deletions | additions       diff --git a/Commutativity.md b/Commutativity.md  index ea9a285..df41ab6 100644  --- a/Commutativity.md  +++ b/Commutativity.md 
...
Commutativity is well known, especially in distributed systems, for enabling important optimizations. Since the 80s, commutativity has been exploited by database systems designers \cite{Weihl:1988,Fekete:90}, within the safe confines of relational models, where knowledge of query plans and complete control of the data structures allows systems to determine when transactions may conflict. Recently, commutativity has seen a resurgence in systems without a predefined data model, such as NoSQL databases and transactional memory.  In the realm of evental eventual  consistency, commutativity has been leveraged for convergence guarantees. RedBlue consistency allows executing commutative ("blue") operations locally, knowing they will eventually converge. Similarly, conflict-free replicated data types (CRDTs) \cite{Shapiro:SSS11} define commutative merge functions for all operations to ensure that replicas will converge.  Lynx \cite{Zhang:SOSP13} uses knowledge of some commutative operations to make tracking serializability cheaper in chained transactions. Doppel \cite{Narula:OSDI14} added several explicitly commutative operations on records which they exploited to better handle common high-contention situations such as counters and "top-k lists" in the context of a single node multicore database. Finally, HyFlow \cite{Kim:EuroPar13}, a distributed transactional memory framework, reorders commutative operations on specific data types to execute before others to allow them to operate concurrently on a single version of a record.  ## Commutativity Specifications  Though *commutativity* is often discussed in terms of an operation commuting with all other operations, it is actually more nuanced. If a pair of operations commute, then executing them in either order will produce the same result. Using the definitions from \cite{Kulkarni:PLDI11}, whether or not a pair of method invocations commute is a property of the data type and is a function of the methods, their arguments, their return values, and the *abstract* state of their target. We call the full set of commutativity rules for a data type its *commutativity specification.* An example specification for a *set* is shown in Table \cite{tab:spec}. \ref{tab:spec}.  ## Transaction Boosting  If two operations in different transactions commute with each other, it means they could execute in any order,