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Show that any $S \in NP$ has 2 different polynomial-time verifiers $V_1, V_2$ such that, for all $x,y$, the following conditions hold: If $V_1(x,y)=1$ then $V_2(x,y)=0$ If $V_2(x,y)=1$ then $V_1(x,y)= …

Given an undirected graph $G(V, E)$, two vertices $u$ and $v$ and a natural number $k$, does a path of at least length $k$ exists between these two vertices? How can we solve this problem? I think m …

Given $A=\left\{n\in \mathbb{N} \mid \text{$n$ is odd}\right\}$, we want to prove that if $S \in P$ then there is a Karp reduction from $S$ to $A$. My attempt: If $S \in P$ we can solve $S$ with a re …

Does $NP^{SAT}=NP^{NP}$? We can see easily that $NP^{SAT}\subseteq NP^{NP}$, because $SAT \in NP$. But is the other side $NP^{NP}\subseteq NP^{SAT}$ also true? If yes, how can we prove it? …