Suppose you have a full binary tree and you do an operation on every node, where the cost of the operation is proportional to the height of that node. That is, the cost for each of the leaves is , for each of the nodes in the next level up the cost is , and so on. We can visualize the scenario like this:
As a function of the total number of nodes , how expensive is this? We can see that is an upper bound, since there are nodes and the height of each node is at most . But it seems like it might actually be faster than this in reality, since, intuitively, most of the nodes have a height which is much smaller than .
(One specific motivation for this scenario is that we can build a binary heap from an arbitrary set of data by looping over the nodes from the bottom up and calling reheapDown
on each; in the worst case reheapDown
takes time proportional to the height of the node, as in this scenario. But it doesn’t matter if you don’t know about binary heaps.)
Let’s take the same tree and put a dollar at every node, for a total of :
Now imagine sliding all the money as far up and to the right as it will go. That is, we take each dollar, and keep moving it up as long as it is a left child. As soon as we reach a node which is a right child we stop. The tree ends up looking like this:
Now take each pile of money and move it up one step to its parent, except the money at the root of the tree, which you can put in your pocket.
And voilà! We now have exactly enough money at each node to pay for the cost of the operations, and we even have a bit left over (which we can use to buy coffee). But we started with and only shuffled money around; this shows that the total cost is actually .
Exercise for the reader: what does this have to do with the number of bit flips needed to count from to with a binary counter?
If you missed seeing me at ICFP, this is why.
In honor of my son’s birth (he will need to learn the alphabet and Haskell soon)—and at the instigation of Kenny Foner—I revived the Haskell Alphabet by converting it to modern Hakyll and updating some of the broken or outdated links. Some of it is a bit outdated (I wrote it seven years ago), but it’s still a fun little piece of Haskell history. Enjoy!
I am always on the lookout for more exercises to add and for more links to interesting further reading. If you know of a cool exercise or a cool paper or blog post that helps explain/illustrate/apply a standard Haskell type class, please let me know (or just add it yourself, it’s a wiki!). And, of course, the same goes if you notice any errors or confusing bits.
Happy Haskelling!
Feel free to use any of the lecture notes, assignments, or even exam questions. I didn’t leave the exams linked by accident; I use an exam format where the students have a week or so to prepare solutions to the exam, using any resources they want, and then have to come in on exam day and write down their solutions without referring to anything (I got this idea from Scott Weinstein). So leaving the exams posted publically on the web isn’t a problem for me.
Please don’t ask for solutions; I won’t give any, even if you are an instructor. But questions, comments, bug reports, etc. are most welcome.
The workshop will be co-located with ICFP in Oxford, and is devoted to the use of “static type information…used effectively in the development of computer programs”, construed broadly (see the CFP for more specific examples of what is in scope). Last year’s workshop drew a relativey large crowd and had a lot of really great papers and talks, and I expect this year to be no different! Andrew Kennedy (Facebook UK) will also be giving an invited keynote talk.
Please consider submitting something! We are looking for both full papers as well as two-page extended abstracts reporting work in progress. The submission deadline for regular papers is 24 May, and 7 June for extended abstracts.
My deep work sessions are still going strong, though having this opportunity to reflect has been good: I realized that over the months I have become more lax about using my computer and about what sort of things I am willing to do during my “deep work” sessions. It’s too easy to let them become just a block of time I can use to get done all the urgent things I think I need to get done. Of course, sometimes there are truly urgent things to get done, and having a big block of time to work on them can be great, especially if they require focused effort. But it pays to be more intentional about using the deep work blocks to work on big, long-term projects. The myriad little urgent things will get taken care of some other time, if they’re truly important (or maybe they won’t, if they’re not).
Since I’m only teaching two classes this semester, both of which I have taught before, I thought I would have more time for deep work sessions this semester than last, but for some reason it seems I have less. I’m not yet sure whether there’s something I could have done about that, or if the semester just looks different than I expected. This semester has also seen more unavoidable conflicts with my deep work blocks. Usually, I try to keep my scheduled deep work blocks sacrosanct, but I have made some exceptions this semester: for example, search committee meetings are quite important and also extremely difficult to schedule, so I let them be scheduled over top of my deep work blocks if necessary. (But it sure does wreak havoc on my work that week.)
I’m also still blocking my email before 4pm. On the one hand, I know this is helping a lot with my productivity and general level of anxiety. Recently I needed to (or thought I needed to!) briefly unblock my email during the day to check whether I had received a certain reply, and I specifically noticed how my anxiety level shot up as soon as I opened my inbox and saw all the messages there—a good reminder of why I have my email blocked in the first place. On the other hand, it can be frustrating, since the hour from 4-5 is often taken up with other things, so email gets pushed to the evening, or to the next day. When this goes on several days in a row it really doesn’t help my anxiety level to know there are emails sitting there that I ought to respond to. So perhaps there might be a better time to process my email than 4-5, but to be honest I am not sure what it would be. I certainly don’t want to do it first thing in the morning, and the middle of the day is not really any better, schedule-wise, than the end. In any case, I intend to keep doing it until a better idea comes along.
When are two species isomorphic? Since species are, by definition, functors , the obvious answer is “when they are isomorphic as functors”, that is, when there is a natural isomorphism between them.
Let’s unpack definitions a bit. Recall that is the groupoid (i.e. category with all morphisms invertible) of finite sets and bijections, and is the category of finite sets and total functions.
Given two functors , a natural isomorphism is some family of bijections such that for any bijection , the following square commutes:
Think of as a relabelling—that is, a 1-1 correspondence between labels in and labels in . By functoriality of and , we can lift to relabel whole structures, via and . The whole square then says that the family of correspondences “commute with relabelling”—that is, intuitively, can’t “look at” the labels at all, because it has to “do the same thing” even when we change the labels out from under it. It operates based solely on the structure present in the – or -structures—which by functoriality must be independent of the labels—and not based on the identity or structure of the set of labels itself.
All this generalizes readily to virtual species: recall that virtual species consist of pairs of species , where we define to be equivalent to if and only if is (naturally) isomorphic to . That is, isomorphism of virtual species already has natural isomorphism of regular species baked into it by definition.
Now, recall the signed involution from my previous post. Since it depends on a chosen linear order, it is not a natural isomorphism: an arbitrary relabelling certainly need not preserve the ordering on the labels, so the involution is not preserved under relabelling.
This brings up the natural (haha) question: is it possible to give a natural isomorphism between signed sets and signed ballots? If so, we would certainly prefer it to this involution that makes use of a linear order. But sadly, the answer, it turns out, is no. Let range from to . We are looking for a natural isomorphism
in the case that is odd, or
when is even.
Notice that in any case, whether positive or negative, the -structure will be fixed by any relabelling which is a permutation (because permuting the elements of a set does not change the set). Hence, any natural isomorphism must send it to some structure which is also fixed by all permutations. But the only such ballot structure is the one consisting of a single part containing all the elements. This ballot has an odd number of parts, so there cannot be a natural isomorphism in the case that is even—we would have to match the structure with some even-sized ballot which is fixed by all permutations of the labels, but there are none. Hence there is no natural isomorphism, which by definition would have to work for all .
But what about for odd ? Can we have a natural isomorphism between the structures restricted to the case when is odd? In fact, no: we can make a more general argument that applies for any . Consider the different ballots consisting of separate singleton parts. Each of these is fixed by no permutations other than the identity. So, under a natural isomorphism they would all have to map to ballots of the opposite parity which are also fixed by no permutations other than the identity: but by the Pigeonhole Principle, any ballot with fewer than parts must have at least one part containing more than one element, and will therefore be fixed under any permutation that only touches the elements in that part.
In this situation—when there is a 1-1 correspondence between species, but not a natural one—we say the species are equipotent but not isomorphic. The species have the same number of structures of each size, and hence exactly the same generating function, but they are not (naturally) isomorphic: it is possible to tell them apart by looking at how structures behave under relabelling. Another classic example of this phenomenon is the species of lists and the species of permutations: each has exactly labelled structures of size , but they are not naturally isomorphic. Lists have no symmetry at all, that is, they are fixed by no permutations other than the identity, but permutations, in general, do have some symmetry. For example, any permutation is fixed when the labels are permuted by itself: the permutation is the same as the permutation . The classic combinatorial proof that there are the same number of lists and permutations also uses an extra linear order on the labels.
A classic lemma in combinatorics states that any nonempty finite set has the same number of even and odd subsets. In fact, I recently wrote a proof of this on my other blog. Since it’s written for a more general audience, it’s spelled out there in quite a lot of detail; but if you understand the idea of signed involutions, the classic combinatorial proof is quite simple to state: given a set , pick some , and define a signed involution on subsets of (signed according to the parity of their size) by “toggling” the presence of . That is, given , if then take it out, and if then add it in. This is clearly an involution and sends even subsets to odd and vice versa.
However, this involution is not natural—it depends on the choice of .^{1} Is it possible to prove it via a natural correspondence? In fact, that’s one of the things Anders Claesson’s original post—the one that got me started down this whole rabbit hole—was trying to do. Unfortunately, hidden in the middle of his proof was an assumed correspondence between signed sets and signed ballots, and as we’ve seen, this correspondence actually cannot be proved naturally. (I should reiterate that in no way do I mean to disparage his post—it’s still a great post with a lot of cool insights, not to mention a nice framework for thinking about multiplicative inverses of species. It just doesn’t quite accomplish one of the things he thought it was accomplishing!)
Now, at this point, all we know is that the particular argument used in that post is not, in fact, natural. But that doesn’t necessarily mean a natural correspondence between even and odd subsets is impossible. However, it turns out that it is (mostly) impossible: we can give a more direct argument that, in fact, there is no natural proof—that is, the species of odd subsets and the species of even subsets are equipotent but not naturally isomorphic.
The proof is quite simple, and along similar lines as the proof for signed sets and ballots. Note that the empty subset is fixed by all permutations, as is the maximal subset—and these are clearly the only such subsets. So any natural correspondence must match these subsets with each other—but when is even they have the same parity.
What about if we restrict to the case when is odd? Unlike the case of signed sets and ballots, it turns out that we actually can give a natural proof in this case! The involution to use is the one that simply sends any to its complement . This is clearly an involution, and since is odd, it reverses the parity as required. It also does not depend at all on the elements of or any assumed structure on them, that is, it commutes perfectly well with relabelling.
This corresponds nicely with an observation we can make about Pascal’s triangle: it is easy to see that the alternating sum of any odd row is , for example, , since the entries are all duplicated, with one positive and one negative version of each. However, for even rows it is not quite so obvious: why is ? To show this cancels to give 0 we must “split up” some of the numbers so that one part of them cancels with one number and another part cancels with another; that is, we cannot necessarily treat all the subsets of a given size uniformly. But subsets of a given size are completely indistinguishable up to relabelling—hence the necessity of some extra structure to allow us to make the necessary distinctions.
So, how should such a proof look? For a given number of labels , there is a single signed set structure, which is just the set of labels itself (with a sign depending on the parity of ). On the other hand, there are lots of ballots on labels; the key is that some are positive and some are negative, since the sign of the ballots depends on the parity of the number of parts, not the number of labels. For example, consider . There is a single (negative) signed set structure:
(I will use a dashed blue line to indicate negative things, and a solid black line for positive things.)
On the other hand, as we saw last time, there are 13 ballot structures on 3 labels, some positive and some negative:
In this example, it is easy to see that most of the positives and negatives cancel, with exactly one negative ballot left over, which corresponds with the one negative set. As another example, when , there is a single positive set, and 75 signed ballots:
This time it is not quite so easy to tell at a glance (at least not the way I have arranged the ballots in the above picture!), but in fact one can verify that there are exactly 37 negative ballots and 38 positive ones, again cancelling to match the one positive set.
What we need to show, then, is that we can pair up the ballots in such a way that positive ballots are matched with negative ballots, with exactly one ballot of the appropriate sign left to be matched with the one signed set. This is known as a signed involution: an involution is a function which is its own inverse, so it matches things up in pairs; a signed involution sends positive things to negative things and vice versa, except for any fixed points.
In order to do this, we will start by assuming the set of labels is linearly ordered. In one sense this is no big deal, since for any finite set of labels we can always just pick an arbitrary ordering, if there isn’t an “obvious” ordering to use already. On the other hand, it means that the correspondence will be specific to the chosen linear ordering. All other things being equal, we would prefer a correspondence that depends solely on the structure of the ballots, and not on any structure inherent to the labels. I will have quite a bit more to say about this in my third and (probably) final post on the topic. But for today, let’s just see how the correspondence works, given the assumption of a linear order on the labels. I came up with this proof independently while contemplating Anders Claesson’s post, though it turns out that the exact same proof is already in a paper by Claesson and Hannah (in any case it is really just a small lemma, the sort of thing you might give as a homework problem in an undergraduate course on combinatorics).
Given some ballot, find the smallest label. For example, if the labels are as in the examples so far, we will find the label .
If the smallest label is contained in some part together with at least one other label, separate it out into its own part by itself, and put it to the right of its former part. Like this:
On the other hand, if the smallest label is in a part by itself, merge it with the part on the left (if one exists). This is clearly the inverse of the above operation.
The only case we haven’t handled is when the smallest label is in a part by itself which is the leftmost part in the ballot. In that case, we leave that part alone, switch to considering the second-smallest label, and recursively carry out the involution on the remainder of the ballot.
For example:
In this case we find the smallest label (1) in a part by itself in the leftmost position, so we leave it where it is and recurse on the remainder of the ballot. Again, we find the smallest remaining label (2) by itself and leftmost, so we recurse again. This time, we find the smallest remaining label (3) in a part with one other label, so we separate it out and place it to the right.
This transformation on ballots is clearly reversible. The only ballots it doesn’t change are ballots with each label in its own singleton part, sorted from smallest to biggest, like this:
In this case the algorithm recurses through the whole ballot and finds each smallest remaining label in the leftmost position, ultimately doing nothing. Notice that a sorted ballot of singletons has the same sign as the signed set on the same labels, namely, . In any other case, we can see that the algorithm matches positive ballots to negative and vice versa, since it always changes the number of parts by 1, either splitting one part into two or merging two parts into one.
Here’s my implementation of the involution in Haskell:
type Ballot = [[Int]]
ballotInv :: Ballot -> Ballot
ballotInv = go 1
where
go _ [] = []
go s ([a]:ps)
| s == a = [a] : go (s+1) ps
go s (p:ps)
| s `elem` p = delete s p : [s] : ps
go s (p:[a]:ps)
| s == a = sort (a:p) : ps
go s (p:ps) = p : go s ps
(The call to sort
is not strictly necessary, but I like to keep each part canonically sorted.)
Here again are the 13 signed ballots for , this time arranged so that the pair of ballots in each row correspond to each other under the involution, with the leftover, sorted ballot by itself at the top.
If you’d like to see an illustration of the correspondence for , you can find it here (I didn’t want to include inline since it’s somewhat large).
This completes the proof that signed sets and signed ballots correspond. But did we really need that linear order on the labels? Tune in next time to find out!
Recall that denotes the species of sets, defined by , that is, the only -structure on a given label set is the set of labels itself. Recall also that the exponential generating function of a species is given by
where counts the number of labelled -structures of size . In the case of , we have for all , so
(This is why is such a good name for the species of sets—though in a fantastic coincidence, it seems to originally come from the French word for set, ensemble, rather than from the fact that (though on the other hand calling it a “coincidence” is probably too strong, since Joyal must surely have picked the notation with the generating function already in mind!).)
Now, from my previous post we know that
Let’s first consider (without the ). This means that we have, for some , a -ary product of structures—in other words, a list of nonempty sets. This is the species of ballots, also known as ordered set partitions, and can also be written . As an example, here is a ballot on the set of labels :
The order of the parts matters, so this is a different ballot:
But the order of labels within each part doesn’t matter (since each part is a set). As another example, here is the complete collection of ballot structures on the labels :
We can see that there are 13 in total: six where the labels are each in their own separate part (corresponding to the six possible permutations of the labels); six where two labels share a part and the other label is a singleton part (corresponding to the three ways to choose the solitary label, times the two ways to order the parts); and one final ballot where all three labels are grouped in the same part. (As an exercise, can you verify that there are 75 different ballot structures on a set of four labels?)
Returning to , we can see that it consists of signed ballots, where the sign of a ballot is the parity of its number of parts, that is, a ballot with parts has sign . The second half of Anders’ post gives a nice combinatorial proof that , via a sign-reversing involution: if we consider -structures, i.e. pairs of sets and signed ballots, there is a natural^{1} way to pair them up, matching positive and negative structures so everything cancels (except in the case of the empty label set, which is why we get instead of ).
However, Anders is trying to do more than that. Note first that since multiplication of EGFs corresponds to multiplication of species, the EGF for is of course . But this ought to also be the EGF for the virtual species , and the rest of his post hinges on identifying and . As Anders and I discovered, however, this is precisely the point where it is worth being more careful.
First of all, what is ? Intuitively, an structure consists of a set of negative atoms; since each set can be thought of as an (unordered) product of atoms, the whole set acquires a sign given by the parity of the number of atoms. In other words, intuitively it seems that should be the species of signed sets, where an even-sized set is considered positive and an odd-sized set negative. That is,
where denotes the species of sets of size exactly . As a sanity check, this makes sense as an EGF equation too, since the EGF of is just and indeed
But hold on a minute, what does really mean, formally? It is the composition of the species with the virtual species , and it turns out that it is not at all a priori obvious how to define composition for virtual species! We can find the definition on p. 127 of Bergeron et al. A special case (which is enough for our present purposes) is
where and are two sorts of atoms, and denotes Cartesian product of species. In our case,
since is the identity for Cartesian product (overlaying an additional structure on a set of labels does not add any structure, since there is only one possible -structure).
All of this is to say, is actually defined as ! So at first glance it may seem we actually have nothing to prove: and are the same by definition, end of story. …but in fact, all we have done is shift the burden of proof elsewhere: now it is our intuitive idea of representing signed sets that requires proof!
To sum up, we know that is the species of signed ballots, with sign given by parity of the number of parts; and intuitively, we also believe that should correspond to parity-signed sets, . So, is there a nice combinatorial proof showing the correspondence between signed sets and signed ballots?
One can use the law of excluded middle to show that the answer must be “yes”: suppose the answer were “no”; but then I would not be writing this blog post, which is a contradiction since I am writing this blog post. But is there a constructive proof? Fear not! This blog post has gotten long enough, so I will stop here for now and let interested readers puzzle over it; in my next post I will explain what I came up with, along with some musings on linear orders and naturality.
Recall that the intuitive idea of a combinatorial species is a family of labelled structures which are invariant under relabelling. If you’ve never seen the formal definition before, don’t worry: just think “data structures” or “algebraic data types” for now.
The basic idea of virtual species is to work with pairs of species where is considered “positive” and “negative”. Formally, we consider equivalence classes of such pairs under the equivalence relation defined by iff .^{1} This parallels the way one typically gives a formal definition of the integers starting from the natural numbers (the “Grothendieck construction”); see my previous post for more details.
How can we build intuition for virtual species, and for additive inverses of species in particular? To be honest I have been struggling with this question for many years.
Multiplicative inverses are much simpler to think about: they are like matter and antimatter. Having both an -structure and an structure is the same as having nothing; they annihilate each other. By “having nothing” we mean “having no information”, that is, having a unit value: .
What about additive inverses? Note first that the species does not correspond to having nothing; the word “nothing” corresponds to the (i.e. unit) species. Instead the (i.e. uninhabited) species corresponds to (logical) impossibility. So to interpret we have to imagine something where having either or is impossible.
…yeah, me neither. This seems deeply strange. If someone says, “I either have an or a ”, you can confidently call them a liar, because it is impossible to have either an or a ; that is, . But surely if you actually have an -structure, it should also be true to say “I have either an or a ”? Well, that works for normal, positive species—in which case we can define a canonical injection . But once we introduce negative species this completely breaks down. As another example, if someone truthfully says, “I have either a tree or a negative non-empty tree”, you should be able to say, “Aha! I know what you have—it must be an empty tree.” In general, it’s strange that expressing a disjunction can cause some possibilities to be ruled out. Normally, we are used to disjunctions only increasing the number of possibilities.
Inspired by James and Sabry’s really cool paper The Two Dualities of Computation: Negative and Fractional Types, I have thought a bit about whether there is some plausible interpretation involving travelling backwards in time, but I haven’t been able to come up with one. I can’t quite seem to make the formalism of the paper match up with my intuition about species (though this may just be a failure of my imagination or intuition).
In any case, let’s see why the ring of virtual species actually has multiplicative inverses—at least, all the ones we could possibly hope for. This is somewhat surprising, since when we build integers from natural numbers by considering equivalence classes of pairs, we certainly don’t get any multiplicative inverses, only additive ones. To get multiplicative inverses we have to do the same process a second time, building the rational numbers as equivalence classes of pairs of integers. But species already have enough extra structure that throwing in additive inverses is all it takes.
First, a caveat: we don’t get multiplicative inverses for all species, but only those species such that : that is, species with only a single structure of size zero, which are of the form . With any constant term other than , we clearly have no hope of finding another species such that , since the constant term of will be a multiple of ’s constant term.
So given such a , write , where denotes “non-empty -structures”. Then we can define the multiplicative inverse of as follows:
That is, a -structure consists of a list of nonempty -structures, except that even-length lists are considered “positive” and odd-length lists considered “negative”.
We can easily check that this indeed defines a multiplicative inverse for :
The infinite sums telescope down to leave only . Notice this really isn’t about species in particular, but really about infinite power series (of which species are the categorification): any infinite power series with integer coefficients and a constant term of has a multiplicative inverse which is also such a power series.
As an example, consider . We know this is “supposed” to be the species of lists (since it results from solving for ), but let’s see what happens. In this case and . So the inverse ought to be
And hey, look at that! Lists!
So what would we need to get a true field, i.e. a multiplicative inverse for every nonzero species? Well, for that we would need to throw in rational coefficients. I forget exactly where I read this—some paper by Baez and Dolan, most likely—but I believe the proper way to interpret this would be as groupoid-valued species, since there is a sense in which the “cardinality” of groupoids can be interpreted as rational numbers. But to be honest I have no idea where this leads.