I can write this: data Team player = ... Today, I can legally write newtype IP player = IP (Team player) instance Foldable IP where foldMap f (IP t) = only fold over players currently in play This would let me "mask" part of the team for folding purposes without actually extracting the part I want to consider. Your law forbids this. Under your regime, I am forced to make IP sufficiently heavy to hide the benched players from *everyone*. I understand that not every law has to be in the nature of a rewrite rule, but it's also true that not every law has to be so strong as to limit instances to a small handful of possible implementations. I think what I'm asking of you is reasonable: a situation where the law you propose would help prove a clearly useful program property. The law is getting more and more complicated to take GADTs and such into account. Such a complex law, in my opinion, needs a truly compelling purpose. On Feb 27, 2015 2:17 PM, "Gershom B" wrote:

On February 27, 2015 at 1:39:10 AM, David Feuer (david.feuer@gmail.com) wrote:

...

I am still struggling to understand why you want this to be a law for Foldable. It seems an interesting property of some Foldable instances, but, unlike Edward Kmett's proposed monoid morphism law, it's not clear to me how you can use this low to prove useful properties of programs. Could you explain?

I think there are a number of purposes for laws. Some can be thought of as “suggested rewrite rules” — and the monoid morphism law is one such, as are many related free approaches.

Note that the monoid morphism law that Edward provides is _not_ a “proposed” law — it is an “almost free theorem” — given a monoid morphism, it follows for free for any Foldable. There is no possible foldable instance that can violate this law, assuming you have an actual monoid morphism.

So Edward may have proposed adding it to the documentation (which makes sense to me) — but it provides absolutely no guidance or constraints as to what an “allowable” instance of Foldable is or is not.

But there are other reasons for laws than just to provide rewrite rules, even though it is often desirable to express laws in such terms. Consider the first Functor law for example — fmap id === id. Now clearly we can use it to go eliminate a bunch of “fmap id” calls in our program, should we have had them. But when would that ever be the case? Instead, the law is important because it _restricts_ the range of allowable instances — and so if you know you have a data type, and you know it has a functor instance, you then know what that functor instance must do, without looking at the source code.

In “An Investigation of the Laws of Traversals” from where we get our current Traversable laws, for example, Jaskelioff and Rypacek explain their motivation as establishing coherence laws that “capture the intuition of traversals” and rule out “bad” traversals (i.e. ones that violate our intuition). That’s the sort of goal I’m after — something that rules out “obviously bad” Foldable instances and lets us know something about the instances provided on structures without necessarily needing to read the code of those instances — and beyond what we can learn “for free” from the type.

Let me give an example. If you had a structure that represented a sequence of ‘a’, and you discovered that its Foldable instance only folded over the second, fifth, and seventeenth elements and no others — you would, I expect, find this to be a surprising result. You might even say “well, that’s a stupid Foldable instance”. However, outside of your gut feeling that this didn’t make sense, there would be no even semi-formal way to say it was “wrong”. Under a law such as I am trying to drive towards, we would be able to rule out such instances, just as our traversable laws now let us rule out instances that e.g. drop elements from the resultant structure. Hence, if you saw such a structure, and saw it had a Foldable instance, you would now know something useful about the behavior of that instance, without having to examine the implementation of Foldable — one of those useful things being, for example, that it could not possibly, lawfully, only fold over the fifth and seventeenth elements contained but no others.

(Now how to specify such a law that permits the _maximum_ amount of useful information while also permitting the _maximum_ number of instances that “match our intuition” is what my last post was driving at — I think I am closer, but certainly there remains work to be done).

In fact, we should note that when a Foldable is also Traversable, we do have a law specifying how the two typeclasses cohere, and because Traversable _does_ have strong laws, this _does_ suffice to well characterize Foldable in such a case. One nice property of the sort of law I have been driving at is that it will _agree_ with the current characterization in that same circumstance — when Foldable and Traversable instances both exist — and it will allow us to extend _those same intuitions_ in a formal way to circumstances when, for whatever reason, a Traversable instance does _not_ exist.

I hope this helps explain what I’m up to?

Cheers, Gershom

...

On Wed, Feb 25, 2015 at 5:40 PM, Gershom B wrote:

...

I think I can now float an amended law that is a further step in the

...

...

direction. First I will give the law, and then I will attempt to both motivate why the amended treatment is needed, and why the weakening of

...

...

initial proposal is a legitimate approach. Finally, I will discuss some remaining issues that it would be nice to examine, including some sketches of related but alternate approaches.

So, the new law:

=== for a lawful Foldable f, and

given a fresh newtype GenericSet = GenericSet Integer deriving (Eq, Ord), and mkGenericSet = GenericSet, where GenericSet is otherwise fully abstract

then

forall (g :: forall a. f a -> Maybe a), (x :: f GenericSet). maybe True (`Foldable.elem` x) (g x) === True ==

You may recall that the prior formulation asked that this condition hold for all a., (x :: f a), and so required an external notion of equality. Here, we ask that it only hold at a _specified type_ -- i.e. "GenericSet".

So, in what sense was the older formulation "too strong"? Here is a minimal case.

data TBox a where TAny :: a -> TBox a TString :: String -> TBox String

TBox is just a box with some optional type information, and it should be able to be equipped with the obvious Foldable instance. However, we can now write a function such as

discrim :: forall a. TBox a -> Maybe a discrim (TAny x) = Just x discrim (TString _) = "zoinks"

This is to say that if we _know_ that TBox contains a string, we can now take _any_ string out of it. Clearly, the existence of such functions means that the obvious Foldable instance cannot satisfy the stronger law.

By contrast, the straightforward Foldable instance for TBox still satisfies the weaker law, as we only test the action on "GenericSet" -- and since we have defined GenericSet as a fresh newtype, we know that it cannot have a tag to be matched on within a GADT we have already defined. Hence, our law, though it gives a universal property, does not give this property itself "universally" _at every type a_ but rather at a "generic type". This is by analogy with the technique of using "generic points" in algebraic geometry [1] to give properties that are not "everywhere true" but instead are

...

...

"almost everywhere", with a precise meaning given to "almost" -- i.e.

...

...

the space for which they do not hold _vanishes_.

This is to say, a Foldable satisfying the proposed law will not have

...

...

property hold "at all types," but rather at "almost all types" for an analogously precise meaning of "almost". I believe that a statement of why GenericSet is indeed "generic" in some topological sense is possible, although I have not fully fleshed this out.

To my knowledge, the introduction of this sort of approach in an FP context is new, but I would welcome any references showing prior art.

Aside from dealing with GADTs in some fashion, the new approach has a few other useful features. First, by formulating things directly in terms of "Foldable.elem" we omit the need for some sort of notion of equality in the metalogic -- instead we can use the "Eq" typeclass directly. Furthermore, this happens to sidestep the "toList" problem in the initial approach -- it directly allows search of potentially infinite structures.

There are still a few things that could potentially be better.

1) The "fully internal" approach means that we are now subject to the "bias" of "||" -- that is to say that _|_ || True -> _|_, but True || _|_ -> True. So we do not fully capture the searchability of infinite structures. This may indicate that a retreat to a metalogical description of "elem" with unbiased "or" is in order.

2) While we can generically characterize a Foldable instance on some f "almost everywhere" this is at the cost of giving it _no_ characterization at the points where our generic characterization fails. It would be nice to establish some sort of relation between the generic characterization and the action at specified points. However, I am not quite sure how to present this. An alternate approach could be to specify a Foldable law for any `f` that first takes `f` (which may be a GADT) to a related type `f1` (which must be an ordinary ADT) that squashes or omits dictionaries and equality constraints, and likewise takes the Foldable instance to a related instance on f1, and then provides a condition on f1. So rather that retreating from universal to generic properties, we instead take hold of the machinery of logical relations directly to establish a law. I would be interested in being pointed to related work along these lines as well.

3) An additional drawback of the "Generic Point" approach as given is

...

...

we chose to derive only two particular typeclasses -- Eq and Ord. An alternate approach would be to quantify over all a, but then give the property in terms of say "newtype Generify a = Generify a deriving (...)" which derives all classes on "a" that do not feature "a" in a positive position. Doing this would also mean a retreat from a fully internal notion of equality, of course...

Anyway, this is clearly still work in progress, but I would appreciate any feedback on the direction this is going, or references that may seem useful.

Cheers, Gershom

[1] https://en.wikipedia.org/wiki/Generic_point

On Thu, Feb 12, 2015 at 6:33 PM, Edward Kmett wrote:

...

With Foldable we do have a very nice law around foldMap.

A monoid homomorphism g is a combinator such that

g mempty = mempty g (mappend a b) = mappend (g a) (mappend (g b)

For any monoid homomorphism g,

foldMap (g . f) = g . foldMap f

We can use that to construct proofs of an analogue to "the banana

split

...

theorem" for foldr, but rephrased in terms of foldMap:

foldMap f &&& foldMap g = foldMap (f &&& g)

Getting there uses the fact that fst and snd are both monoid homomorphisms.

There are also laws relating the behavior of all of the other combinators in Foldable to foldMap.

Ultimately the reasons for the other members of the class are a sop to efficiency concerns: asymptotic factors in terms of time or stack usage matter.

-Edward

On Thu, Feb 12, 2015 at 5:22 PM, Kim-Ee Yeoh wrote:

...

(removing Libraries, since not everyone on HC is on that list)

I do not know all of the context of Foldable, but one I do know

...

...

...

...

not been mentioned is the implicit rule-of-thumb that every type class should have a law.

So the undercurrent is that if Foldable doesn't have a law, should it even be a type class? This has led to efforts to uncover laws for Foldable.

Worth discussing in a separate thread is the criterion itself of "if it doesn't have a law, it's not a type class". Useful sometimes, but

...

...

...

...

the sine qua non of type classes.

-- Kim-Ee

On Fri, Feb 13, 2015 at 2:47 AM, Gershom B wrote:

...

For a long time, many people, including me, have said that

"Foldable has

...

no laws" (or Foldable only has free laws) -- this is true, as it stands, with the exception that Foldable has a non-free law in interaction with Traversable (namely that it act as a proper specialization of Traversable methods). However, I believe that there is a good law we can give for Foldable.

I earlier explored this in a paper presented at IFL 2014 but (rightfully) rejected from the IFL post-proceedings. (http://gbaz.github.io/slides/buildable2014.pdf). That paper got

...

the

...

...

...

...

way there, but I believe now have a better approach on the question of a Foldable law -- as sketched below.

I think I now (unlike in the paper) can state a succinct law for Foldable that has desired properties: 1) It is not "free" -- it can be violated, and thus stating it adds semantic content. 2) We typically expect it to be true. 3) There are no places where I can see an argument for violating it.

If it pans out, I intend to pursue this and write it up more

...

...

...

...

...

but given the current FTP discussion I thought it was worth documenting this earlier rather than later. For simplicity, I will state this

...

...

...

...

...

terms of `toList` although that does not properly capture the infinite cases. Apologies for what may be nonstandard notation.

Here is the law I think we should discuss requiring:

* * * Given Foldable f, then forall (g :: forall a. f a -> Maybe a), (x :: f a). case g x of Just a --> a `elem` toList x * * *

Since we do not require `a` to be of type `Eq`, note that the `elem` function given here is not internal to Haskell, but in the

...

...

...

...

...

Furthermore, note that the use of parametricity here lets us make an "end run" around the usual problem of giving laws to Foldable --

rather than

...

providing an interaction with another class, we provide a claim about _all_ functions of a particular type.

Also note that the functions `g` we intend to quantify over are functions that _can be written_ -- so we can respect the property of data structures to abstract over information. Consider

data Funny a = Funny {hidden :: a, public :: [a]}

instance Foldable Funny where foldMap f x = foldMap f (public x)

Now, if it is truly impossible to ever "see" hidden (i.e. it is not exported, or only exported through a semantics-breaking "Internal" module), then the Foldable instance is legitimate. Otherwise, the Foldable instance is illegitimate by the law given above.

I would suggest the law given is "morally" the right thing for Foldable -- a Foldable instance for `f` should suggest that it gives us "all

right the true that the that that's that's not part of formally, property in metalogic. the as

...

...

...

...

...

in any `f a`", and so it is, in some particular restricted sense, initial among functions that extract as.

I do not suggest we add this law right away. However, I would like to suggest considering it, and I believe it (or a cleaned-up variant) would help us to see Foldable as a more legitimately lawful class that not only provides conveniences but can be used to aid reasoning.

Relating this to adjointness, as I do in the IFL preprint, remains future work.

Cheers, Gershom

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On February 27, 2015 at 5:55:42 PM, David Feuer (david.feuer@gmail.com) wrote:

Here's a better way to express what I mean: According to your law, if I write a newtype whose Foldable instance "masks" part of the underlying structure, then I must make the newtype abstract, and must not offer any way to remove the mask. This does not strike me as an intuitively obvious requirement.

Welp, it seems intuitively obvious to me :-) “a fold should fold over all the elements”. Using newtypes for masking seems a rather strange idiom that I haven’t encountered. Especially so if the _core_ use of the newtype is _just_ to pick a “nonstandard” Foldable instance. A clearer semantic thing to do is simply to have a function that produces the structure you want rather than masking it. If you want to capture “all” of the foldable surface, then you can, for example, use a filtered FMList. Or, you can just provide a particular, distinctly named function with the signature of foldr, foldMap, or whatever, that only operates on the desired subset. That’s in fact less “noisy” than using a newtype wrapper, and effectively as convenient. Or, for that matter, you can just use lens mechanisms to produce a Fold over that “view” on data. So there are many other, less verbose and more idiomatic ways to accomplish the “masking” you describe, and it seems to me to not restrict any essential expressive power for a law to forbid it. In return, we do indeed gain a useful reasoning principle, which I would suggest that most people intuitively expect even if we have not been able to state — that a Foldable instance for a structure can be understood without having to read the source or haddocks to confirm if it indeed does any strange masking or not. Cheers, Gershom

On Fri, Feb 27, 2015 at 5:35 PM, David Feuer wrote:

...

I can write this:

data Team player = ...

Today, I can legally write

newtype IP player = IP (Team player) instance Foldable IP where foldMap f (IP t) = only fold over players currently in play

This would let me "mask" part of the team for folding purposes without actually extracting the part I want to consider. Your law forbids this. Under your regime, I am forced to make IP sufficiently heavy to hide the benched players from *everyone*.

I understand that not every law has to be in the nature of a rewrite rule, but it's also true that not every law has to be so strong as to limit instances to a small handful of possible implementations. I think what I'm asking of you is reasonable: a situation where the law you propose would help prove a clearly useful program property. The law is getting more and more complicated to take GADTs and such into account. Such a complex law, in my opinion, needs a truly compelling purpose.

On Feb 27, 2015 2:17 PM, "Gershom B" wrote:

...

On February 27, 2015 at 1:39:10 AM, David Feuer (david.feuer@gmail.com) wrote:

...

I am still struggling to understand why you want this to be a law for Foldable. It seems an interesting property of some Foldable instances, but, unlike Edward Kmett's proposed monoid morphism law, it's not clear to me how you can use this low to prove useful properties of programs. Could you explain?

I think there are a number of purposes for laws. Some can be thought of as “suggested rewrite rules” — and the monoid morphism law is one such, as are many related free approaches.

Note that the monoid morphism law that Edward provides is _not_ a “proposed” law — it is an “almost free theorem” — given a monoid morphism, it follows for free for any Foldable. There is no possible foldable instance that can violate this law, assuming you have an actual monoid morphism.

So Edward may have proposed adding it to the documentation (which makes sense to me) — but it provides absolutely no guidance or constraints as to what an “allowable” instance of Foldable is or is not.

But there are other reasons for laws than just to provide rewrite rules, even though it is often desirable to express laws in such terms. Consider the first Functor law for example — fmap id === id. Now clearly we can use it to go eliminate a bunch of “fmap id” calls in our program, should we have had them. But when would that ever be the case? Instead, the law is important because it _restricts_ the range of allowable instances — and so if you know you have a data type, and you know it has a functor instance, you then know what that functor instance must do, without looking at the source code.

In “An Investigation of the Laws of Traversals” from where we get our current Traversable laws, for example, Jaskelioff and Rypacek explain their motivation as establishing coherence laws that “capture the intuition of traversals” and rule out “bad” traversals (i.e. ones that violate our intuition). That’s the sort of goal I’m after — something that rules out “obviously bad” Foldable instances and lets us know something about the instances provided on structures without necessarily needing to read the code of those instances — and beyond what we can learn “for free” from the type.

Let me give an example. If you had a structure that represented a sequence of ‘a’, and you discovered that its Foldable instance only folded over the second, fifth, and seventeenth elements and no others — you would, I expect, find this to be a surprising result. You might even say “well, that’s a stupid Foldable instance”. However, outside of your gut feeling that this didn’t make sense, there would be no even semi-formal way to say it was “wrong”. Under a law such as I am trying to drive towards, we would be able to rule out such instances, just as our traversable laws now let us rule out instances that e.g. drop elements from the resultant structure. Hence, if you saw such a structure, and saw it had a Foldable instance, you would now know something useful about the behavior of that instance, without having to examine the implementation of Foldable — one of those useful things being, for example, that it could not possibly, lawfully, only fold over the fifth and seventeenth elements contained but no others.

(Now how to specify such a law that permits the _maximum_ amount of useful information while also permitting the _maximum_ number of instances that “match our intuition” is what my last post was driving at — I think I am closer, but certainly there remains work to be done).

In fact, we should note that when a Foldable is also Traversable, we do have a law specifying how the two typeclasses cohere, and because Traversable _does_ have strong laws, this _does_ suffice to well characterize Foldable in such a case. One nice property of the sort of law I have been driving at is that it will _agree_ with the current characterization in that same circumstance — when Foldable and Traversable instances both exist — and it will allow us to extend _those same intuitions_ in a formal way to circumstances when, for whatever reason, a Traversable instance does _not_ exist.

I hope this helps explain what I’m up to?

Cheers, Gershom

...

...

I think I can now float an amended law that is a further step in the right direction. First I will give the law, and then I will attempt to both motivate why the amended treatment is needed, and why the weakening of the initial proposal is a legitimate approach. Finally, I will discuss some remaining issues that it would be nice to examine, including some sketches of related but alternate approaches.

So, the new law:

=== for a lawful Foldable f, and

given a fresh newtype GenericSet = GenericSet Integer deriving (Eq, Ord), and mkGenericSet = GenericSet, where GenericSet is otherwise fully abstract

then

forall (g :: forall a. f a -> Maybe a), (x :: f GenericSet). maybe True (`Foldable.elem` x) (g x) === True ==

You may recall that the prior formulation asked that this condition hold for all a., (x :: f a), and so required an external notion of equality. Here, we ask that it only hold at a _specified type_ -- i.e. "GenericSet".

So, in what sense was the older formulation "too strong"? Here is a minimal case.

data TBox a where TAny :: a -> TBox a TString :: String -> TBox String

TBox is just a box with some optional type information, and it should be able to be equipped with the obvious Foldable instance. However, we can now write a function such as

discrim :: forall a. TBox a -> Maybe a discrim (TAny x) = Just x discrim (TString _) = "zoinks"

This is to say that if we _know_ that TBox contains a string, we can now take _any_ string out of it. Clearly, the existence of such functions means that the obvious Foldable instance cannot satisfy the stronger law.

By contrast, the straightforward Foldable instance for TBox still satisfies the weaker law, as we only test the action on "GenericSet" -- and since we have defined GenericSet as a fresh newtype, we know that it cannot have a tag to be matched on within a GADT we have already defined. Hence, our law, though it gives a universal property, does not give this property itself "universally" _at every type a_ but rather at a "generic type". This is by analogy with the technique of using "generic points" in algebraic geometry [1] to give properties that are not "everywhere true" but instead are true "almost everywhere", with a precise meaning given to "almost" -- i.e. that the space for which they do not hold _vanishes_.

This is to say, a Foldable satisfying the proposed law will not have the property hold "at all types," but rather at "almost all types" for an analogously precise meaning of "almost". I believe that a statement of why GenericSet is indeed "generic" in some topological sense is possible, although I have not fully fleshed this out.

To my knowledge, the introduction of this sort of approach in an FP context is new, but I would welcome any references showing prior art.

Aside from dealing with GADTs in some fashion, the new approach has a few other useful features. First, by formulating things directly in terms of "Foldable.elem" we omit the need for some sort of notion of equality in the metalogic -- instead we can use the "Eq" typeclass directly. Furthermore, this happens to sidestep the "toList" problem in the initial approach -- it directly allows search of potentially infinite structures.

There are still a few things that could potentially be better.

1) The "fully internal" approach means that we are now subject to the "bias" of "||" -- that is to say that _|_ || True -> _|_, but True || _|_ -> True. So we do not fully capture the searchability of infinite structures. This may indicate that a retreat to a metalogical description of "elem" with unbiased "or" is in order.

2) While we can generically characterize a Foldable instance on some f "almost everywhere" this is at the cost of giving it _no_ characterization at the points where our generic characterization fails. It would be nice to establish some sort of relation between the generic characterization and the action at specified points. However, I am not quite sure how to present this. An alternate approach could be to specify a Foldable law for any `f` that first takes `f` (which may be a GADT) to a related type `f1` (which must be an ordinary ADT) that squashes or omits dictionaries and equality constraints, and likewise takes the Foldable instance to a related instance on f1, and then provides a condition on f1. So rather that retreating from universal to generic properties, we instead take hold of the machinery of logical relations directly to establish a law. I would be interested in being pointed to related work along these lines as well.

3) An additional drawback of the "Generic Point" approach as given is that we chose to derive only two particular typeclasses -- Eq and Ord. An alternate approach would be to quantify over all a, but then give the property in terms of say "newtype Generify a = Generify a deriving (...)" which derives all classes on "a" that do not feature "a" in a positive position. Doing this would also mean a retreat from a fully internal notion of equality, of course...

Anyway, this is clearly still work in progress, but I would appreciate any feedback on the direction this is going, or references that may seem useful.

Cheers, Gershom

[1] https://en.wikipedia.org/wiki/Generic_point

On Thu, Feb 12, 2015 at 6:33 PM, Edward Kmett wrote:

...

With Foldable we do have a very nice law around foldMap.

A monoid homomorphism g is a combinator such that

g mempty = mempty g (mappend a b) = mappend (g a) (mappend (g b)

For any monoid homomorphism g,

foldMap (g . f) = g . foldMap f

We can use that to construct proofs of an analogue to "the banana split theorem" for foldr, but rephrased in terms of foldMap:

foldMap f &&& foldMap g = foldMap (f &&& g)

Getting there uses the fact that fst and snd are both monoid homomorphisms.

There are also laws relating the behavior of all of the other combinators in Foldable to foldMap.

Ultimately the reasons for the other members of the class are a sop to efficiency concerns: asymptotic factors in terms of time or stack usage matter.

-Edward

On Thu, Feb 12, 2015 at 5:22 PM, Kim-Ee Yeoh wrote: > > (removing Libraries, since not everyone on HC is on that list) > > I do not know all of the context of Foldable, but one I do know > that's > not been mentioned is the implicit rule-of-thumb that every type > class > should have a law. > > So the undercurrent is that if Foldable doesn't have a law, should > it > even be a type class? This has led to efforts to uncover laws for > Foldable. > > Worth discussing in a separate thread is the criterion itself of "if > it > doesn't have a law, it's not a type class". Useful sometimes, but > that's not > the sine qua non of type classes. > > > > -- Kim-Ee > > On Fri, Feb 13, 2015 at 2:47 AM, Gershom B wrote: >> >> For a long time, many people, including me, have said that >> "Foldable has >> no laws" (or Foldable only has free laws) -- this is true, as it >> stands, >> with the exception that Foldable has a non-free law in interaction >> with >> Traversable (namely that it act as a proper specialization of >> Traversable >> methods). However, I believe that there is a good law we can give >> for >> Foldable. >> >> I earlier explored this in a paper presented at IFL 2014 but >> (rightfully) rejected from the IFL post-proceedings. >> (http://gbaz.github.io/slides/buildable2014.pdf). That paper got >> part of

On Wed, Feb 25, 2015 at 5:40 PM, Gershom B wrote: the

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>> way there, but I believe now have a better approach on the question >> of a >> Foldable law -- as sketched below. >> >> I think I now (unlike in the paper) can state a succinct law for >> Foldable that has desired properties: 1) It is not "free" -- it can >> be >> violated, and thus stating it adds semantic content. 2) We >> typically expect >> it to be true. 3) There are no places where I can see an argument >> for >> violating it. >> >> If it pans out, I intend to pursue this and write it up more >> formally, >> but given the current FTP discussion I thought it was worth >> documenting this >> earlier rather than later. For simplicity, I will state this >> property in >> terms of `toList` although that does not properly capture the >> infinite >> cases. Apologies for what may be nonstandard notation. >> >> Here is the law I think we should discuss requiring: >> >> * * * >> Given Foldable f, then >> forall (g :: forall a. f a -> Maybe a), (x :: f a). case g x of >> Just a >> --> a `elem` toList x >> * * * >> >> Since we do not require `a` to be of type `Eq`, note that the >> `elem` >> function given here is not internal to Haskell, but in the >> metalogic. >> >> Furthermore, note that the use of parametricity here lets us make >> an >> "end run" around the usual problem of giving laws to Foldable -- >> rather than >> providing an interaction with another class, we provide a claim >> about _all_ >> functions of a particular type. >> >> Also note that the functions `g` we intend to quantify over are >> functions that _can be written_ -- so we can respect the property >> of data >> structures to abstract over information. Consider >> >> data Funny a = Funny {hidden :: a, public :: [a]} >> >> instance Foldable Funny where >> foldMap f x = foldMap f (public x) >> >> Now, if it is truly impossible to ever "see" hidden (i.e. it is not >> exported, or only exported through a semantics-breaking "Internal" >> module), >> then the Foldable instance is legitimate. Otherwise, the Foldable >> instance >> is illegitimate by the law given above. >> >> I would suggest the law given is "morally" the right thing for >> Foldable >> -- a Foldable instance for `f` should suggest that it gives us "all >> the as >> in any `f a`", and so it is, in some particular restricted sense, >> initial >> among functions that extract as. >> >> I do not suggest we add this law right away. However, I would like >> to >> suggest considering it, and I believe it (or a cleaned-up variant) >> would >> help us to see Foldable as a more legitimately lawful class that >> not only >> provides conveniences but can be used to aid reasoning. >> >> Relating this to adjointness, as I do in the IFL preprint, remains >> future work. >> >> Cheers, >> Gershom >> >> _______________________________________________ >> Haskell-Cafe mailing list >> Haskell-Cafe@haskell.org >> http://www.haskell.org/mailman/listinfo/haskell-cafe >> > > > _______________________________________________ > Haskell-Cafe mailing list > Haskell-Cafe@haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe >

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On February 27, 2015 at 6:43:17 PM, David Feuer (david.feuer@gmail.com) wrote:

Suppose you convince me that hiding elements from foldMap is bad (I'm easy to persuade!). There is still, I believe, a much more serious problem. Specifically, you claim that your new law somehow takes care of potentially infinite containers, but I do not see how it does.   If the first child of the root is an infinite tree, your search will never reach the root of the second child! Generally, foldMap on any potentially infinite tree will be forced to work (at least approximately) breadth-first.

You are absolutely right that this is a more serious problem. I noted it under point 1) in my list of things that needed further thought or repair. In a sense, the fault is with the definition of `elem` which in turn is in terms of `||`, and the fact that things like `||` in Haskell (absent an “unamb” operator) need to be biased in their treatment of bottoms. At the time I suggested moving back to an `elem` in the metalogic to work around this. But I just thought of a much nicer repair! The law, as I had it, had (with some preconditions) forall (g :: forall a. f a -> Maybe a), (x :: f GenericSet). maybe True (`Foldable.elem` x) (g x) === True the repair is just to instead replace the expression with: maybe True (`Foldable.elem` x) (g x) =/= False By flipping the test, that should suffice to put us on the “right side” of domain semantics. (By the way, vis-a-vis hiding elements, even if we buy a “masking newtype” as a decent idiom, the requirement that such a newtype be _genuinely_ abstract if it is to admit Foldable seems a good one to enforce anyway. While it might not feel immediately evident why we have the requirement, the upshot is that it holds as a better, clearer abstraction, and again no expressive power is lost, since we don’t need to “unwrap” the type when we have original lying around to begin with). Cheers, Gershom

So consider. data Thing a = OneThing a instance Monoid a => Monoid (Thing a) where mempty = OneThing mempty… In this case, I think a) we can equip thing with a lawful “mappend” in the obvious fashion. b) there is no reason to expect that `toList mempty = []` either with or without a law such as I’m looking for. Cheers, Gershom On February 27, 2015 at 6:00:07 PM, Daniel Díaz (diaz.carrete@gmail.com) wrote:

Hi,

Sorry for the slight derail, but I wanted to ask the following doubt: if a Foldable type also happens to be a Monoid (say, like Set) does that automatically imply that toList mempty = [] ?

On Friday, February 27, 2015 at 8:18:05 PM UTC+1, Gershom B wrote:

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On February 27, 2015 at 1:39:10 AM, David Feuer (david...@gmail.com ) wrote:

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I am still struggling to understand why you want this to be a law for Foldable. It seems an interesting property of some Foldable instances, but, unlike Edward Kmett's proposed monoid morphism law, it's not clear to me how you can use this low to prove useful properties of programs. Could you explain?

I think there are a number of purposes for laws. Some can be thought of as “suggested rewrite rules” — and the monoid morphism law is one such, as are many related free approaches.

Note that the monoid morphism law that Edward provides is _not_ a “proposed” law — it is an “almost free theorem” — given a monoid morphism, it follows for free for any Foldable. There is no possible foldable instance that can violate this law, assuming you have an actual monoid morphism.

So Edward may have proposed adding it to the documentation (which makes sense to me) — but it provides absolutely no guidance or constraints as to what an “allowable” instance of Foldable is or is not.

But there are other reasons for laws than just to provide rewrite rules, even though it is often desirable to express laws in such terms. Consider the first Functor law for example — fmap id === id. Now clearly we can use it to go eliminate a bunch of “fmap id” calls in our program, should we have had them. But when would that ever be the case? Instead, the law is important because it _restricts_ the range of allowable instances — and so if you know you have a data type, and you know it has a functor instance, you then know what that functor instance must do, without looking at the source code.

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