Math and stuff

Proposition Let $U$ be a quorum in FBAS $\ev{V, Q}$, let $B \subset V$ be a set of nodes, and let $U’ = U \setminus B$. If $U’ \ne \emptyset$, then $U’$ is a quorum in $\ev{V, Q}^B$. Solution Since $U’ \ne \emptyset$, it suffices to show that $\forall v...

Proposition If $B_1$ and $B_2$ are DSets in an FBAS $\ev{V, Q}$ enjoying quorum intersection, then $B = B_1 \cap B_2$ is a DSet, too. Solution If $B_1 = V$ or $B_2 = V$, then we are done. Suppose otherwise. For any $v \in V$, \[\begin{align*} v \in V \setminus...

Proposition In an FBAS with quorum intersection, the set of befouled nodes is a DSet. Solution Let $\ev{V, Q}$ be an FBAS with quorum intersection. Let $B$ be the intersection of all DSets that contain all ill-behaved nodes. Since the intersection of two DSets is a DSet and $V$ is...

Proposition 1 In an FBAS $(V, Q)$, the union of two quorums is a quorum. Proof Let $U_1, U_2$ be two quorums. Let $v \in U_1 \cup U_2$. Then $v \in U_i$ for $i = 1$ or $i = 2$. Then $q \subset U_i$ for some $q \in Q(v)$. Therefore,...

A node in an FBAS can fail in many ways. The following Venn diagram copied from P.7 of the SDF white paper) shows different types of node failures. Byzantine failures Examples A malicious user modifies the software and attempts to double spend money. A computer crashes due to a power...