S→R and binary operator ⊕ Î R´R→R.
In fact, Richard Bird’s first homomorphism lemma3 says that any function
h Î M<S>→R is a homomorphism
with respect to ∪ if and only if it can
be factored into a map followed by a
reduce: h(xs) = xs.Select(f).Ag-gregate(⊕). Mathematics dictates
that coSQL queries are performed using MapReduce.
Practical implementations of MapReduce usually slightly generalize
Bird’s lemma to use SelectMany
instead of Select so that the map
phase can return a collection instead
of a single value, and insert an intermediate GroupBy as a way to “write”
equivalence classes of values from the
map phase into the key-value store for
subsequent processing in the reduce
phase, and then aggregate over each
For example, DryadLINQ15 uses the
yable<S> to represent the partitioned
input collection for LINQ queries and
then implements MapReduce over the
partitioned collection using the function illustrated in Figure 15.
In an open world where collections
are distributed across the network, it is
much harder for a query optimizer to
perform a global optimization taking
into account latency, errors, etc. Hence,
most coSQL databases rely on explicit
programmatic queries of a certain pattern such as MapReduce that can be
executed reliably on the target physical
machine configuration or cluster.
The nascent noSQL market is extremely fragmented, with many competing vendors and technologies. Programming, deploying, and managing
noSQL solutions requires specialized
and low-level knowledge that does not
easily carry over from one vendor’s
product to another.
A necessary condition for the net-
work effect to take off in the noSQL
database market is the availability of a
common abstract mathematical data
model and an associated query lan-
guage for noSQL that removes product
differentiation at the logical level and
instead shifts competition to the phys-
ical and operational level. The avail-
ability of such a common mathemati-
cal underpinning of all major noSQL
databases can provide enough critical
mass to convince businesses, develop-
ers, educational institutions, etc. to
invest in noSQL.
Many thanks to Brian Beckman, Jimmy “the aggregator” Nilsson, Bedarra-Dave Thomas, Ralf Lämmel, Torsten
Grust, Maarten Fokkinga, Rene Bouw,
Alexander Stojanovic, and the anonymous referee for their comments that
drastically improved the presentation
of this paper, and of course to Dave
Campbell for supporting work on all
cool things LINQ.
A Conversation with
Erik Meijer and Jose Blakeley
BASE: An Acid Alternative
Bridging the Object-Relational Divide
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Erik Meijer ( email@example.com) has been working
on “democratizing the cloud” for the past 15 years.
he is perhaps best known for his work on the haskell
language and his contributions to lInQ and the reactive
Gavin Bierman ( firstname.lastname@example.org) is a senior
researcher at Microsoft research cambridge focusing
on database query languages, type systems, semantics,
programming language design and implementation, data
model integration, separation logic, and dynamic software
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