

**Basic Representation –**

**Assumption:** We will replace 0 by Y and 1 by X.

**Approach Used –**

First check the first symbol, if it’s 0 then replace it by Y and by X if it’s 1. Then go to the end of string. So last symbol is same as first. We replace it also by X or Y depending on it.

Now again come back to the position next to the symbol replace from the starting and repeat the same process as told above.

One important thing to note is that since w^{r} is reverse of w of both of them will have equal number of symbols. Every time replace a nth symbol from beginning of string, replace a corresponding nth symbol from the end.

**Step-1:**

If symbol is 0 replace it by Y and move right, Go to state Q2

If symbol is 1 replace it by X and move right, Go to state Q1**Step-2:**

If symbol is 0 replace it by 0 and move right, remain on same state

If symbol is 1 replace it by 1 and move right, remain on same state

——————————————————————-

If symbol is X replace it by X and move right, Go to state Q3

If symbol is Y replace it by Y and move right, Go to state Q3

If symbol is $ replace it by $ and move right, Go to state Q3**Step-3:**

If symbol is 1 replace it by X and move left, Go to state Q4

If symbol is 0 replace it by Y and move left, Go to state Q5**Step-4:**

If symbol is 1 replace it by 1 and move left

If symbol is 0 replace it by 0 and move left

Remain on same state**Step-5:**

If symbol is X replace it by X and move right

If symbol is Y replace it by Y and move right

Go to state Q0**Step-6:**

If symbol is X replace it by X and move right

If symbol is Y replace it by Y and move right

Go to state Q6

ELSE

Go to step 1**Step-7:**

If symbol is X replace it by X and move right, Remain on same state

If symbol is Y replace it by Y and move right, Remain on same state

If symbol is $ replace it by $ and move left, STRING IS ACCEPTED, GO TO FINAL STATE Q7

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