LetΣ ={a, b}. LetL={aibai|i≥0}.Give a Turing machine (TM) that accepts the languageL.Assume (as in the examples done in our course videos) that, when theTM starts, the head is on a blank symbol,∆, and the input string isimmediately after that blank symbol on the tape. For example, if theinput string wereaaabaaa, then the inital tape configuration would be∆aaabaaa

Prove that L2 = {w = {0,1}+ | w starts and ends with the same symbol} is regular. string of length 1 (e.g., w = 0) starts and ends with the same symbol.

Let Σ={0,1}. Give a regular expression r over Σ such that L(r) = { w in Σ* | (w contains at least 2 symbols, that is, the length of w is at least 2) and (the first symbol of w is different from the last symbol of w) and (centre (w) is mimsy according to the following two definitions)}. For any string y containing at least 2 symbols, let centre (y) be the substring of y obtained by removing the first symbol and the last symbol of y. A string x over Σ is called mimsy if x is made up of either • at least two 0-symbols (and no 1-symbols), or an even number of 1-symbols (and no 0-symbols).

Recall a set A is countable if |N| ≥ |A|. Recall that Cantor’s theorem shows that P(N) = {X ⊆ N}is uncountable. Prove this.Show that F = {X ⊆ N | |X| < ∞} is countable, with an onto functiong : N → F that is computable.   Write a program/function (e.g., in Python) that runs forever and prints every element of F,e.g., if run it would print something like{0}{1}{0,2}{3}{0,2}{0,5,7}{1}{0,1}{}{2}...(Note that you must prove that the program works, i.e., it prints every element of F, and onlyelements of F.)

We are going to write a Turing machine that reads a string of 1's as input and produces a 1 for output if the string is a length that is a multiple of 3 and outputs a O otherwise. The 1 or O should be spaced 1 position to the right of the end of the input. You may assume that the input contains only zero or more 1's. For example here are four before and after snapshots. Before After B1 В Input is zero 1's (0 is a multiple of 3) <== 1 1 0 11 11 0 111 111 1 Here is a Turing machine that does most of the work but is missing one instruction. Fill the blank with the instruction needed. Follow the format of the other instructions (ie, current-state current- head head-instruction new-state, with exactly one space between each). qO 1 R q1 q1 1 R q2 q2 1 R q0 q0 BR q3 q1 BR q4 93 В 1 q5 q4 B0 q6 Hint: I suggest you draw what is given and then figure out what single instruction will make the machine behave as described.