With an obvious notion of negation of graphs, wouldn't the `connect` operation be \g1 g2 -> not $ (not g1) `overlay` (not g2)? (I've already asked about lattice theory elsewhere (https://news.ycombinator.com/item?id=13125334 ); this thought makes me wonder if there's any connection to modal logic https://en.wikipedia.org/wiki/Modal_logic, where modalities are often defined in these sort of "de Morgan pairs".)
EDIT: I originally typo'd and proposed the above as a definition of the `overlay` operation, which would be both a pointless circularity and wrong.
> JadeNB: This works but I believe only in the case with undirected graphs and if the graphs being connected do not have common vertices.
Agreed; I thought that your `overlay` was the "(forced) disjoint union". I had missed that "(vertex x) `overlay` (vertex y)" was not isomorphic to "(vertex x) `overlay` (vertex x)". I don't even know what negation would mean for directed graphs.
(By the way, your code doesn't seem to give any way of producing inhabitants of `Vertex g`. Since the displayed code only ever uses `Vertex g` in order to promote it immediately to `Graph g`, does it make sense just to have them as a separate type? (Maybe I'm misreading; it's been a while since I've Haskell'd.) Also, am I wrong to read `vertex` as a kind of `return`? (I hope that I am not wrong to think that graphs should serve as just as good instances of monads as lists, if not better.))
While we're talking: thanks for this article; I enjoyed it a lot. My one suggestion is that I think that what you call a "clique" is usually called a "complete graph", with "clique" (implicitly, in G) reserved for subgraphs of G that happen to be complete. https://en.wikipedia.org/wiki/Clique_(graph_theory)
> By the way, your code doesn't seem to give any way of producing inhabitants of `Vertex g`.
In fully polymorphic code (like clique) you indeed can't produce new inhabitants, or do anything else with the values of type `Vertex g`. However, if you know something about `Vertex g` then you can do certain things. For example, if we know that the type of vertices has `Ord` instance then we can compare them (as we do in `Relation`). And in very concrete code, e.g. operating on `Basic Int` values, you can create new inhabitants freely -- any number will do.
> Also, am I wrong to read `vertex` as a kind of `return`?
It is indeed very much like `return` or `pure`! We use it to inject a type into a sort of container type, in this case, into a graph type.
> I hope that I am not wrong to think that graphs should serve as just as good instances of monads as lists, if not better
Many `Graph` instances are `Monad`s. For example, `Basic` is a `Monad` (and many other things too). But on the other hand `Relation` is not a `Monad`, because it imposes the `Ord` constraint on the underlying type (and Haskell monads are not allowed that).
> While we're talking: thanks for this article; I enjoyed it a lot.
Thank you! I didn't expect so much interest to it, so I'm a bit overwhelmed to be honest :)
> My one suggestion is that I think that what you call a "clique" is usually called a "complete graph", with "clique" (implicitly, in G) reserved for subgraphs of G that happen to be complete.
Thanks, indeed "clique" may be not the best name... I decided to use it, because I would like to reserve "complete graph" to mean something like `clique [1..]` i.e. the complete graph on the whole universe. If we look from this perspective then something like `clique [1..10]` may look like a clique within some bigger graph. Anyway, I admit this may be confusing.
> Many `Graph` instances are `Monad`s. For example, `Basic` is a `Monad` (and many other things too). But on the other hand `Relation` is not a `Monad`, because it imposes the `Ord` constraint on the underlying type (and Haskell monads are not allowed that).
Indeed, I did say that it was a monad and not a `Monad`; math doesn't suffer from these implementation constraints. :-)
> Thanks, indeed "clique" may be not the best name... I decided to use it, because I would like to reserve "complete graph" to mean something like `clique [1..]` i.e. the complete graph on the whole universe. If we look from this perspective then something like `clique [1..10]` may look like a clique within some bigger graph. Anyway, I admit this may be confusing.
Oh, that makes more sense. Maybe it's worth even making just a throwaway remark to that effect?
> Indeed, I did say that it was a monad and not a `Monad`; math doesn't suffer from these implementation constraints. :-)
Ah, I see :-) Indeed, from the mathematical standpoint we can both inject a vertex into the graph type with `vertex`, and flatten a graph whose nodes are graphs (the monad `join`, not to be confused with graph `join` operation that I'm using). I haven't given this much thought before, thanks for noticing this.
> Oh, that makes more sense. Maybe it's worth even making just a throwaway remark to that effect?
I've added the following remark: "we will call such cliques covering the whole universe complete graphs". Hopefully this helps to clarify the naming convention.
EDIT: I originally typo'd and proposed the above as a definition of the `overlay` operation, which would be both a pointless circularity and wrong.