Original question from the textbook for reference:

1. (8 points) Exercise 7.14 from the text book (3rd edition), modified with k as part of the input. A permutation on the set {1...k} is a one-to-one, onto function on this set. When p is a permutation, p' means the composition of p with itself t times. Consider the language PERM-POWER = {(k, p, q, t) | p =q}. (a) (2 points) discuss a suitable encoding scheme for over the alphabet S = {0,1, #; } (b) (6 points) show that PERM-POWER E P by giving the implementation details of a Turing machine that decides it in polynomial time. Note that because t may have up to O(n) bits, the value of t may be as large as 20(n). This means a poly-time solution should run in time (poly) logarithmic in the value of t. 7.14 A permutation on the set {1,..., k} is a one-to-one, onto function on this set. When pis a permutation, p' means the composition of p with itself t times. Let PERM-POWER = {(p,q, t) p=q' where p and q are permutations on {1,..., k} and t is a binary integer}. Show that PERM-POWER E P. (Note that the most obvious algorithm doesn't run within polynomial time. Hint: First try it where t is a power of 2.) 1. (8 points) Exercise 7.14 from the text book (3rd edition), modified with k as part of the input. A permutation on the set {1...k} is a one-to-one, onto function on this set. When p is a permutation, p' means the composition of p with itself t times. Consider the language PERM-POWER = {(k, p, q, t) | p =q}. (a) (2 points) discuss a suitable encoding scheme for over the alphabet S = {0,1, #; } (b) (6 points) show that PERM-POWER E P by giving the implementation details of a Turing machine that decides it in polynomial time. Note that because t may have up to O(n) bits, the value of t may be as large as 20(n). This means a poly-time solution should run in time (poly) logarithmic in the value of t. 7.14 A permutation on the set {1,..., k} is a one-to-one, onto function on this set. When pis a permutation, p' means the composition of p with itself t times. Let PERM-POWER = {(p,q, t) p=q' where p and q are permutations on {1,..., k} and t is a binary integer}. Show that PERM-POWER E P. (Note that the most obvious algorithm doesn't run within polynomial time. Hint: First try it where t is a power of 2.)