From ffc10b03c22c56fc18958a047055f14d46af0e94 Mon Sep 17 00:00:00 2001
From: Michael Whittaker <mjwhittaker@berkeley.edu>
Date: Sun, 17 Jan 2021 19:50:54 -0800
Subject: [PATCH] Updated notes.

---
 notes/references.bib              | 17 ++++++++
 notes/sections/quorum_systems.tex | 71 ++++++++++++++-----------------
 2 files changed, 50 insertions(+), 38 deletions(-)

diff --git a/notes/references.bib b/notes/references.bib
index 40b71e3..afaffce 100644
--- a/notes/references.bib
+++ b/notes/references.bib
@@ -1,3 +1,14 @@
+@article{ibaraki1993theory,
+    title={A theory of coteries: Mutual exclusion in distributed systems},
+    author={Ibaraki, Toshihide and Kameda, Tiko},
+    journal={IEEE Transactions on Parallel and Distributed Systems},
+    volume={4},
+    number={7},
+    pages={779--794},
+    year={1993},
+    publisher={IEEE}
+}
+
 @article{naor1998load,
     title={The load, capacity, and availability of quorum systems},
     author={Naor, Moni and Wool, Avishai},
@@ -9,6 +20,12 @@
     publisher={SIAM}
 }
 
+@article{neilsen1991general,
+    title={A general method to define quorums},
+    author={Neilsen, Mitchell and Mizuno, Masaaki and Raynal, Michel},
+    year={1991}
+}
+
 @article{vukolic2013origin,
     title={The origin of quorum systems},
     author={Vukoli{\'c}, Marko and others},
diff --git a/notes/sections/quorum_systems.tex b/notes/sections/quorum_systems.tex
index 0e08f49..5db5415 100644
--- a/notes/sections/quorum_systems.tex
+++ b/notes/sections/quorum_systems.tex
@@ -1,52 +1,42 @@
 \section{Quorum Systems}
-These definitions are taken from \cite{naor1998load} and
-\cite{vukolic2013origin}.
-%
 Given a set $X = \set{x_1, \ldots, x_n}$, a \defword{quorum system} over $X$ is
 a set $Q = \set{q_1, \ldots, q_m}$ of subsets of $X$, called \defword{quorums},
 such that every pair of quorums intersect. That is, for every $q_1, q_2 \in
 Q$, $q_1 \cap q_2 \neq \emptyset$.
-
+%
 A quorum system $Q$ is a \defword{coterie} if there does not exist quorums
 $q_1, q_2 \in Q$ such that $q_1 \subset q_2$. In other words, a coterie is a
 quorum system that does not contain some quorum $q_1$ that is a strict subset
 of some other quorum $q_2$.
+%
+Let $P, Q$ be two quorum systems over the same set $X$. $P$ \defword{dominates}
+$Q$, denoted $P > Q$, if $P \neq Q$ and for every $q \in Q$, there exists some
+$p \in P$ such that $p \subseteq q$. A quorum system $Q$ is \defword{dominated}
+if there exists some quorum system $P$ that dominates it.
 
-Let $P, Q$ be two coteries over the same set $X$. $P$ \defword{dominates} $Q$,
-denoted $P > Q$, if $P \neq Q$ and for every $q \in Q$, there exists some $p
-\in P$ such that $p \subseteq q$. A coterie $Q$ is \defword{dominated} if there
-exists some coterie $P$ that dominates it.
+We can associate every quorum system $Q$ with a monotone boolean function
+$f_Q$. For example, the majority quorum system $Q = \set{\set{a, b}, \set{b,
+c}, \set{a, c}}$ corresponds to the function $f_Q = ab + bc + ac$. The prime
+implicants of the boolean function correspond to the minimal sets of the quorum
+system. We say $f \leq g$ if for every $\vec{x}$, $f(\vec{x}) \implies
+g(\vec{x})$. In other words, $f \leq g$ if $\setst{\vec{x}}{g(\vec{x})}
+\subseteq \setst{\vec{x}}{f(\vec{x})}$. Consider two quorum systems $Q_1$ and
+$Q_2$, $Q_1$ dominates $Q_2$ if and only if $f_{Q_2} < f_{Q_1}$.
 
-\begin{example}
-  Let $P$ be the majority quorum system over the set $X = \set{a, b, c}$. That
-  is, $P = \set{\set{a, b}, \set{a, c}, \set{b, c}}$. Let $Q = \set{\set{a, b,
-  c}}$. Note that both quorum systems are coteries. $P$ dominates $Q$. To see
-  this, we first confirm that $P \neq Q$. Next, we consider every $q \in Q$ and
-  find a corresponding $p \in P$ where $p \subseteq q$. Here, we only have one
-  choice for $q$ (i.e. $q = \set{a, b, c}$), and for every $p \in P$, $p
-  \subseteq q$.
-\end{example}
-
-There is an intuitive way to think about non-dominated coteries. A coterie $Q$
-is non-dominated if (1) removing any element from any quorum would make $Q$ no
-longer a coterie and (2) there are no other quorums that we can add to $Q$
-while preserving the fact that $Q$ is a coterie. Here's why. (1) Assume for
-contradiction that we remove some element from some quorum in $Q$ to
-form a new coterie $P$. $P$ dominates $Q$ in the obvious way, but $Q$ is
-non-dominated. (2) Assume for contradiction that we add some quorum to $Q$ to
-form a new coterie $P$. Again, $P$ dominates $Q$ in the obvious way, but $Q$ is
-non-dominated. Intuitively, a non-dominated coterie is a maximal coterie (we
-cannot add any more quorums) with minimal quorums (we cannot remove any
-elements from any quorums).
-
-\NOTE[michael]{%
-  I think domination will be an important property for recursive quorum
-  systems. I think there are some quorum systems that we cannot form as a
-  recursive quorum system, but we can form a recursive quorum system that
-  dominates it. The definition of domination that I found is only defined on
-  coteries. Can we define this without coteries? I think so, but we have to
-  double check.
-}
+\newcommand{\dual}[1]{#1^d}
+Let the dual $\dual{f}$ of $f$ be $\dual{f}(x) = \bar{f}(\bar{x})$. $\dual{f}$
+is the function we get if we swap and with or. Note that $\dual{f}$ corresponds
+to the sets that intersect every set in $f$. We say a monotone function $f$
+is \defword{dual-minor} if $f \leq \dual{f}$, \defword{dual-major} if $f \geq
+\dual{f}$, and \defword{self-dual} if $f = \dual{f}$. A function $f$
+corresponds to a quorum system if and only if $f$ is dual-minor.  A quorum
+system $Q$ is non-dominated if and only if $f_Q$ is self-dual. Every
+non-dominated coterie can be represented as a composition of the simple
+majority function (with duplicates). Every coterie can be represented by a
+composition of and, or, and majority. \cite{ibaraki1993theory} also talks about
+when a function can be decomposed without duplicates as well as other things
+involving decomposition. \cite{neilsen1991general} talks about composition as
+well, and even includes examples for things like grids and trees.
 
 Let $\sigma: Q \to [0, 1]$ be a discrete probability distribution over the
 quorums of $Q$ (i.e., $\sum_{q \in Q} \sigma(q) = 1$). We call $\sigma$ a
@@ -78,6 +68,11 @@ higher the throughput it can support.
   which is optimal.
 \end{example}
 
+\cite{naor1998load} prove that if a quorum system dominates another, it has
+lower or equal load. This shows that there always is a non-dominated coterie
+that has the lowest possible load. If we're trying to optimize for load, this
+tells us that we limit ourselves to non-dominated coteries.
+
 The \defword{resilience} or \defword{fault tolerance} of a quorum system $Q$ is
 the largest number $f$ such that for every subset $F \subseteq X$ with $|F| =
 f$, there still exists some quorum $q \in Q$ such that $q \cap F = \emptyset$.