# The Universe of Discourse

Sun, 04 Dec 2022
• I revised my chart of Haskell's numbers to include a few missing things, uncrossed some of the arrows, and added an explicit public domain notice,

The article contained a typo, a section titled “Shuff that don't work so good”. I decided this was a surprise gift from the Gods of Dada, and left it uncorrected.

• Similar to my suggested emoji for U.S. presidents, a Twitter user suggested emoji for UK prime ministers, some of which I even understand.

I added some discussion of why I did not use a cat emoji for President Garfield. A reader called January First-of-May suggested a tulip for Dutch-American Martin Van Buren, which I gratefully added.

• In my article on adaptive group testing, Sam Dorfman and I wondered if there wasn't earlier prior art in the form of coin-weighing puzzles. M. January brought to my attention that none is known! The earliest known coin-weighing puzzles date back only to 1945. See the article for more details.

• Some time ago I wrote an article on “What was wrong with SML?”. I said “My sense is that SML is moribund” but added a note back in April when a reader (predictably) wrote in to correct me.

However, evidence in favor of my view appeared last month when the Haskell Weekly News ran their annual survey, which included the question “Which programming languages other than Haskell are you fluent in?”, and SML was not among the possible choices. An oversight, perhaps, but a rather probative one.

• I wondered if my earlier article was the only one on the Web to include the phrase “wombat coprolites”. It wasn't.

This complaint is a little stale, but maybe it will still be interesting. A while back I was traveling to California on business several times a year, and the company I worked for required that I use SAP Concur expense management software to submit receipts for reimbursement.

At one time I would have had many, many complaints about Concur. But today I will make only one. Here I am trying to explain to the Concur phone app where my expense occurred, maybe it was a cab ride from the airport or something.

I had to interact with this control every time there was another expense to report, so this is part of the app's core functionality.

There are a lot of good choices about how to order this list. The best ones require some work. The app might use the phone's location feature to figure out where it is and make an educated guess about how to order the place names. (“I'm in California, so I'll put those first.”) It could keep a count of how often this user has chosen each location before, and put most commonly chosen ones first. It could store a list of the locations the user has selected before and put the previously-selected ones before the ones that had never been selected. It could have asked, when the expense report was first created, if there was an associated location, say “California”, and then and then used that to put California places first, then United States places, then the rest. It could have a hardwired list of the importance of each place (or some proxy for that, like population) and put the most important places at the top.

The actual authors of SAP Concur's phone app did none of these things. I understand. Budgets are small, deadlines are tight, product managers can be pigheaded. Sometimes the programmer doesn't have the resources to do the best solution.

But this list isn't even alphabetized.

There are two places named Los Alamos; they are not adjacent. There are two places in Spain; they are also not adjacent. This is inexcusable. There is no resource constraint that is so stringent that it would prevent the programmers from replacing

    displaySelectionList(matches)


with

    displaySelectionList(matches.sorted())


They just didn't.

And then whoever reviewed the code, if there was a code review, didn't say “hey, why didn't you use displaySortedSelectionList here?”

And then the product manager didn't point at the screen and say “wouldn't it be better to alphabetize these?”

And the UX person, if there was one, didn't raise any red flag, or if they did nothing was done.

I don't know what Concur's software development and release process is like, but somehow it had a complete top-to-bottom failure of quality control and let this shit out the door.

I would love to know how this happened. I said a while back:

Assume that bad technical decisions are made rationally, for reasons that are not apparent.

I think this might be a useful counterexample. And if it isn't, if the individual decision-makers all made choices that were locally rational, it might be an instructive example on how an organization can be so dysfunctional and so filled with perverse incentives that it produces a stack of separately rational decisions that somehow add up to a failure to alphabetize a pick list.

### Addendum : A possible explanation

Dennis Felsing, a former employee of SAP working on their HANA database, has suggested how this might have come about. Suppose that the app originally used a database that produced the results already sorted, so that no sorting in the client was necessary, or at least any omitted sorting wouldn't have been noticed. Then later, the backend database was changed or upgraded to one that didn't have the autosorting feature. (This might have happened when Concur was acquired by SAP, if SAP insisted on converting the app to use HANA instead of whatever it had been using.)

This change could have broken many similar picklists in the same way. Perhaps there was large and complex project to replace the database backend, and the unsorted picklist were discovered relatively late and were among the less severe problems that had to be overcome. I said “there is no resource constraint that is so stringent that it would prevent the programmers from (sorting the list)”. But if fifty picklists broke all at the same time for the same reason? And you weren't sure where they all were in the code? At the tail end of a large, difficult project? It might have made good sense to put off the minor problems like unsorted picklists for a future development cycle. This seems quite plausible, and if it's true, then this is not a counterexample of “bad technical decisions are made rationally for reasons that are not apparent”. (I should add, though, that the sorting issue was not fixed in the next few years.)

In the earlier article I said “until I got the correct explanation, the only explanation I could think of was unlimited incompetence.” That happened this time also! I could not imagine a plausible explanation, but M. Felsing provided one that was so plausible I could imagine making the decision the same way myself. I wish I were better at thinking of this kind of explanation.

Sun, 27 Nov 2022

One day I asked Lorrie if she thought that Schroeder actually grew up to be a famous concert pianist. We agreed that he probably did. Or at least Schroeder has as good a chance as anyone does. To become a famous concert pianist, you need to have talent and drive. Schroeder clearly has talent (he can play all that Beethoven and Mozart on a toy piano whose black keys are only painted on) and he clearly has drive. Not everyone with talent and drive does succeed, of course, but he might make it, whereas some rando like me has no chance at all.

That led to a longer discussion about what became of the other kids. Some are easier than others. Who knows what happens to Violet, Sally, (non-Peppermint) Patty, and Shermy? I imagine Violet going into realty for some reason.

As a small child I did not understand that Lucy's “psychiatric help 5¢” lemonade stand was hilarious, or that she would have been the literally worst psychiatrist in the world. (Schulz must have known many psychiatrists; was Lucy inspired by any in particular?) Surely Lucy does not become an actual psychiatrist. The world is cruel and random, but I refuse to believe it is that cruel. My first thought for Lucy was that she was a lawyer, perhaps a litigator. Now I like to picture her as a union negotiator, and the continual despair of the management lawyers who have to deal with her.

Her brother Linus clearly becomes a university professor of philosophy, comparative religion, Middle-Eastern medieval literature, or something like that. Or does he drop out and work in a bookstore? No, I think he's the kind of person who can tolerate the grind of getting a graduate degree and working his way into a tenured professorship, with a tan corduroy jacket with patches on the elbows, and maybe a pipe.

Peppermint Patty I can imagine as a high school gym teacher, or maybe a yoga instructor or massage therapist. I bet she'd be good at any of those. Or if we want to imagine her at the pinnacle of achievement, coach of the U.S. Olympic softball team. Marcie is calm and level-headed, but a follower. I imagine her as a highly competent project manager.

In the conversation with Lorrie, I said “But what happens to Charlie Brown?”

“No, why?”

“To everyone's great surprise, Charlie Brown grows up to be a syndicated cartoonist and a millionaire philanthropist.”

Of course she was right. Charlie Brown is good ol' Charlie Schulz, whose immense success suprised everyone, and nobody more than himself.

Charles M. Schulz was born 100 years ago last Saturday.

[ Addendum 20221204: I forgot Charlie Brown's sister Sally. Unfortunately, the vibe I get from Sally is someone who will be sucked into one of those self-actualization cults like Lifespring or est. ]

Sat, 26 Nov 2022

I was delighted to learn some time ago that there used to be giant wombats, six feet high at the shoulders, unfortunately long extinct.

It's also well known (and a minor mystery of Nature) that wombats have cubical poop.

Today I wondered, did the megafauna wombat produce cubical megaturds? And if so, would they fossilize (as turds often do) and leave ten-thousand-year-old mineral cubescat littering Australia? And if so, how big are these and where can I see them?

A look at Intestines of non-uniform stiffness mold the corners of wombat feces (Yang et al, Soft Matter, 2021, 17, 475–488) reveals a nice scatter plot of the dimensions of typical wombat scat, informing us that for (I think) the smooth-nosed (common) wombat:

• Length: 4.0 ± 0.6 cm
• Height: 2.3 ± 0.3 cm
• Width: 2.5 ± 0.3 cm

Notice though, not cubical! Clearly longer than they are thick. And I wonder how one distinguishes the width from the height of a wombat turd. Probably the paper explains, but the shitheads at Soft Matter want £42.50 plus tax to look at the paper. (I checked, and Alexandra was not able to give me a copy.)

Anyway the common wombat is about 40 cm long and 20 cm high, while the extinct giant wombats were nine or ten times as big: 400 cm long and 180 cm high, let's call it ten times. Then a propportional giant wombat scat would be a cuboid approximately 24 cm (9 in) wide and tall, and 40 cm (16 in) long. A giant wombat poop would be as long as… a wombat!

But not the imposing monoliths I had been hoping for.

Yang also wrote an article Duration of urination does not change with body size, something I have wondered about for a long time. I expected bladder size (and so urine quantity) to scale with the body volume, the cube of the body length. But the rate of urine flow should be proportional to the cross-sectional area of the urethra, only the square of the body length. So urination time should be roughly proportional to body size. Yang and her coauthors are decisive that this is not correct:

we discover that all mammals above 3 kg in weight empty their bladders over nearly constant duration of 21 ± 13 s.

What is wrong with my analysis above? It's complex and interesting:

This feat is possible, because larger animals have longer urethras and thus, higher gravitational force and higher flow speed. Smaller mammals are challenged during urination by high viscous and capillary forces that limit their urine to single drops. Our findings reveal that the urethra is a flow-enhancing device, enabling the urinary system to be scaled up by a factor of 3,600 in volume without compromising its function.

Wow. As Leslie Orgel said, evolution is cleverer than you are.

However, I disagree with the conclusion: 21±13 is not “nearly constant duration”. This is a range of 8–34s, with some mammals taking four times as long as others.

The appearance of the fibonacci numbers here is surely coincidental, but wouldn't it be awesome if it wasn't?

[ Addendum: I wondered if this was the only page on the web to contain the bigram “wombat coprolites”, but Google search produced this example from 2018:

Have wombats been around for enough eons that there might be wombat coprolites to make into jewelry? I have a small dinosaur coprolite that is kind of neat but I wouldn't make that turd into a necklace, it looks just like a piece of poop.

]

Tue, 08 Nov 2022

I haven't done one of these in a while. And there have been addenda. I thought hey, what if I ask Git to give me a list of commits from October that contain the word ‘Addendum’. And what do you know, that worked pretty well. So maybe addenda summaries will become a regular thing again, if I don't forget by next month.

Most of the addenda resulted in separate followup articles, which I assume you will already have seen. ([1] [2] [3]) I will not mention this sort of addendum in future summaries.

• In my discussion of lazy search in Haskell I had a few versions that used do-notation in the list monad, but eventually abandoned it n favor of explicit concatMap. For example:

      s nodes = nodes ++ (s $concatMap childrenOf nodes)  I went back to see what this would look like with do notation:  s nodes = (nodes ++) . s$ do
n <- nodes
childrenOf n


Meh.

• Regarding the origin of the family name ‘Hooker’, I rejected Wiktionary's suggestion that it was an occupational name for a maker of hooks, and speculated that it might be a fisherman. I am still trying to figure this out. I asked about it on English Language Stack Exchange but I have not seen anything really persuasive yet. One of the answers suggests that it is a maker of hooks, spelled hocere in earlier times.

(I had been picturing wrought-iron hooks for hanging things, and wondered why the occupational term for a maker of these wasn't “Smith”. But the hooks are supposedly clothes-fastening hooks, made of bone or some similar finely-workable material. )

The OED has no record of hocere, so I've asked for access to the Dictionary of Old English Corpus of the Bodleian library. This is supposedly available to anyone for noncommercial use, but it has been eight days and they have not yet answered my request.

I will post an update, if I have anything to update.

Fri, 04 Nov 2022

I keep getting lost in the maze of Haskell's numeric types. Here's the map I drew to help myself out. (I think there might have been something like this in the original Haskell 1998 report.)

(PNG version) (Original DOT file (The SVG above is hand-edited graphviz output))

Ovals are typeclasses. Rectangles are types. Black mostly-straight arrows show instance relationships. Most of the defined functions have straightforward types like !!\alpha\to\alpha!! or !!\alpha\to\alpha\to\alpha!! or !!\alpha\to\alpha\to\text{Bool}!!. The few exceptions are shown by wiggly colored arrows.

## Basic plan

After I had meditated for a while on this picture I began to understand the underlying organization. All numbers support !!=!! and !!\neq!!. And there are three important properties numbers might additionally have:

• Ord : ordered; supports !!\lt\leqslant\geqslant\gt!! etc.
• Fractional : supports division
• Enum: supports ‘pred’ and ‘succ’

Integral types are both Ord and Enum, but they are not Fractional because integers aren't closed under division.

Floating-point and rational types are Ord and Fractional but not Enum because there's no notion of the ‘next’ or ‘previous’ rational number.

Complex numbers are numbers but not Ord because they don't admit a total ordering. That's why Num plus Ord is called Real: it's ‘real’ as constrasted with ‘complex’.

## More stuff

That's the basic scheme. There are some less-important elaborations:

Real plus Fractional is called RealFrac.

Fractional numbers can be represented as exact rationals or as floating point. In the latter case they are instances of Floating. The Floating types are required to support a large family of functions like !!\log, \sin,!! and π.

You can construct a Ratio a type for any a; that's a fraction whose numerators and denominators are values of type a. If you do this, the Ratio a that you get is a Fractional, even if a wasn't one. In particular, Ratio Integer is called Rational and is (of course) Fractional.

## Shuff that don't work so good

Complex Int and Complex Rational look like they should exist, but they don't really. Complex a is only an instance of Num when a is floating-point. This means you can't even do 3 :: Complex Int — there's no definition of fromInteger. You can construct values of type Complex Int, but you can't do anything with them, not even addition and subtraction. I think the root of the problem is that Num requires an abs function, and for complex numbers you need the sqrt function to be able to compute abs.

Complex Int could in principle support most of the functions required by Integral (such as div and mod) but Haskell forecloses this too because its definition of Integral requires Real as a prerequisite.

You are only allowed to construct Ratio a if a is integral. Mathematically this is a bit odd. There is a generic construction, called the field of quotients, which takes a ring and turns it into a field, essentially by considering all the formal fractions !!\frac ab!! (where !!b\ne 0!!), and with !!\frac ab!! considered equivalent to !!\frac{a'}{b'}!! exactly when !!ab' = a'b!!. If you do this with the integers, you get the rational numbers; if you do it with a ring of polynomials, you get a field of rational functions, and so on. If you do it to a ring that's already a field, it still works, and the field you get is trivially isomorphic to the original one. But Haskell doesn't allow it.

I had another couple of pages written about yet more ways in which the numeric class hierarchy is a mess (the draft title of this article was "Haskell's numbers are a hot mess") but I'm going to cut the scroll here and leave the hot mess for another time.

[ Addendum: Updated SVG and PNG to version 1.1. ]

Mon, 31 Oct 2022

Content warning: something here to offend almost everyone

A while back I complained that there were no emoji portraits of U.S. presidents. Not that there a Chester A. Arthur portrait would see a lot of use. But some of the others might come in handy.

I couldn't figure them all out. I have no idea what a Chester Arthur emoji would look like. And I assigned 🧔🏻 to all three of Garfield, Harrison, and Hayes, which I guess is ambiguous but do you really need to be able to tell the difference between Garfield, Harrison, and Hayes? I don't think you do. But I'm pretty happy with most of the rest.

 George Washington 💵 John Adams Thomas Jefferson 📜 James Madison James Monroe John Quincy Adams 🍐 Andrew Jackson Martin Van Buren 🌷 William Henry Harrison 🪦 John Tyler James K. Polk Zachary Taylor Millard Fillmore ⛽ Franklin Pierce James Buchanan Abraham Lincoln 🎭 Andrew Johnson 💩 Ulysses S. Grant 🍸 Rutherford B. Hayes 🧔🏻 James Garfield 🧔🏻 Chester A. Arthur Grover Cleveland 🔂 Benjamin Harrison 🧔🏻 Grover Cleveland 🔂 William McKinley Theodore Roosevelt 🧸 William Howard Taft 🛁 Woodrow Wilson 🎓 Warren G. Harding 🫖 Calvin Coolidge 🙊 Herbert Hoover ⛺ Franklin D. Roosevelt 👨‍🦽 Harry S. Truman 🍄 Dwight D. Eisenhower 🪖 John F. Kennedy 🍆 Lyndon B. Johnson 🗳️ Richard M. Nixon 🐛 Gerald R. Ford 🏈 Jimmy Carter 🥜 Ronald Reagan 💸 George H. W. Bush 👻 William J. Clinton 🎷 George W. Bush 👞 Barack Obama 🇰🇪 Donald J. Trump 🍊 Joseph R. Biden 🕶️

Honorable mention: Benjamin Franklin 🪁

Dishonorable mention: J. Edgar Hoover 👚

If anyone has better suggestions I'm glad to hear them. Note that I considered, and rejected 🎩 for Lincoln because it doesn't look like his actual hat. And I thought maybe McKinley should be 🏔️ but since they changed the name of the mountain back I decided to save it in case we ever elect a President Denali.

(Thanks to Liam Damewood for suggesting Harding, and to Colton Jang for Clinton's saxophone.)

[ Addendum 20221106: Twitter user Simon suggests emoji for UK prime ministers. ]

[ Addendum 20221108: Rasmus Villemoes makes a good suggestion of 😼 for Garfield. I had considered this angle, but abandoned it because there was no way to be sure that the cat would be orange, overweight, or grouchy. Also the 🧔🏻 thing is funnier the more it is used. But I had been unaware that there is CAT FACE WITH WRY SMILE until M. Villemoes brought it to my attention, so maybe. (Had there been an emoji resembling a lasagna I would have chosen it instantly.) ]

[ Addendum 20221108: January First-of-May has suggested 🌷 for Maarten van Buren, a Dutch-American whose first language was not English but Dutch. Let it be so! ]

Sun, 30 Oct 2022

A while back, discussing Vladimir Putin (not putain) I said

In English we don't seem to be so quivery. Plenty of people are named “Hoare”. If someone makes a joke about the homophone, people will just conclude that they're a boor.

Today I remembered Frances Trollope and her son Anthony Trollope. Where does the name come from? Surely it's not occupational?

Happily no, just another coincidence. According to Wikipedia it is a toponym, referring to a place called Troughburn in Northumberland, which was originally known as Trolhop, “troll valley”. Sir Andrew Trollope is known to have had the name as long ago as 1461.

According to the Times of London, Joanna Trollope, a 6th-generation descendant of Frances, once recalled

a night out with a “very prim and proper” friend who had the surname Hoare. The friend was dismayed by the amusement she caused in the taxi office when she phoned to book a car for Hoare and Trollope.

I guess the common name "Hooker" is occupational, perhaps originally referring to a fisherman.

[ Frances Trollope previously on this blog: [1] [2] ]

[ Addendum: (Wiktionary says that Hooker is occupational, a person who makes hooks. I find it surprising that this would be a separate occupattion. And what kind of hooks? I will try to look into this later. ]

Sun, 23 Oct 2022

Yesterday I described an algorithm that locates the ‘bad’ items among a set of items, and asked:

does this technique have a name? If I wanted to tell someone to use it, what would I say?

The answer is: this is group testing, or, more exactly, the “binary splitting” version of adaptive group testing, in which we are allowed to adjust the testing strategy as we go along. There is also non-adaptive group testing in which we come up with a plan ahead of time for which tests we will perform.

I felt kinda dumb when this was pointed out, because:

• A typical application (and indeed the historically first application) is for disease testing
• My previous job was working for a company doing high-throughput disease testing
• I found out about the job when one of the senior engineers there happened to overhear me musing about group testing
• Not only did I not remember any of this when I wrote the blog post, I even forgot about the disease testing application while I was writing the post!

Oh well. Thanks to everyone who wrote in to help me! Let's see, that's Drew Samnick, Shreevatsa R., Matt Post, Matt Heilige, Eric Harley, Renan Gross, and David Eppstein. (Apologies if I left out your name, it was entirely unintentional.)

Is the history of this algorithm lost in time, or do we know who first invented it, or at least wrote it down?

The concept of group testing was first introduced by Robert Dorfman in 1943 in a short report published in the Notes section of Annals of Mathematical Statistics. Dorfman's report – as with all the early work on group testing – focused on the probabilistic problem, and aimed to use the novel idea of group testing to reduce the expected number of tests needed to weed out all syphilitic men in a given pool of soldiers.

Eric Harley said:

[It] doesn't date back as far as you might think, which then makes me wonder about the history of those coin weighing puzzles.

Yeah, now I wonder too. Surely there must be some coin-weighing puzzles in Sam Loyd or H.E. Dudeney that predate Dorfman?

Dorfman's original algorithm is not the one I described. He divides the items into fixed-size groups of n each, and if a group of n contains a bad item, he tests the n items individually. My proposal was to always split the group in half. Dorfman's two-pass approach is much more practical than mine for disease testing, where the test material is a body fluid sample that may involve a blood draw or sticking a swab in someone's nose, where the amount of material may be limited, and where each test offers a chance to contaminate the sample.

Wikipedia has an article about a more sophisticated of the binary-splitting algorithm I described. The theory is really interesting, and there are many ingenious methods.

Thanks to everyone who wrote in. Also to everyone who did not. You're all winners.

[ Addendum 20221108: January First-of-May has brought to my attention section 5c of David Singmaster's Sources in Recreational Mathematics, which has notes on the known history of coin-weighing puzzles. To my surprise, there is nothing there from Dudeney or Loyd; the earliest references are from the American Mathematical Monthly in 1945. I am sure that many people would be interested in further news about this. ]

Fri, 21 Oct 2022

A year or two ago I wrote about what you do if you already have a Monad and you need to define an Applicative instance for it. This comes up in converting old code that predates the incorporation of Applicative into the language: it has these monad instance declarations, and newer compilers will refuse to compile them because you are no longer allowed to define a Monad instance for something that is not an Applicative. I complained that the compiler should be able to infer this automatically, but it does not.

My current job involves Haskell programming and I ran into this issue again in August, because I understood monads but at that point I was still shaky about applicatives. This is a rough edit of the notes I made at the time about how to define the Applicative instance if you already understand the Monad instance.

pure is easy: it is identical to return.

Now suppose we have >>=: how can we get <*>? As I eventually figured out last time this came up, there is a simple solution:

    fc <*> vc = do
f <- fc
v <- vc
return $f v  or equivalently:  fc <*> vc = fc >>= \f -> vc >>= \v -> return$ f v


And in fact there is at least one other way to define it is just as good:

    fc <*> vc = do
v <- vc
f <- fc
return $f v  (Control.Applicative.Backwards provides a Backwards constructor that reverses the order of the effects in <*>.) I had run into this previously and written a blog post about it. At that time I had wanted the second <*>, not the first. The issue came up again in August because, as an exercise, I was trying to implement the StateT state transformer monad constructor from scratch. (I found this very educational. I had written State before, but StateT was an order of magnitude harder.) I had written this weird piece of code:  instance Applicative f => Applicative (StateT s f) where pure a = StateT$ \s -> pure (s, a)
stf <*> stv = StateT $\s -> let apf = run stf s apv = run stv s in liftA2 comb apf apv where comb = \(s1, f) (s2, v) -> (s1, f v) -- s1? s2?  It may not be obvious why this is weird. Normally the definition of <*> would look something like this:  stf <*> stv = StateT$
\s0 ->  let (s1, f) = run stf s0
let (s2, v) = run stv s1
in (s2, f v)


This runs stf on the initial state, yielding f and a new state s1, then runs stv on the new state, yielding v and a final state s2. The end result is f v and the final state s2.

Or one could just as well run the two state-changing computations in the opposite order:

  stf <*> stv = StateT $\s0 -> let (s1, v) = run stv s0 let (s2, f) = run stf s1 in (s2, f v)  which lets stv mutate the state first and gives stf the result from that. I had been unsure of whether I wanted to run stf or stv first. I was familiar with monads, in which the question does not come up. In v >>= f you must run v first because you will pass its value to the function f. In an Applicative there is no such dependency, so I wasn't sure what I neeeded to do. I tried to avoid the question by running the two computations ⸢simultaneously⸣ on the initial state s0:  stf <*> stv = StateT$
\s0 ->  let (sf, f) = run stf s0
let (sv, v) = run stv s0
in (sf, f v)


Trying to sneak around the problem, I was caught immediately, like a small child hoping to exit a room unseen but only getting to the doorway. I could run the computations ⸢simultaneously⸣ but on the very next line I still had to say what the final state was in the end: the one resulting from computation stf or the one resulting from computation stv. And whichever I chose, I would be discarding the effect of the other computation.

My co-worker Brandon Chinn opined that this must violate one of the applicative functor laws. I wasn't sure, but he was correct. This implementation of <*> violates the applicative ”interchange” law that requires:

    f <*> pure x  ==  pure ($x) <*> f  Suppose f updates the state from !!s_0!! to !!s_f!!. pure x and pure ($ x), being pure, leave it unchanged.

My proposed implementation of <*> above runs the two computations and then updates the state to whatever was the result of the left-hand operand, sf discarding any updates performed by the right-hand one. In the case of f <*> pure x the update from f is accepted and the final state is !!s_f!!. But in the case of pure (\$ x) <*> f the left-hand operand doesn't do an update, and the update from f is discarded, so the final state is !!s_0!!, not !!s_f!!. The interchange law is violated by this implementation.

(Of course we can't rescue this by yielding (sv, f v) in place of (sf, f v); the problem is the same. The final state is now the state resulting from the right-hand operand alone, !!s_0!! on the left side of the law and !!s_f!! on the right-hand side.)

### Stack Overflow discussion

I worked for a while to compose a question about this for Stack Overflow, but it has been discussed there at length, so I didn't need to post anything:

That first thread contains this enlightening comment:

• Functors are generalized loops

[ f x | x <- xs];

• Applicatives are generalized nested loops

[ (x,y) | x <- xs, y <- ys];

• Monads are generalized dynamically created nested loops

[ (x,y) | x <- xs, y <- k x].

That middle dictum provides another way to understand why my idea of running the effects ⸢simultaneously⸣ was doomed: one of the loops has to be innermost.

The second thread above (“How arbitrary is the ap implementation for monads?”) is close to what I was aiming for in my question, and includes a wonderful answer by Conor McBride (one of the inventors of Applicative). Among other things, McBride points out that there are at least four reasonable Applicative instances consistent with the monad definition for nonempty lists. (There is a hint in his answer here.)

Another answer there sketches a proof that if the applicative ”interchange” law holds for some applicative functor f, it holds for the corresponding functor which is the same except that its <*> sequences effects in the reverse order.

Thu, 20 Oct 2022

Last week I was in the kitchen and Katara tried to tell Toph a secret she didn't want me to hear. I said this was bad opsec, told them that if they wanted to exchange secrets they should do it away from me, and without premeditating it, I uttered the following:

You shouldn't talk about things you shouldn't talk about while I'm in the room while I'm in the room.

I suppose this is tautological. But it's not any sillier than Tarski's observation that "snow is white" is true exactly if snow is white, and Tarski is famous.

I've been trying to think of more examples that really work. The best I've been able to come up with is:

You shouldn't eat things you shouldn't eat because they might make you sick, because they might make you sick.

I'm trying to decide if the nesting can be repeated. Is this grammatical?

You shouldn't talk about things you shouldn't talk about things you shouldn't talk about while I'm in the room while I'm in the room while I'm in the room.

I think it isn't. But if it is, what does it mean?

Wed, 19 Oct 2022

Consider this problem:

Input: A set !!S!! of items, of which an unknown subset, !!S_{\text{bad}}!!, are ‘bad’, and a function, !!\mathcal B!!, which takes a subset !!S'!! of the items and returns true if !!S'!! contains at least one bad item:

$$\mathcal B(S') = \begin{cases} \mathbf{false}, & \text{if S'\cap S_{\text{bad}} = \emptyset} \\ \mathbf{true}, & \text{otherwise} \\ \end{cases}$$

Think of a boxful of electronic components, some of which are defective. You can test any subset of components simultaneously, and if the test succeeds you know that each of those components is good. But if the test fails all you know is that at least one of the components was bad, not how many or which ones.

The obvious method is simply to test the components one at a time:

$$S_{\text{bad}} = \{ x\in S \mid \mathcal B(\{x\}) \}$$

This requires exactly !!|S|!! calls to !!\mathcal B!!.

But if we expect there to be relatively few bad items, we may be able to do better:

• Call !!\mathcal B(S)!!. That is, test all the components at once. If none is bad, we are done.
• Otherwise, partition !!S!! into (roughly-equal) halves !!S_1!! and !!S_2!!, and recurse.

In the worst case this takes (nearly) twice as many calls as just calling !!\mathcal B!! on the singletons. But if !!k!! items are bad it requires only !!O(k\log |S|)!! calls to !!\mathcal B!!, a big win if !!k!! is small compared with !!|S|!!.

My question is: does this technique have a name? If I wanted to tell someone to use it, what would I say?

It's tempting to say "binary search" but it's not very much like binary search. Binary search finds a target value in a sorted array. If !!S!! were an array sorted by badness we could use something like binary search to locate the first bad item, which would solve this problem. But !!S!! is not a sorted array, and we are not really looking for a target value.

Is the history of this algorithm lost in time, or do we know who first invented it, or at least wrote it down? I think it sometimes pops up in connection with coin-weighing puzzles.

[ Addendum 20221023: this is the pure binary-splitting variation of adaptive group testing. I wrote a followup. ]