Concurrent TM

Concurrency is all the rage in modern programming languages. If we add to a standard TM a second, independently controlled read/write head, controlled by a separate DFA, does that increase the power of the TM?

Unsolvable problems

Can you demonstrate that the following problems are unsolvable? (The general technique is to show that, if the TM exists, you can use it to solve the Halting Problem.)- Does a given Turing machine M halt on
*all*inputs?

- Does Turing machine M halt for
*any*input? (That is, is L(M)=?)

- Does Turing machine M halt when given a blank input tape?

- Do two Turing machines M1 and M2 accept the same language?

- Is the language L(M) finite?

- Does L(M) contain any two strings of the same length?

- Does L(M) contain a string of length k, for some given k?