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JadeNB: This works but I believe only in the case with undirected graphs and if the graphs being connected do not have common vertices. So, indeed there is a s

JadeNB: This works but I believe only in the case with undirected graphs and if the graphs being connected do not have common vertices. So, indeed there is a sort of de Morgan law: a -- b = !(!a + !b) where ! is the graph edge-complement, a and b do not have common vertices, and -- is a commutative version of ->. (Responding so late, because I was also "submitting too fast"!)

JadeNB 10y ago
> 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) https://en.wikipedia.org/wiki/Clique_(graph_theory)
- sn0wleopard 10y ago
  > 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.
  - JadeNB 10y ago
    > 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?
    - sn0wleopard 10y ago
      > 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.

JadeNB 10y ago

> 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) https://en.wikipedia.org/wiki/Clique_(graph_theory)

sn0wleopard 10y ago
> 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.
- JadeNB 10y ago
  > 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?
  - sn0wleopard 10y ago
    > 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.