two question

1. Prove that if SAT ∈ BPP then there exists a מכונת ימת זא SAT ∈ BPPםא כי כיחו .2 randomized polynomial time TM with the םעהתכונות לינומית הסתברותית ורינג properties listed in item 1 of this question. .זו לששאלה 1 בסעיף פורטות Guidance: Recall the self-reducibility of SAT from class. רכה: היזכרו ברדוקציה העצמית של SAT אינו

2. Prove that if there exists a randomized טיורינג ונת םאמתי כי הוכיחו .1 polynomial time TM 𝑀 such that given a formula 𝜑: סהתברותית פולינומית הינתןנוסחה יימת את התכונות הבאות:  If 𝜑 is satisfiable: with probability at least 2, 𝑀 finds a satisfying , �� 2 יקה: בהסתברות לפחות משהה מספקת עבור ��, ואחרת  אם�� צאת ריזה ״ נה ספיקה״; otherwise and , 𝜑for assignment  אם�� נה ספיקה: ריזה ״ ;“unsatisfiable is “𝜑 declares  If 𝜑 is unsatisfiable: 𝑀 declares “𝜑 is unsatisfiable” with probability 1; Then SAT ∈ RP

extra creditLet Even-IS be the following language: :הבאה פה Even-IS י D⟨𝐺, 𝑘⟩ ∶ 𝐺 = (𝑉, 𝐸) is a simple graph that has independent set of size 𝑘, and for every 𝑣 ∈ 𝑉, deg 𝑣 is even 1. Prove that Even-IS is NP-hard.

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