deletions | additions       diff --git a/Commutativity.md b/Commutativity.md  index 1ab1735..d871953 100644  --- a/Commutativity.md  +++ b/Commutativity.md 
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- *discuss formal representations of commutativity from \cite{Herlihy:PPoPP08} and \cite{Kulkarni:PLDI11}, tradeoff between maximum concurrency and cost (runtime and implementation) of tracking it*   Commutativity is well known, especially in distributed systems, for enabling important optimizations. In classic databases literature from Since  the 70s and 80s, commutativity was has been  exploited by database systems designers designers, particularly Weihl et al.\cite{Weihl:1988,Fekete:90},  within the safe confines of relational models. They were able to use their deep, complete models, where  knowledge ofthe data model and  query plans and complete control of the data structures allows systems  to determine which when  transactions conflict with one another. *(citations?)* may conflict. Recently, commutativity has seen a resurgence in systems without a predefined data model, such as NoSQL databases and transactional memory.  Since then, In the realm of evental consistency,  commutativity has seen a resurgence in modern NoSQL systems which have forgone complex data models been leveraged  for increased flexibility and scalability. In an eventually consistent model, 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} (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. 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.