Description 1. Prove that if SAT ? BPP then there exists a ????? ??? ?? SAT ? BPP?? ...

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Description 1. Prove that if SAT ? BPP then there exists a ????? ??? ?? SAT ? BPP?? ?? ???? .2 randomized polynomial time TM with the ????????? ??????? ????????? ????? properties listed in item 1 of this question. .?? ?????? 1 ????? ?????? Guidance: Recall the self-reducibility of SAT from class. ???: ?????? ???????? ?????? ?? SAT ???? 2. Prove that if there exists a randomized ??????? ??? ????? ?? ?????? .1 polynomial time TM ???? such that given a formula ????: ????????? ????????? ?????????? ???? ?? ??????? ?????: ? If ???? is satisfiable: with probability at least 2, ???? finds a satisfying , ?? 2 ???: ???????? ????? ???? ????? ???? ??, ????? ? ???? ??? ???? ? ?? ??????; otherwise and , ????for assignment ? ???? ?? ?????: ???? ? ;“unsatisfiable is “???? declares ? If ???? is unsatisfiable: ???? declares “???? is unsatisfiable” with probability 1; Then SAT ? RP extra creditLet Even-IS be the following language: :???? ?? Even-IS ? D?????, ????? ? ???? = (????, ????) is a simple graph that has independent set of size ????, and for every ???? ? ????, deg ???? is even 1. Prove that Even-IS is NP-hard.