\begin{itemize}
  \item
    Recursive quorum systems are all non-dominated bicoteries. We can view the
    tree as a way to generate an arbitrary set of read quorums. Then we just
    bubble down dual to compute write quorums.

  \item
    The theory of coteries and decomposition has been very well studied. The
    one thing that seems a little less studied is the relationship between load
    and bicoteries. Can we find a nice way to find an optimal load bicoterie?
    This is probably hard, but maybe something about bicoteries makes this
    easier than for coteries.

  \item
    If we wanted to develop a library for bicoteries, we would like to (a)
    allow people to form arbitrary bicoteries, (b) compute an optimal strategy
    for a bicotier, (c) efficiently pick quorums based on the strategy, and (d)
    efficiently determine if something is a superset of a quorum. We can do (a)
    with a library of dual-able operators like or and and along with some
    builtins for things like grids. (b) we can use LP with specializations. We
    might have to do something smart to get ``good' strategies (e.g.,
    balanced). For (d), we can just evaluate the tree or its dual or maybe use
    something more efficient like BDDs? For (c) I'm not completely sure. For
    some quorum systems, the number of quorums is exponential, so we wouldn't
    want to represent them explicitly. Ideally we could also find an optimal
    load bicoterie, but that's probably not going to happen. The library can
    also help compute fault tolerance and maybe load under failure and whatnot.
    How to compute fault tolerance with duplicates?

  \item
    Maybe we can run some practical experiments to show how important a good
    bicoterie is. Maybe we'll find it's not that important.

  \item
    Is there anything smart about reconfiguring or changing coteries on the
    fly? For example, grids are not good to go from 6 to 7 nodes. With
    bicoteries, we probably want to adjust on the fly as the read and write
    ratios change.

  \item
    Maybe we can introduce the idea that not all nodes have the same capacity.
    Some have higher load than others. We could bake that into the nodes like,
    a = Node("a", cap=1000).

  \item
    Maybe some nodes are also co-located, so the fault tolerance is a little
    different. How can we compute fault tolerance with duplicates anyway?

  \item
    Just for fun, we could probably cross compile to javascript and show how to
    subsume other quorum systems with our library. We should be able to subsume
    all if we use duplicates.
\end{itemize}

\TODO[michael]{Prove that for every read-write quorum system, there exists a coterie with at least as good load.}
\TODO[michael]{The above shows that the optimal load quorum system is a non-dominated coterie. Maybe this is useful? We can generate every NDC?}
\TODO[michael]{Understand how domination relates to subsumption.}
\TODO[michael]{Understand dual-major, dual-minor, and self-dual.}
\TODO[michael]{Extend these notions to read-write quorums.}
\TODO[michael]{Enumerate all read-write quorums on four nodes and see the ones we can and can't subsume.}
\TODO[michael]{Prove that if I have the read quorums R, then the write quorums are bar(R).}