I'm way too discombobulated from getting next month's release of Logic for Programmers ready, so I'm pulling a idea from the slush pile. Basically I wanted to come up with a mental model of arrays as a concept that explained APL-style multidimensional arrays and tables but also why there weren't multitables. So, arrays. In all languages they are basically the same: they map a sequence of numbers (I'll use 1..N)1 to homogeneous values (values of a single type). This is in contrast to the other two foundational types, associative arrays (which map an arbitrary type to homogeneous values) and structs (which map a fixed set of keys to heterogeneous values). Arrays appear in PLs earlier than the other two, possibly because they have the simplest implementation and the most obvious application to scientific computing. The OG FORTRAN had arrays. I'm interested in two structural extensions to arrays. The first, found in languages like nushell and frameworks like Pandas, is the table. Tables have string keys like a struct and indexes like an array. Each row is a struct, so you can get "all values in this column" or "all values for this row". They're heavily used in databases and data science. The other extension is the N-dimensional array, mostly seen in APLs like Dyalog and J. Think of this like arrays-of-arrays(-of-arrays), except all arrays at the same depth have the same length. So [[1,2,3],[4]] is not a 2D array, but [[1,2,3],[4,5,6]] is. This means that N-arrays can be queried on any axis. ]x =: i. 3 3 0 1 2 3 4 5 6 7 8 0 { x NB. first row 0 1 2 0 {"1 x NB. first column 0 3 6 So, I've had some ideas on a conceptual model of arrays that explains all of these variations and possibly predicts new variations. I wrote up my notes and did the bare minimum of editing and polishing. Somehow it ended up being 2000 words. 1-dimensional arrays A one-dimensional array is a function over 1..N for some N. To be clear this is math functions, not programming functions. Programming functions take values of a type and perform computations on them. Math functions take values of a fixed set and return values of another set. So the array [a, b, c, d] can be represented by the function (1 -> a ++ 2 -> b ++ 3 -> c ++ 4 -> d). Let's write the set of all four element character arrays as 1..4 -> char. 1..4 is the function's domain. The set of all character arrays is the empty array + the functions with domain 1..1 + the functions with domain 1..2 + ... Let's call this set Array[Char]. Our compilers can enforce that a type belongs to Array[Char], but some operations care about the more specific type, like matrix multiplication. This is either checked with the runtime type or, in exotic enough languages, with static dependent types. (This is actually how TLA+ does things: the basic collection types are functions and sets, and a function with domain 1..N is a sequence.) 2-dimensional arrays Now take the 3x4 matrix i. 3 4 0 1 2 3 4 5 6 7 8 9 10 11 There are two equally valid ways to represent the array function: A function that takes a row and a column and returns the value at that index, so it would look like f(r: 1..3, c: 1..4) -> Int. A function that takes a row and returns that column as an array, aka another function: f(r: 1..3) -> g(c: 1..4) -> Int.2 Man, (2) looks a lot like currying! In Haskell, functions can only have one parameter. If you write (+) 6 10, (+) 6 first returns a new function f y = y + 6, and then applies f 10 to get 16. So (+) has the type signature Int -> Int -> Int: it's a function that takes an Int and returns a function of type Int -> Int.3 Similarly, our 2D array can be represented as an array function that returns array functions: it has type 1..3 -> 1..4 -> Int, meaning it takes a row index and returns 1..4 -> Int, aka a single array. (This differs from conventional array-of-arrays because it forces all of the subarrays to have the same domain, aka the same length. If we wanted to permit ragged arrays, we would instead have the type 1..3 -> Array[Int].) Why is this useful? A couple of reasons. First of all, we can apply function transformations to arrays, like "combinators". For example, we can flip any function of type a -> b -> c into a function of type b -> a -> c. So given a function that takes rows and returns columns, we can produce one that takes columns and returns rows. That's just a matrix transposition! Second, we can extend this to any number of dimensions: a three-dimensional array is one with type 1..M -> 1..N -> 1..O -> V. We can still use function transformations to rearrange the array along any ordering of axes. Speaking of dimensions: What are dimensions, anyway Okay, so now imagine we have a Row × Col grid of pixels, where each pixel is a struct of type Pixel(R: int, G: int, B: int). So the array is Row -> Col -> Pixel But we can also represent the Pixel struct with a function: Pixel(R: 0, G: 0, B: 255) is the function where f(R) = 0, f(G) = 0, f(B) = 255, making it a function of type {R, G, B} -> Int. So the array is actually the function Row -> Col -> {R, G, B} -> Int And then we can rearrange the parameters of the function like this: {R, G, B} -> Row -> Col -> Int Even though the set {R, G, B} is not of form 1..N, this clearly has a real meaning: f[R] is the function mapping each coordinate to that coordinate's red value. What about Row -> {R, G, B} -> Col -> Int? That's for each row, the 3 × Col array mapping each color to that row's intensities. Really any finite set can be a "dimension". Recording the monitor over a span of time? Frame -> Row -> Col -> Color -> Int. Recording a bunch of computers over some time? Computer -> Frame -> Row …. This is pretty common in constraint satisfaction! Like if you're conference trying to assign talks to talk slots, your array might be type (Day, Time, Room) -> Talk, where Day/Time/Room are enumerations. An implementation constraint is that most programming languages only allow integer indexes, so we have to replace Rooms and Colors with numerical enumerations over the set. As long as the set is finite, this is always possible, and for struct-functions, we can always choose the indexing on the lexicographic ordering of the keys. But we lose type safety. Why tables are different One more example: Day -> Hour -> Airport(name: str, flights: int, revenue: USD). Can we turn the struct into a dimension like before? In this case, no. We were able to make Color an axis because we could turn Pixel into a Color -> Int function, and we could only do that because all of the fields of the struct had the same type. This time, the fields are different types. So we can't convert {name, flights, revenue} into an axis. 4 One thing we can do is convert it to three separate functions: airport: Day -> Hour -> Str flights: Day -> Hour -> Int revenue: Day -> Hour -> USD But we want to keep all of the data in one place. That's where tables come in: an array-of-structs is isomorphic to a struct-of-arrays: AirportColumns( airport: Day -> Hour -> Str, flights: Day -> Hour -> Int, revenue: Day -> Hour -> USD, ) The table is a sort of both representations simultaneously. If this was a pandas dataframe, df["airport"] would get the airport column, while df.loc[day1] would get the first day's data. I don't think many table implementations support more than one axis dimension but there's no reason they couldn't. These are also possible transforms: Hour -> NamesAreHard( airport: Day -> Str, flights: Day -> Int, revenue: Day -> USD, ) Day -> Whatever( airport: Hour -> Str, flights: Hour -> Int, revenue: Hour -> USD, ) In my mental model, the heterogeneous struct acts as a "block" in the array. We can't remove it, we can only push an index into the fields or pull a shared column out. But there's no way to convert a heterogeneous table into an array. Actually there is a terrible way Most languages have unions or product types that let us say "this is a string OR integer". So we can make our airport data Day -> Hour -> AirportKey -> Int | Str | USD. Heck, might as well just say it's Day -> Hour -> AirportKey -> Any. But would anybody really be mad enough to use that in practice? Oh wait J does exactly that. J has an opaque datatype called a "box". A "table" is a function Dim1 -> Dim2 -> Box. You can see some examples of what that looks like here Misc Thoughts and Questions The heterogeneity barrier seems like it explains why we don't see multiple axes of table columns, while we do see multiple axes of array dimensions. But is that actually why? Is there a system out there that does have multiple columnar axes? The array x = [[a, b, a], [b, b, b]] has type 1..2 -> 1..3 -> {a, b}. Can we rearrange it to 1..2 -> {a, b} -> 1..3? No. But we can rearrange it to 1..2 -> {a, b} -> PowerSet(1..3), which maps rows and characters to columns with that character. [(a -> {1, 3} ++ b -> {2}), (a -> {} ++ b -> {1, 2, 3}]. We can also transform Row -> PowerSet(Col) into Row -> Col -> Bool, aka a boolean matrix. This makes sense to me as both forms are means of representing directed graphs. Are other function combinators useful for thinking about arrays? Does this model cover pivot tables? Can we extend it to relational data with multiple tables? Systems Distributed Talk (will be) Online The premier will be August 6 at 12 CST, here! I'll be there to answer questions / mock my own performance / generally make a fool of myself. Sacrilege! But it turns out in this context, it's easier to use 1-indexing than 0-indexing. In the years since I wrote that article I've settled on "each indexing choice matches different kinds of mathematical work", so mathematicians and computer scientists are best served by being able to choose their index. But software engineers need consistency, and 0-indexing is overall a net better consistency pick. ↩ This is right-associative: a -> b -> c means a -> (b -> c), not (a -> b) -> c. (1..3 -> 1..4) -> Int would be the associative array that maps length-3 arrays to integers. ↩ Technically it has type Num a => a -> a -> a, since (+) works on floats too. ↩ Notice that if each Airport had a unique name, we could pull it out into AirportName -> Airport(flights, revenue), but we still are stuck with two different values. ↩

The excellent-but-defunct blog Programming in the 21st Century defines "puzzle languages" as languages were part of the appeal is in figuring out how to express a program idiomatically, like a puzzle. As examples, he lists Haskell, Erlang, and J. All puzzle languages, the author says, have an "escape" out of the puzzle model that is pragmatic but stigmatized. But many mainstream languages have escape hatches, too. Languages have a lot of properties. One of these properties is the language's capabilities, roughly the set of things you can do in the language. Capability is desirable but comes into conflicts with a lot of other desirable properties, like simplicity or efficiency. In particular, reducing the capability of a language means that all remaining programs share more in common, meaning there's more assumptions the compiler and programmer can make ("tractability"). Assumptions are generally used to reason about correctness, but can also be about things like optimization: J's assumption that everything is an array leads to high-performance "special combinations". Rust is the most famous example of mainstream language that trades capability for tractability.1 Rust has a lot of rules designed to prevent common memory errors, like keeping a reference to deallocated memory or modifying memory while something else is reading it. As a consequence, there's a lot of things that cannot be done in (safe) Rust, like interface with an external C function (as it doesn't have these guarantees). To do this, you need to use unsafe Rust, which lets you do additional things forbidden by safe Rust, such as deference a raw pointer. Everybody tells you not to use unsafe unless you absolutely 100% know what you're doing, and possibly not even then. Sounds like an escape hatch to me! To extrapolate, an escape hatch is a feature (either in the language itself or a particular implementation) that deliberately breaks core assumptions about the language in order to add capabilities. This explains both Rust and most of the so-called "puzzle languages": they need escape hatches because they have very strong conceptual models of the language which leads to lots of assumptions about programs. But plenty of "kitchen sink" mainstream languages have escape hatches, too: Some compilers let C++ code embed inline assembly. Languages built on .NET or the JVM has some sort of interop with C# or Java, and many of those languages make assumptions about programs that C#/Java do not. The SQL language has stored procedures as an escape hatch and vendors create a second escape hatch of user-defined functions. Ruby lets you bypass any form of encapsulation with send. Frameworks have escape hatches, too! React has an entire page on them. (Does eval in interpreted languages count as an escape hatch? It feels different, but it does add a lot of capability. Maybe they don't "break assumptions" in the same way?) The problem with escape hatches In all languages with escape hatches, the rule is "use this as carefully and sparingly as possible", to the point where a messy solution without an escape hatch is preferable to a clean solution with one. Breaking a core assumption is a big deal! If the language is operating as if its still true, it's going to do incorrect things. I recently had this problem in a TLA+ contract. TLA+ is a language for modeling complicated systems, and assumes that the model is a self-contained universe. The client wanted to use the TLA+ to test a real system. The model checker should send commands to a test device and check the next states were the same. This is straightforward to set up with the IOExec escape hatch.2 But the model checker assumed that state exploration was pure and it could skip around the state randomly, meaning it would do things like set x = 10, then skip to set x = 1, then skip back to inc x; assert x == 11. Oops! We eventually found workarounds but it took a lot of clever tricks to pull off. I'll probably write up the technique when I'm less busy with The Book. The other problem with escape hatches is the rest of the language is designed around not having said capabilities, meaning it can't support the feature as well as a language designed for them from the start. Even if your escape hatch code is clean, it might not cleanly integrate with the rest of your code. This is why people complain about unsafe Rust so often. It should be noted though that all languages with automatic memory management are trading capability for tractability, too. If you can't deference pointers, you can't deference null pointers. ↩ From the Community Modules (which come default with the VSCode extension). ↩

I'm a big (neo)vim buff. My config is over 1500 lines and I regularly write new scripts. I recently ported my neovim config to a new laptop. Before then, I was using VSCode to write, and when I switched back I immediately saw a big gain in productivity. People often pooh-pooh vim (and other assistive writing technologies) by saying that writing code isn't the bottleneck in software development. Reading, understanding, and thinking through code is! Now I don't know how true this actually is in practice, because empirical studies of time spent coding are all over the place. Most of them, like this study, track time spent in the editor but don't distinguish between time spent reading code and time spent writing code. The only one I found that separates them was this study. It finds that developers spend only 5% of their time editing. It also finds they spend 14% of their time moving or resizing editor windows, so I don't know how clean their data is. But I have a bigger problem with "writing is not the bottleneck": when I think of a bottleneck, I imagine that no amount of improvement will lead to productivity gains. Like if a program is bottlenecked on the network, it isn't going to get noticeably faster with 100x more ram or compute. But being able to type code 100x faster, even with without corresponding improvements to reading and imagining code, would be huge. We'll assume the average developer writes at 80 words per minute, at five characters a word, for 400 characters a minute.What could we do if we instead wrote at 8,000 words/40k characters a minute? Writing fast Boilerplate is trivial Why do people like type inference? Because writing all of the types manually is annoying. Why don't people like boilerplate? Because it's annoying to write every damn time. Programmers like features that help them write less! That's not a problem if you can write all of the boilerplate in 0.1 seconds. You still have the problem of reading boilerplate heavy code, but you can use the remaining 0.9 seconds to churn out an extension that parses the file and presents the boilerplate in a more legible fashion. We can write more tooling This is something I've noticed with LLMs: when I can churn out crappy code as a free action, I use that to write lots of tools that assist me in writing good code. Even if I'm bottlenecked on a large program, I can still quickly write a script that helps me with something. Most of these aren't things I would have written because they'd take too long to write! Again, not the best comparison, because LLMs also shortcut learning the relevant APIs, so also optimize the "understanding code" part. Then again, if I could type real fast I could more quickly whip up experiments on new apis to learn them faster. We can do practices that slow us down in the short-term Something like test-driven development significantly slows down how fast you write production code, because you have to spend a lot more time writing test code. Pair programming trades speed of writing code for speed of understanding code. A two-order-of-magnitude writing speedup makes both of them effectively free. Or, if you're not an eXtreme Programming fan, you can more easily follow the The Power of Ten Rules and blanket your code with contracts and assertions. We could do more speculative editing This is probably the biggest difference in how we'd work if we could write 100x faster: it'd be much easier to try changes to the code to see if they're good ideas in the first place. How often have I tried optimizing something, only to find out it didn't make a difference? How often have I done a refactoring only to end up with lower-quality code overall? Too often. Over time it makes me prefer to try things that I know will work, and only "speculatively edit" when I think it be a fast change. If I could code 100x faster it would absolutely lead to me trying more speculative edits. This is especially big because I believe that lots of speculative edits are high-risk, high-reward: given 50 things we could do to the code, 49 won't make a difference and one will be a major improvement. If I only have time to try five things, I have a 10% chance of hitting the jackpot. If I can try 500 things I will get that reward every single time. Processes are built off constraints There are just a few ideas I came up with; there are probably others. Most of them, I suspect, will share the same property in common: they change the process of writing code to leverage the speedup. I can totally believe that a large speedup would not remove a bottleneck in the processes we currently use to write code. But that's because those processes are developed work within our existing constraints. Remove a constraint and new processes become possible. The way I see it, if our current process produces 1 Utils of Software / day, a 100x writing speedup might lead to only 1.5 UoS/day. But there are other processes that produce only 0.5 UoS/d because they are bottlenecked on writing speed. A 100x speedup would lead to 10 UoS/day. The problem with all of this that 100x speedup isn't realistic, and it's not obvious whether a 2x improvement would lead to better processes. Then again, one of the first custom vim function scripts I wrote was an aid to writing unit tests in a particular codebase, and it lead to me writing a lot more tests. So maybe even a 2x speedup is going to be speed things up, too. Patreon Stuff I wrote a couple of TLA+ specs to show how to model fork-join algorithms. I'm planning on eventually writing them up for my blog/learntla but it'll be a while, so if you want to see them in the meantime I put them up on Patreon.

I released Logic for Programmers exactly one year ago today. It feels weird to celebrate the anniversary of something that isn't 1.0 yet, but software projects have a proud tradition of celebrating a dozen anniversaries before 1.0. I wanted to share about what's changed in the past year and the work for the next six+ months. The Road to 0.1 I had been noodling on the idea of a logic book since the pandemic. The first time I wrote about it on the newsletter was in 2021! Then I said that it would be done by June and would be "under 50 pages". The idea was to cover logic as a "soft skill" that helped you think about things like requirements and stuff. That version sucked. If you want to see how much it sucked, I put it up on Patreon. Then I slept on the next draft for three years. Then in 2024 a lot of business fell through and I had a lot of free time, so with the help of Saul Pwanson I rewrote the book. This time I emphasized breadth over depth, trying to cover a lot more techniques. I also decided to self-publish it instead of pitching it to a publisher. Not going the traditional route would mean I would be responsible for paying for editing, advertising, graphic design etc, but I hoped that would be compensated by much higher royalties. It also meant I could release the book in early access and use early sales to fund further improvements. So I wrote up a draft in Sphinx, compiled it to LaTeX, and uploaded the PDF to leanpub. That was in June 2024. Since then I kept to a monthly cadence of updates, missing once in November (short-notice contract) and once last month (Systems Distributed). The book's now on v0.10. What's changed? A LOT v0.1 was very obviously an alpha, and I have made a lot of improvements since then. For one, the book no longer looks like a Sphinx manual. Compare! Also, the content is very, very different. v0.1 was 19,000 words, v.10 is 31,000.1 This comes from new chapters on TLA+, constraint/SMT solving, logic programming, and major expansions to the existing chapters. Originally, "Simplifying Conditionals" was 600 words. Six hundred words! It almost fit in two pages! The chapter is now 2600 words, now covering condition lifting, quantifier manipulation, helper predicates, and set optimizations. All the other chapters have either gotten similar facelifts or are scheduled to get facelifts. The last big change is the addition of book assets. Originally you had to manually copy over all of the code to try it out, which is a problem when there are samples in eight distinct languages! Now there are ready-to-go examples for each chapter, with instructions on how to set up each programming environment. This is also nice because it gives me breaks from writing to code instead. How did the book do? Leanpub's all-time visualizations are terrible, so I'll just give the summary: 1180 copies sold, $18,241 in royalties. That's a lot of money for something that isn't fully out yet! By comparison, Practical TLA+ has made me less than half of that, despite selling over 5x as many books. Self-publishing was the right choice! In that time I've paid about $400 for the book cover (worth it) and maybe $800 in Leanpub's advertising service (probably not worth it). Right now that doesn't come close to making back the time investment, but I think it can get there post-release. I believe there's a lot more potential customers via marketing. I think post-release 10k copies sold is within reach. Where is the book going? The main content work is rewrites: many of the chapters have not meaningfully changed since 1.0, so I am going through and rewriting them from scratch. So far four of the ten chapters have been rewritten. My (admittedly ambitious) goal is to rewrite three of them by the end of this month and another three by the end of next. I also want to do final passes on the rewritten chapters; as most of them have a few TODOs left lying around. (Also somehow in starting this newsletter and publishing it I realized that one of the chapters might be better split into two chapters, so there could well-be a tenth technique in v0.11 or v0.12!) After that, I will pass it to a copy editor while I work on improving the layout, making images, and indexing. I want to have something worthy of printing on a dead tree by 1.0. In terms of timelines, I am very roughly estimating something like this: Summer: final big changes and rewrites Early Autumn: graphic design and copy editing Late Autumn: proofing, figuring out printing stuff Winter: final ebook and initial print releases of 1.0. (If you know a service that helps get self-published books "past the finish line", I'd love to hear about it! Preferably something that works for a fee, not part of royalties.) This timeline may be disrupted by official client work, like a new TLA+ contract or a conference invitation. Needless to say, I am incredibly excited to complete this book and share the final version with you all. This is a book I wished for years ago, a book I wrote because nobody else would. It fills a critical gap in software educational material, and someday soon I'll be able to put a copy on my bookshelf. It's exhilarating and terrifying and above all, satisfying. It's also 150 pages vs 50 pages, but admittedly this is partially because I made the book smaller with a larger font. ↩