Algebraic patterns – Semigroup
philipnilsson.github.io
philipnilsson.github.io
http://mathworld.wolfram.com/Monoid.html
Most of his examples have an identity, giving a monoid.
* (string, concat): identity ''
* numbers with addition/multiplication: identity 0/1
* numbers with min/max: Numbers in computers are
finite, so this is quite useful. stdint.h defines
INT32_MAX, INT32_MIN, etc.
* booleans with && / || : true / false
* function composition: identity function f(x) = x
* frequency map: {} or {A: 0, B:0, ...} depending on
how you define it
Other examples I can think of:
* Python's dict with respect to member .update()
(if you think of it as an operator rather than
mutating member). Identity is {}
* Unix Pipes: identity 'cat' (note that the shell
supports point-free programming)http://math.stackexchange.com/questions/460/are-there-any-in...
If I take everyone's meanings correctly, what you describe (a monoid with the identity removed) is different from what cottonseed (https://news.ycombinator.com/item?id=12110936) describes, which is a monoid with the identity forgotten. That is, it's still there, but we don't require it to be there axiomatically.
Note that every monoid is a semigroup (because it satisfies a stronger collection of axioms), but not every monoid with identity removed is a semigroup (because it's possible to combine two non-identity elements to get the identity element): consider, for example, the integers without 0.
instance Monoid a => Monoid (b -> a) where
f <> g = \x -> f x <> g x
If you have two comparators, `lastNameComparator` and `firstNameComparator` and compose them into a comparator that compares first by firstName, then by lastName, surely that is a closed operation. It is also associative, and comparators do have an identity, so, yes they are also monoids.
Floating point numbers added in the wrong order can result in catastrophic cancellation. So for example, as far as JavaScript is concerned, numbers aren't associative.
Integers may overflow. Twos-complement addition is associative, but only because if you get the same bogus answer in the case of overflow. If you actually detect overflow, you may get overflow or not depending on the order in which you add the numbers.
Oftentimes we ignore these issues, but it seems like if you're talking about taking advantage of associative rules to reorder calculations, you have to think about it. Thinking in terms of semigroups instead of concrete datatypes seems like a likely source of bugs when your model makes assumptions that aren't actually true.
So, basically, semigroups and other abstractions are a-ok where they are typically used.
Note that both the full integers as well as the integers modulo n are groups under addition. As groups are more restrictive than semigroups, all groups are also semigroups (groups are semigroups that also have an identity element and where every element is invertible).
Integer multiplication also forms a semigroup (and in fact a monoid, due to identity), but not a full group due to zero not having an inverse. So, it's usually said that "the non-zero integers" form a group.
Wait, what? Very few integers have multiplicative inverses (in the integers). If you meant to speak of modular inverses, then even that is false for composite moduli like 2^n (with n > 1), which may be why im4w1 (https://news.ycombinator.com/item?id=12113934) asked if you meant to work modulo a prime.
"The form of parallelism induced by the associativity of the semigroup operation is heavily relied upon in the Map-Reduce programming model. We’ll expound on this in more detail in a later article, as this is more easily developed algebraically after defining the concept of a Monoid."
Although OP, philipnilsson is taking it step by step which I greatly appreciate as well
Note that CRDT stands for Conflict free Replicated Datatype and there are a few sub-acronym expansions: Convergent Replicated Datatype and Commutative Replicated Datatype.
[EDIT] I wanted to add that CRDTs were mentioned here because in order for them to provide the kind of guarantees they do they must satisfy the law of associativity (or, be a semigroup), commutativity, and idempotency. So, technically, you have to go further than semigroups with commutative semigroups. [/EDIT]
At Plum we made heavy use of CRDT's on-top of an eventually consistent whole-state replication technology called Plumtree, no relation to the company's name, just coincidence, within the internet-connected dimmer we built called the Lightpad.
The primary design goal was to enable advanced configurations: Arbitrary groups of Lightpads could be controlled from any other one on the same network by binding it to a specific finger gesture. We also did not want this configuration to be dependent upon any single master, they had to be truly master-less.
In-general, Eventual Consistency is pretty scary when you're deploying it to embedded internet-connected devices where you can't immediately shell onto the device like you can with a server in a data center. I have lots of stories here I'll write about sometime of the pains I encountered while developing this solution and how I ended up on CRDTs.
Strong Eventual Consistency (typically, CRDTs deployed on-top of an eventually consistent substrate), though, is very safe and provides a lot of guarantees about the concurrent behavior of the algorithm as it acts upon your data.
Using a CRDT (specifically the ORSWOT - Observed Removal Set Without Tombstones) eliminated the majority of our pains and concerns with data-loss, conflicting concurrent writes, and integrity in the face of flaky consumer-grade home networks (Plumtree is highly-available and partition tolerant - perfect for our needs where we have to be able to continue moving forward even if a majority or more of the Lightpads go offline and never come back up).
To elaborate more on your point, CRDTs are super helpful while trying to distribute a lot of devices, but they only get you so far. While building Lasp[0][1], a distributed data-flow language using CRDTs, we found out a lot of scalability problems with naive usage of CRDTs. We are aiming to reach 10k-20k nodes in the near future, so we are focusing a lot on reducing network usage.
State-based CRDTs send their full state to all their peers, which works ok when your states are small, but they introduce lots of overhead in any other case. Operation-based CRDTs only send the actions performed on it (add 1, or rmv 2, for example), but these are not idempotent and require a lot of guarantees from the underlying distribution backend.
We are focusing on using Delta CRDTs, that combine the low network usage of operation-based structures, with the idempotence of state-based approaches.
Using plumtree for your backend makes it resilient to network failures, but using the default full membership protocol makes it almost unusable when you're dealing with a big number of nodes. Using alternative protocols like Hyparview greatly reduces the number of protocol messages in your network.
Finally, since Lasp is a data-flow language, we are applying control-flow analysis to select and remove unused or intermediate values in the program, thus also reducing network usage.
[0]: http://lasp-lang.org/ [1]: https://github.com/lasp-lang/lasp
Our node cluster sizes are fairly manageable so we can tolerate the inefficiency right now but it will be nice to have a solution in-place when we want to optimize.
I did not know about HyParView. Thank you!
I've never read that paper before, it's really good. I like how the author's objective is to formalize map-side aggregations. But I'm wondering if that's a natural extension of map-reduce itself? ie. don't move data if it's not required given that's the constraint?
Do you have more of such papers handy?
As it is, it works more or less exactly like interfaces from other languages. When you implement Comparable in Java you implicitly expect certain laws (ie that it's a valid ordering), but there is also no way to specify or verify them.
I learned groups and fields first and then learned about the more relaxed versions. I'm not sure which approach is better though. Groups are certainly more fun to play with but monoids probably show up in more contexts.