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Chapter 10: Q2E (page 393)
Show that 12 is no pseudoprime because it fails some Fermat test.
The above problem can be solved using the Fermat鈥檚 Little Theorem which says that if n is a prime number, then for every 鈥榓鈥� , , i.e. .
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Recall that is the class of languages that are decided by nondeterministic polynomial time Turing machines with an oracle for the satisfiability problem. Show that .
Define a ZPP-machine to be a probabilistic Turning machine that is permitted three types of output on each of its branches: accept, reject, and? A ZPP. If M outputs the correct answer on every input string w with probability at least machine M decides a language A (Accept it and reject if ),and M never outputs the wrong answer. On every input, M may output? with probability at most. Furthermore, the average running time over all branches of M on w must be bounded by a polynomial in the length of w. Show that , where ZPP is the collection of languages that are recognized by ZPP - machines.
Let| and equivalent branching programs.Show that is coNP-complete.
Prove that if Ais a language in L, a family of branching programsexists wherein each Bnaccepts exactly the strings in A of length nand is bounded in size by a polynomial in n.
Show that the majority function with n inputs can be computed by a branching program that hasnodes.
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