3. (9 points) The goal of this problem is to show that our formal definitions of P and NP imply that P = NP cannot have a non-constructive proof. Throughout this problem we assume a consistent, binary, encoding of both Turing machines and 3-SAT instances. Note that the complexities are intentionally not tight to address potential overheads from scan- ning the tape linearly in order to look for certain symbols. (a) (3 points) Show that for any constant c, there is a constant c2 such that if Mi is a Turing machine encoded by at most bits that correctly decides 3-SAT in O(n10) time (aka. 3-SAT E TIME(n10), then there is a Turing machine M, encoded by at most most c2 bits that runs in O(n100) time such that if the 3-SAT instance has a satisfying assignment, M, accepts and writes a satisfying assignment at the end of the tape. (b) (3 points) Let M ...Mbe Turing machines show that the language {(k, M... Mk,t,w) | w is an instance of 3-SAT and one of M, writes a satisfying assignment to w in at most t steps can be decided by a Turing machine in (k2107 10) time. (c) (3 points) Under the assumption that there exists an unknown Turing machine encodable in c bits that decides 3-SAT is in O(n10) time, give implementation details for a Turing Machine M that decides 3-SAT in O(n10000) time. Hint: the constant factor in the big-O is able to hide any function related ci and c2, which independent of input sizes, and thus constants. 3. (9 points) The goal of this problem is to show that our formal definitions of P and NP imply that P = NP cannot have a non-constructive proof. Throughout this problem we assume a consistent, binary, encoding of both Turing machines and 3-SAT instances. Note that the complexities are intentionally not tight to address potential overheads from scan- ning the tape linearly in order to look for certain symbols. (a) (3 points) Show that for any constant c, there is a constant c2 such that if Mi is a Turing machine encoded by at most bits that correctly decides 3-SAT in O(n10) time (aka. 3-SAT E TIME(n10), then there is a Turing machine M, encoded by at most most c2 bits that runs in O(n100) time such that if the 3-SAT instance has a satisfying assignment, M, accepts and writes a satisfying assignment at the end of the tape. (b) (3 points) Let M ...Mbe Turing machines show that the language {(k, M... Mk,t,w) | w is an instance of 3-SAT and one of M, writes a satisfying assignment to w in at most t steps can be decided by a Turing machine in (k2107 10) time. (c) (3 points) Under the assumption that there exists an unknown Turing machine encodable in c bits that decides 3-SAT is in O(n10) time, give implementation details for a Turing Machine M that decides 3-SAT in O(n10000) time. Hint: the constant factor in the big-O is able to hide any function related ci and c2, which independent of input sizes, and thus constants
