Q 1)Check all and only those of the following languages that are recursive (decidable).

1)The set of Turing-machine codes for Turing machines with more states than tape symbols

2)The set of Turing-machine codes for Turing machines that accept all strings over their input alphabet

3) The set of Turing machine codes for Turing machines that both accept and reject the same input.

Q 2)

In the following, we use comp(L) to represent the complement of the language L. We are given the following facts:

There is a polynomial-time reduction of L1 to L2.

There is a polynomial-time reduction of comp(L2) to L3

L3 is in NP.

Given what we know about the relationships among P, NP, and co-NP, which of the following is DEFINITELY true?

A) L2 is in NP.

B) comp(L2) is in co-NP.

C) L1 is not in P.

D) comp(L1) is in NP.

Q3)

Here is an instance of Post's Correspondence Problem:

List A	List B
011	01
110	10
01	1011

What is the length of the shortest solution to this instance of PCP? Note: we are asking for the number of occurrences of pairs, not the length of the string that is formed by concatenating the first or second components of the pairs.