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Serializable data types for eventually consistent systems.



Set union is commutative and convergent; hence it is always safe to have simultaneous writes to a set which only allows addition. You cannot remove an element of a G-Set.


  'type': 'g-set',
  'e': ['a', 'b', 'c']


2-phase sets consist of two g-sets: one for adding, and another for removing. To add an element, add it to the add set A. To remove e, add e to the remove set R. An element can only be added once, and only removed once. Elements can only be removed if they are present in the set. Removes take precedence over adds.


  'type': '2p-set',
  'a': ['a', 'b'],
  'r': ['b']

In this set, only 'a' is present.


LWW-Element-Set is like 2P-Set: it comprises an add G-Set (A) and a remove G-Set (R), with a timestamp for each element. To add an element e, add (e, timestamp) to the add set A. To remove e, add (e, timestamp) to the remove set R. An element is present iff it is in A, and no newer element exists in R. Merging is accomplished by taking the union of all A and all R, respectively.

Since the last write wins, we can safely take only the largest add, and the largest delete. All others can be pruned.

When A and R have equal timestamps, the direction of the inequality determines whether adds or removes win. {'bias': 'a'} indicates that adds win. {'bias': 'r'} indicates that removes win. The default bias is 'a'.

Timestamps may be any ordered primitive: integers, floats, strings, etc. If a coordinated unique timestamp service is used, LWW-Element-Set behaves like a traditional consistent Set structure. If non-unique timestamps are used, the resolution of the timestamp determines the window under which conflicts will be resolved by the bias towards adds or deletes.

TODO: define sorting strategies for strings. By byte value, UTF-8 ordering, numeric, etc...

In JSON, we write the set as a list of 2- or 3-tuples: [element, add-time] or [element, add-time, delete-time]


  'type': 'lww-e-set',
  'bias': 'a',
  'e': [
    ['a', 0],
    ['b', 1, 2],
    ['c', 2, 1],
    ['d', 3, 3]

In this set:

  • a was created at 0 and still exists.
  • b was deleted after creation; it does not exist.
  • c was created after deletion; it exists
  • d was created and deleted at the same time. Bias a means we prefer adds, so it exists.


Observed-Removed Sets support adding and removing elements in a causally consistent manner. It resembles LWW-Set, except that instead of times, unique tags are associated with each insertion or deletion. In the case of conflicting add and delete, add wins. An element is a member of the set iff the set of insertion tags less the set of deletion tags is nonempty.

We write the set as a list of 2- or 3- tuples: [element, [add-tags]] or [element, [add-tags], [remove-tags]]

To insert e, generate a unique tag, and add it to the insertion tag set for e.

To remove e, take all insertion tags for e, and insert them into the deletion tags for e.

To merge two OR-Sets, for each element in either set, take the union of the insertion tags and the union of the deletion tags.

Tags may be any primitive: strings, ints, floats, etc.


  'type': 'or-set',
  'e': [
    ['a', [1]],
    ['b', [1], [1]],
    ['c', [1, 2], [2, 3]]
  • a exists.
  • b's only insertion was deleted, so it does not exist.
  • c has two insertions, only one of which was deleted. It exists.


MC-Sets resolve divergent histories for an element by choosing the value which has changed the most. You cannot delete an element which is not present, and cannot add an element which is already present. MC-sets are compact and do the right thing when changes to elements are infrequent compared to the conflict resolution window, but behave arbitrarily when divergent histories each include many changes.

Each element e is associated with an integer n, implicitly assumed to be zero. When n is even, the element is absent from the set. When n is odd, the element is present. To add an element to the set, increment n from an even value by one; to remove an element, increment n from an odd value by one. To merge sets, take each element and choose the maximum value of n from each history.

When n is limited to [0, 2], Max-Change-Sets collapse to 2P-Sets. Unlike 2P-Sets, however, one can add and remove an arbitrary number of times. The disadvantage is that there is no bias towards preserving adds or removes. Instead, whichever history has incremented further (undergone more changes) is preferred.

In JSON, max-change sets are represented as a list of [element, n] tuples.


  'type': 'mc-set',
  'e': [
    ['a', 1],
    ['b', 2],
    ['c', 3]
  • a is present
  • b is absent
  • c is present



A G-Counter is a grow-only counter (inspired by vector clocks) in which only increment and merge are possible. Incrementing the counter adds 1 to the count for the current actor. Divergent histories are resolved by taking the maximum count for each actor (like a vector clock merge). The value of the counter is the sum of all actor counts.


  'type': 'g-counter',
  'e': {
    'a': 1,
    'b': 5,
    'c': 2
  • The counter value is 8.


PN-Counters allow the counter to be decreased by tracking the increments (P) separate from the decrements (N), both represented as internal G-Counters. Merge is handled by merging the internal P and N counters. The value of the counter is the value of the P counter minus the value of the N counter.


  'type': 'pn-counter',
  'p': {
    'a': 10,
    'b': 2
  'n': {
    'c': 5,
    'a': 1
  • P=12, N=6, so the value is 6.
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