From 47257ffbaa4b01510720cfaf93fb31ea6a38fb46 Mon Sep 17 00:00:00 2001
From: Michael Whittaker <mjwhittaker@berkeley.edu>
Date: Thu, 14 Jan 2021 11:48:02 -0800
Subject: [PATCH] Added read-write quorum systems section to notes.

---
 notes/notes.tex                              |  1 +
 notes/sections/read_write_quorum_systems.tex | 58 ++++++++++++++++++++
 2 files changed, 59 insertions(+)
 create mode 100644 notes/sections/read_write_quorum_systems.tex

diff --git a/notes/notes.tex b/notes/notes.tex
index 9601201..3124cb3 100644
--- a/notes/notes.tex
+++ b/notes/notes.tex
@@ -10,6 +10,7 @@
 \end{center}
 
 {\input{sections/quorum_systems.tex}}
+{\input{sections/read_write_quorum_systems.tex}}
 
 \bibliographystyle{plain}
 \bibliography{references}
diff --git a/notes/sections/read_write_quorum_systems.tex b/notes/sections/read_write_quorum_systems.tex
new file mode 100644
index 0000000..9c4e5d5
--- /dev/null
+++ b/notes/sections/read_write_quorum_systems.tex
@@ -0,0 +1,58 @@
+\section{Read-Write Quorum Systems}
+\NOTE[mwhittaker]{%
+  These definitions are adapted from the quorum system definitions. We should
+  double check that they all make sense or find the definitions in some paper.
+}
+
+Given a set $X = \set{x_1, \ldots, x_n}$, a \defword{read-write quorum system}
+over $X$ is a pair of sets $Q = (R, W)$ of subsets of $X$ such that every $r
+\in R$ intersects every $w \in W$. The elements $r \in R$ are called
+\defword{read quorums}, and the elements $w \in W$ are called \defword{write
+quorums}.
+
+\TODO[mwhittaker]{%
+  How do we define coteries for read-write quorums? Obviously, we don't want
+  one read quorum being a strict subset of another read quorum, and we don't
+  want a write quorum being a strict subset of another write quorum, but can a
+  read quorum be a strict subset of a write quorum and vice-versa?
+}
+
+\TODO[mwhittaker]{%
+  Once we define coteries, we can define domination.
+}
+
+Let $\sigma_R: R \to [0, 1]$ and $\sigma_W: W \to [0, 1]$ be a discrete
+probability distribution over the read and write quorums of $Q$. We call
+$\sigma$ a \defword{strategy}. Let $0 \leq p_r \leq 1$ be the probability of
+performing a read and $p_w = 1 - p_r$ be the probability of performing a write.
+We have the following definitions.
+\begin{align*}
+  l_{\sigma,p_r,p_w}(x)
+    &\defeq p_r \cdot \parens*{\sum_{\setst{r \in R}{x \in r}} \sigma_R(r)} +
+            p_w \cdot \parens*{\sum_{\setst{w \in W}{x \in w}} \sigma_W(w)} \\
+  L_{\sigma,p_r,p_w}(Q) &\defeq \max_{x \in X} l_{\sigma,p_r,p_w}(x) \\
+  L_{p_r,p_w}(Q)        &\defeq \min_\sigma L_{\sigma,p_r,p_w}(Q)
+\end{align*}
+$l_{\sigma,p_r,p_w}(x)$ is the load on $x$ given some strategy $\sigma$ and
+some workload $p_r,p_w$. $L_{\sigma,p_r,p_w}(Q)$ is the load on most loaded
+element $x$. $L{p_r,p_w}(Q)$ is the \defword{load} of the best possible
+strategy.
+
+The \defword{read resilience} or \defword{read fault tolerance} of a quorum
+system $Q$ is the largest number $f_r$ such that for every subset $F \subseteq
+X$ with $|F| = f_r$, there still exists some read quorum $r \in R$ such that $r
+\cap F = \emptyset$.
+%
+The \defword{write resilience} or \defword{write fault tolerance} of a quorum
+system $Q$ is the largest number $f_w$ such that for every subset $F \subseteq
+X$ with $|F| = f_w$, there still exists some write quorum $w \in W$ such that $w
+\cap F = \emptyset$.
+%
+The \defword{resilience} or \defword{fault tolerance} of a quorum system is the
+minimum of its read resilience and write resilience. Intuitively, a quorum
+system has read fault tolerance $f_r$ if we can fail an arbitrary set of $f_r$
+elements and still have some read quorum left; a quorum system has write fault
+tolerance $f_w$ if we can fail an arbitrary set of $f_w$ elements and still
+have some write quorum left; and a quorum system has fault tolerance $f$ if we
+can fail an arbitrary set of $f$ elements and still have some read quorum and
+some write quorum left.