Let Σ = {0, 1}. Define the language L0000 = { ⟨M1,M2⟩ | M1 and M2 are deterministic Turing machines, and L(M1)∩L(M2) contains only the string ‘0000’ }. Show that L0000 is an undecidable language.

For a Turing machine M, (M) refers to the binary representation of M. For a Turing machine M, L(M) contains the set of all strings accepted by M. For a Turing machine M and an input x = {0,1}*, Steps(M, x) refers to the number of steps taken by M to execute on x before it halts. Here, one step of execution of M on x = one movement (left or right) of the tape head. For a Turing machine M and an input x = {0,1}*, we define the following: ReachCells(M,x) = {i : M reaches ith tape cell when M is executed on x} Informally, it contains all locations on the tape that are visited when M is ecuted on x. The leftmost location on the tape is the first tape cell, the location next to it is the second tape cell, and so on. A string w₁ is an anagram of w2 if w₁ can be obtained by rearranging the alphabets of w2. Formally, if w₁ is an n length string, wê is called an anagram of w₁ if there exists a permutation à on n elements such that π(w₁) = W2.

draw a Turing Machine that accepts strings in each of the languages below. Includethe septuple of the language with transition functions as well. a . L = { a n bn an bn : n  ≥ 1 } b . L = { a n b m an+m: n  ≥ 0, m ≥ 1} c . L = {w : w ∈ { a ,b}+ , |w| is a multiple of 4} d. L = {a n bm  bn  am  : m  ≥ 1 , n > m  } e. L = {wcw : w ∈ { a ,b}* }

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