This should be marked as "Homework" I think.
Ǝ
is "there exists"
Σ
is "the set of symbols in the language" ({a, b,c}
) in this case
∈
is "element of"
Now that we have that, we can read this language. So L
is the set of words w
in {a, b, c}*
such that there exists a word v
and there exists a n >= 2
such that w
is a repetition of v
n
times. E.g. ababab = (ab)^3 ∈ L
.
Now you want to come up with a Turing machine, M
, to represent this language, so you have to consider:
- When do we reject a word (what is our rejection state, what is on the stack)
- When do we accept a word (what is our accepting state, what is on the stack)
- How do we guarantee that
M
terminates.
We can see that a
is not in L
because n >= 2
which implies that the length of v^n
is at least 2
(0
in the case of the empty string though, which is an outlier). Similarly for b
and c
. With that consideration and the knowledge that n >= 2
, figure out what words are not accepted (e.g. consider b
, abc
, cab
, cca
, etc.).