What’s hard about grad school?

Part of what’s hard about grad school is that things are undirected. In undergrad, you have well delimited homework assignments, maybe a project or two. But in grad school everything is open ended, and you’re lost at sea!

And that broad scope—confronting the unknown and feeling stupid—really is part of what’s hard. But in this lecture about writing, Larry McEnerney makes the point that there’s something even harder about grad school than its broad scope. Grad school is hard because it’s about adopting new values.


Undergrad is really all about skills. You’re paying money to get a guided tour through certain skills, with an experienced master paying attention to you, i.e., observing, helping, grading, mentoring, and so on. As Larry says (with perhaps a little too much glee), you’re paying for faculty to pay attention—your work product is not something they want, but rather they are paid to evaluate it and help you improve.

Graduate school teaches you skills, too. In a more intense way—you’re one of a few apprentices at the foot of a master. At least in CS, or at least the way I’ve seen it work best.

Beyond skills

But graduate school isn’t just skills. Graduate school is also the more complex process of joining a community—getting your “union card”. To be successful, grad students must learn and adopt the unwritten codes of their new community. It is hard work to discover and understand what the community values, and early grad students often confuse “novel and difficult application of a tool or technique” for “valuable contribution to the community”.

For example, in programming languages (PL) it’s common for early grad students to revel in complicated notation, subtle formalism, and the details of proofs. Many choose grad school for the novelty—to be the first to know something! And, to be sure, formal notation and novel proofs are fun things. And they are important things! If you get these things wrong, you’re sunk. But caring about these things most is missing the point, and will lead to a frustrating experience writing the paper, reading reviews, writing the response, figuring out what to do next, and so on.

Larry’s point of view is a realist one. Disturbingly realist: he says he’s accused of ‘fascism’ for asking you to identify powerful stakeholders in the community and to cater to them. But he is right: you can only publish papers that reviewers think are valuable, and you will have to cater to their needs to get them to feel that way.

For a PL-specific take, Laurie Tratt’s essay on “What Makes a Good Research Proposal?” uses a value-oriented framework:

…a good proposal must address three fundamental questions:

1. What is the problem being tackled?
2. Why is the problem worth being tackled?
3. What is the insight that makes tackling the problem plausible?

Although the third of these is by far the most important, the first two set up the necessary context to understand the third.

First, note that (2) asks, “Why is it valuable?”. I would go further and say that (2) and (3) are equally important. A plausible approach on a pointless problem doesn’t get you very far.

As someone who recoiled in disgust at even the notion of applied research in grad school, I’ve come completely around to Larry McEnerney’s point of view. None of the formalism matters if it doesn’t do something. Proofs aren’t worth the wildly overengineered LaTeX macros they’re written in if the properties they ensure don’t matter to anyone. I hated this kind of talk as a graduate student, and yet here I am. (In related news, I tell my undergraduate students to take notes… and I didn’t take a single note in class in undergrad. If I had a lawn, I would be asking people to get off it.)

Good news

The good news is that it is within your power to move the field. You have to accommodate the community where it is, but you have a say in where it’s going. In 2017, I told Benjamin Pierce that I was planning to work on the POSIX shell. He laughed, saying he thought it was “irredeemably bad”. At an OBT talk at POPL that year, Phil Wadler said most of my planned work wasn’t even PL. In 2020, I published Smoosh at POPL; in 2021, we had a paper and a panel at HotOS on the shell; in 2022, we have a paper at OSDI.

You don’t have to give up your vision or your voice. In fact, I think my own voice is a key asset in moving the community. Compelling writing and presentations are persuasive! And writing is exactly where Larry picks up.

an early Moog modular synthesizer

When is program synthesis worthwhile?

Program synthesis is an appealing notion: I give you a pretty good (but incomplete) description of a program, and you give me a working program back out. Flash Fill—program synthesis by example that ships in Excel—is the standard example that PL people trot out, and it is an incredible success. (Check out Sumit Gulwani’s POPL 2015 keynote (video in supplemental material) if you’ve never heard of it.)

Fiat Cryptography is another great example of program synthesis success. Where Flash Fill is programming by example, Fiat Cryptography is deductive synthesis: here, an expert manually refines a specification of arithmetic operations used in elliptic curve cryptography to achieve fast and verified implementations of the necessary primitives. Fiat Cryptography is used in BoringSSL, a part of Chrome and Android (check out the paper).

These two approaches are quite different: Flash Fill offers nearly instantaneous feedback as part of an existing interface; Fiat Cryptography is effectively a set of Coq lemmas and tactics, only approachable by experts. (You might wonder why they’re both called program synthesis! For that, I’d follow Sam Tobin-Hochstadt’s second-hand quote: “a synthesizer is just a compiler that doesn’t work”, that is, these tools are both synthesis because they take specifications as input and produce valid implementations as output, when they produce output… but they don’t always succeed in producing output.)

Flash Fill and Fiat Cryptography are tech-transfer success stories working on starkly different timescales: Flash Fill is an end-user synthesis that takes examples and produces Excel formulae in a fraction of a second; Fiat Cryptography is a tool for experts to generate extremely specialized code in a more interactive, involved way, possibly taking days to generate something acceptable. Given these remarkable differences, I wonder… when is program synthesis worthwhile?

A theory of effort

Suppose you’re trying to write some program, and you could (a) use program synthesis or (b) write the program by hand. Let’s say it takes E time to come up with an appropriate specification, S time to synthesize a solution, and I time to inspect and approve a solution. Conversely, suppose it takes P time to write the program and T time to test and approve it.

In general, synthesis is only worthwhile when E + S + I < P + T. Considering that most programmers are more familiar with conventional programming tasks and users are often reluctant to install and use a complex new tool, it’s likely that synthesis is only reasonable when E + S + I <<< P + T (we read <<< as “is much less than”). Let’s take this theory for a spin.

Flash Fill

Flash Fill is example based, where simple string transformations are canonical examples. Writing up some example string transformations might take a few seconds, i.e., E ~ 10. Synthesis is very fast—say, S ~ 1. Inspection might take longer—from a few seconds to a minute or so. Let’s be careful and say I ~ 30 (though your typical Excel user may be a little more YOLO about things).

Conversely, programming string manipulations in Excel are fiddly; I’d certainly have to spend a minute to find the right functions, figure out the annoying indexing bits (another minute—or more), and then go through and verify that I got it right. A user with less programming expertise might take longer. Let’s say P ~ 120 and T ~ 20—though many users will give up before figuring it out at all.

Here synthesis is much faster than writing the program by hand: we have 31 ~ 140. It’s a huge savings in time for an experienced programmer to use Flash Fill, never mind a user less familiar with programming.

Fiat Cryptography

Fiat Cryptography bakes in a set of techniques for deriving very efficient implementations of cryptographic arithmetic primitives. In this case E amounts to choosing an appropriate prime and its representation and encoding it in Coq. I pored over their commit logs, but I couldn’t find an example of new primes being introduced. Let’s say it takes a day, though I suspect it’s much less. (For example, it takes a few days for them to implement new optimizations—a new prime amounts to simply picking a representation.) S is the time it takes for Coq to compile the code—I haven’t tried it, but it takes about 1.5hrs to compile in their CI. Inspection is… free, or rather, the low, low price of trusting their trusted computing base (TCB)! Okay, okay: their validation in CI takes another 1.5hrs. Trust but verify. Verify but test.

On the other hand, what does it take to implement and verify operations for a new prime by hand? Months! And at the end of those months, one has less confidence in the final product compared to an unverified one; verifying the handwritten code will take months more.

Again, synthesis is substantially faster: a day or two, compared to many months. As a bonus, the output of synthesis is substantially more trustworthy than what you’d get rolling your own.


Another example of well situated, worthwhile synthesis is RLIBM, a collection of fast and correct elementary floating point functions. RLIBM doesn’t advertise itself as synthesis, but that’s exactly what it is: given a specification (e.g., log2f) and a correct oracle, they generate appropriate polynomials for approximation. A number of the talks are available online.

RLIBM is less of a ‘drop in’ replacement than Fiat Cryptography, but they’re already seeing plenty of success in getting their code into LLVM. Synthesis is slow—it can take quite some time for them to generate and validate polynomials (i.e., E, S, and I are on the order of days judging from their interactions with LLVM devs). But the value proposition is huge: their approach yields more efficient code, offers a single polynomial for multiple rounding modes and bits of precision, and is more correct than the state of the art. What’s more, the start of the art is also very expensive: Intel has a whole team of mathematical experts working on this problem (P is very, very high); for float32, I = T, since both cases use exhaustive checking.

New frontiers

Synthesis is worthwhile when its costs (coming up with a specification/examples E, running the synthesizer S, and inspecting the solution I) are substantially less than the cost of programming by hand (programming the solution P and testing it T). That is, synthesis is worthwhile when E + S + I <<< P + T.

My estimate is conservative: the framing assumes that all we get from synthesis is a replacement for programming, no better than what we’d produce ourselves. But for users without programming experience, Flash Fill doesn’t just speed things up, it makes new things possible. Similarly, Fiat Cryptography doesn’t just produce a fast C implementation of arithmetic primitives… its proofs add confidence that the result is correct! RLIBM is faster and more precise than existing solutions.

Performance changes how users use software. Fabian Giesen is quoted to similar effect in Dan Luu’s post on productivity and velocity; Shriram Krishnamurthi has made similar comments, too (Thanks to Sam Tobin-Hochstadt and Dan Luu for the links!) Such a maxim certainly applies here: synthesis opens new doors. Non-programmers get help writing simple programs; programmers save substantial time and effort. Either way, synthesis boosts productivity.

When is synthesis not worthwhile?

We’ve seen three examples of program synthesis that have clearly provided value in the real world. Flash Fill, Fiat Cryptography, and RLIBM are clearly worthwhile. When is synthesis not worthwhile?

Let’s consider the case of general-purpose programming. A variety of program synthesizers can generate code in functional or imperative languages. A common example in these settings is linked list reversal. A quadratic solution in a functional language is fairly simple if you know about append or snoc, but harder without it; efficient, linear versions in functional and imperative languages are a bit tougher.

Many academic tools are example based, in which case it’d be reasonable to say that E ~ T. For synthesis to be worthwhile, then, we would need S + I <<< P. That is, the time to run the synthesizer and inspect the resulting program must be substantially less than the time to write the program itself.

Writing a linear-time linked list reversal in OCaml takes me about 30s; less in Haskell. Imperative linked list reversal is a bit slower—say, 2.5min. Testing either of these won’t take long—a few seconds suffices in an FP REPL, or as much as a minute to write the JUnit in Java.

Existing synthesis tools can handily beat my programming time, so… what gives? Why aren’t program synthesis tools everywhere? I think there are a few reasons:

  • In fact, E > T. Looking at synthesis benchmarks, the tools that beat my time take many more examples than I would bother with. Tail-recursive reversal takes 14 examples in an older tool I know of… and that’s to generate the two-argument version, not the outer function that one might care about. If I know to generate the accumulator-passing type of the inner function, why do I need help writing list reverse?
  • I is large. I looked for research on how I and P relate, i.e., how long it takes to write a function as compared to understanding and agreeing with someone else’s implementation, but I couldn’t find much concrete information. I suspect that if the standard is ‘believing a program to be correct’, then I is on the same order as P when T is small, i.e., programs that can be easily tested are as hard to check as they are to write. As T increases, then I is on the order of T. (But who cares what I think? It’s a testable hypothesis!)
  • S should include the cost of invoking the tool… and maybe the amortized cost of acquiring the tool. Flash Fill just comes with Excel and is quite discoverable in the UI, but existing tools need to be downloaded, compiled, and installed, and they don’t have much in the way of editor integration.

Alternatively, consider GitHub’s Copilot. You write a little prose explanation and a function header, and it does the rest. Copilot seems to set E quite low, but I is comparatively high. What’s worse, E != T when you give Copilot textual descriptions rather than input/output examples. Copilot addresses HCI issues nicely—it’s easy to install and easy to use—but it produces particularly low-confidence code. For Copilot code, I’d say I = T, i.e., I’d want to thoroughly test anything that came out of Copilot. (And more importantly, for licensing reasons, I wouldn’t want to use anything that came out of Copilot.)

To sum up: synthesizing code for general purpose languages has E + S + I less than P + T in some cases, but not so much less than P + T that it seems worthwhile.

My examples of worthwhile synthesis offer two primary ways for synthesis to be worthwhile: either focus on fast-turnarounds and excellent usability (very low E, S, and I), or specialize in a domain where the state of the art is excruciatingly hard (P and T are huge and you have a story—verification, validation, etc.—for keeping I low). Flash Fill’s successful tech transfer depended heavily on HCI improvements. Fiat Cryptography’s and RLIBM’s success did not—but they target specialized domains.

Don’t be discouraged

Program synthesis offers a novel relationship between programmers people and computers. I am not at all saying to give up on forms of synthesis that don’t meet my criteria of E + S + I <<< P + T. Some forms of synthesis may not be worthwhile yet, but we don’t know what changes and opportunities the future holds!

Program synthesis is an exciting new area, and there’s lots of PL work on it. It’s unreasonable to expect every academic paper to be immediately applicable, so I don’t expect every proposed synthesizer to meet my criteria. It is important, however, to be realistic about where we stand. If you’re working on synthesis, think about how E, S, and I compare to P and T for you. If your approach doesn’t yet hit my criteria, what needs to change to make that happen?


There’s some great discussion on the Twitter thread announcing this post.

Bridging the gradual typing gap at OOPSLA 2021

I want to believe in a future where the lion will lie down with the lamb; we’ll beat our swords into plowshares; and developers will migrate dynamic prototypes to robust static systems with confidence. But these Aquarian visions are elusive. Having a map of the road to paradise in theory doesn’t mean we know how to get there in practice. Let me tell you about two papers at OOPSLA that shuffle us a few steps forward on this long pilgrim’s trail.

A vintage poster of "Hair", the American Tribal Love Rock Musical, with a trippy inverted head. This poster advertises a performance at the Aquarius Theatre in Los angeles.

Migrating programs

How do you actually get a program from Scheme into ML? Or from JavaScript into TypeScript? The theory of gradual typing goes far beyond these pedestrian questions. In principle, we know how to reconcile dynamism with much more complex systems, like information flow or refinement types or effect systems. But there’s very little tooling to support moving any particular Scheme program into ML. (If your program is a Racket program, then you’re in some luck.)

People have studied program migration before, under a variety of names. Papers go back at least to 2009, arguably even earlier. There are lots of different approaches, and most comprise some form of type inference and custom constraint solving—complex! Worse still, there’s been no consensus on how to evaluate these systems. Luna Phipps-Costin, Carolyn Jane Anderson, me, and Arjun Guha dug into program migration. Our paper, “Solver-based Gradual Type Migration”, tries to build a map of the known territory so far:

  1. There are competing desiderata: maximal type precision, compatibility with code at different types, and preserving the existing semantics of your program, i.e., safety.
  2. We evaluate a variety of past techniques on prior benchmarks, and we devise a novel set of “challenge” problems. Our evaluation framework is robust, and you could plug in other approaches to type migration and evaluate them easily.
  3. We introduce a new, very simple approach to type migration, which we call TypeWhich. TypeWhich uses an off-the-shelf SMT solver. You can choose how compatible/precise you want it to be, but it’ll always be safe.

I’m excited about each of these contributions, each for its own reason.

For (1), I’m excited to formally explain that what you’re actually trying to do with your code matters. “Gradual typing” sensu lato is pretty latus indeed. Are you migrating a closed system, module by module? Or are you coming up with type annotations for a library that might well be called by untyped clients? These are very different scenarios, and you probably want your type migration algorithm to do different things! Bringing in these competing concerns—precision, compatibility, and safety—gives researchers a way to contextualize their approaches to type migration. (All that said, to me, safety is paramount. I’m not at all interested in a type migration that takes a dynamic program that runs correctly on some input and produces a statically typed program that fails on the same input… or won’t even compile! That doesn’t sound very gradual to me.)

For (2), I’m excited to be building a platform for other researchers. To be clear, there’s a long way to go. Our challenge problems are tiny toys. There’s a lot more to do here.

For (3), I’m excited to have an opportunity to simplify things. The TypeWhich constraint generator is simple, classic PL; the constraints it generates for SMT are straightforward; the models that SMT generates are easy to understand. It’s a cool approach!

One tiny final note: Luna has done a tremendous amount of incredibly high quality work on this project, both in code and concept. She’s just now starting her third-year of undergraduate study. So: watch out! You ain’t ready.

Typed functional programming isn’t about functions

If there’s a single defining ‘killer’ feature of typed functional programming, it isn’t first-class functions at all: it’s algebraic datatypes. Algebraic datatypes help make illegal states unrepresentable and ASTs easy to work with. They’re a powerful tool, and their uptake in a variety of new-hotness languages (Kotlin, Rust, Swift) speaks to their broad appeal.

Moving Scheme code to ML is an old goal, and it’s the bread and butter of the introductory sections of gradual typing papers. But are we any closer than we were fifteen years ago? (I’d say “yes”, and point at Typed Racket, or “nobody knows what’s happening anyway” and point at Idris’s Chez Scheme runtime.)

Stefan Malewski, me, and Éric Tanter tried to figure out how algebraic datatypes play with dynamic features. Our paper, “Gradually Structured Data“, uses AGT to ‘compute’ static and dynamic semantics for a language with possibly open algebraic datatypes and the unknown type in a few flavors (?, the unknown type; a new ground type for “datatype”, the same way int and bool and ?->? are ground; and a new type for “any open datatype”). The features gel in a nice way, letting us express some cool behaviors (see Section 2 for how one might evolve a simple JSON API) and sit in a novel space (see Section 5 for a thorough comparison to related features).

I’m particularly pleased that we’ve found a new place in the design spectrum (per our feature chart in Section 5) that seems to support incremental program migration (per our examples in Section 2)—and it’s formally grounded (by using AGT in the middle, formal sections).

This paper came out of conversations with Éric after my screed about gradual typing’s two lineages at SNAPL (see also my followup blogpost, “What to Define When You’re Defining Gradual Type Systems”). There’s plenty more to do: what about separate compilation? What are the right representation choices? How should runtime checks really go, and how can programmers control the costs?

I remember a question I was asked after giving the talk for “Contracts Made Manifest” at POPL 2010 with some panic fondly. That paper compares the latent approach to contracts in Racket-then-Scheme (well structured runtime checks at module boundaries) to the manifest approach (runtime checks are a form of type coercion, occurring anywhere) in the emerging refinement types literature (Sage, Liquid Types, etc.). I had shown that the two aren’t equivalent in the presence of dependency, and I concluded by talking about how the two implementation approaches differed. So: somebody asked, “Which approach should you use?” To be honest, I had hardly even thought about it.

So, suppose you wanted use algebraic datatypes and dynamic features today: which language should you use? I’ve thought about it, and the answer, sadly, is, “It depends”. OCaml’s polymorphic variants get you a long way; Haskell’s Dynamic could work great, but it’s badly in need of usable surface syntax. (I’ve tried to get Richard Eisenberg to help me with the fancy work to make that happen, but he’s justifiably worried that the Haskell community would run him out of town.) Scala, Haskell, and OCaml are your best bets if you want true algebraic datatypes. If you’re more relaxed about things, Typed Racket or TypeScript could work well for you. If what you’re looking for is a type system expressive enough to capture interesting dynamic idioms, then I think there’s a clear choice: CDuce. Ever since un bel recensore anonimo at SNAPL 2019 showed me that CDuce can type flatten, I’ve been impressed. Check this out:

let flatten ( Any -> [ (Any\[Any*])* ] )  (* returns a list of non-lists ???? *)
  | [] -> []                              (* nil *)
  | (h,t) -> (flatten h)@(flatten t)      (* cons *)
  | x -> [x]                              (* anything else *)

Look at that type! In just a few lines of CDuce, we can show that flatten produces not just a list of elements, but a list of things that are not themselves lists. The price here is that CDuce’s types are set-theoretic, which means things are a touch different from what people are used to in OCaml or Haskell. But if you’re okay with that, CDuce is a serious contender!

Coda: see you at OOPSLA?

I’m planning on going to OOPSLA 2021 in Chicago, given the twoopsla and the opportunity to present a paper from OOPSLA 2020, “Formulog: Datalog for SMT-based static analysis”, with Aaron Bembenek and Steve Chong. I’ve already blogged about it, but I’m excited to get to give an in-person version of the talk, too. You can still watch Aaron’s excellent recorded talk on YouTube and enjoy the cabin vibes. There won’t be cabin vibes at my OOPSLA 2020 talk, but there will be terrible jokes. So: think about it. Will I see you at OOPSLA? I hope so!

I’m looking for PhD students!

I’m looking for PhD students in the Fall 2021 application cycle, to start in Fall 2022. Come work with me at Stevens CS in Hoboken, NJ!

I work in Gateway South (the left-hand side of this photo). You could, too! (Photo credit: Stevens Alumni.)

What will we work on?

I’m interested in applying formalism — all those pretty Greek letters in program semantics, type systems, and static analysis — directly to real systems — all that nitty gritty code that makes these beautiful, horrible machines do their thing. I’m working on a few projects that variously emphasize theoretical or practical aspects. My main goal these days is to provide better support for the POSIX shell and its ecosystem, but here’s a sampling from recent papers:

  • Smoosh (POPL 2020): I’m interested in improving and supporting the shell. Smoosh is a formal model of the POSIX shell that can be executed and passes the POSIX test suite. Continuing work on Smoosh means hacking in Lem, OCaml, and Coq (and maybe Rust or C or JS or Elm), and thinking about virtualization, symbolic execution, fuzzing, and how specifications and implementations interact. Or maybe it just means building cool tools for the POSIX world!
  • Formulog (OOPSLA 2020): Datalog, functional programming, and SMT combine to let you write down and run things that look a lot like your formal spec. Continuing work in this line means hacking in Rust (and maybe C++ or Java), and thinking about SMT and how we can be confident that the formalism we write is the code that we run—and that our code is efficient.
  • Gradual types (OOPSLA 2021) and type migration (OOPSLA 2021): People have been trying to combine the benefits of dynamic and static types for years. Work in this line will mean hacking in Rust (and maybe JS or TS or Haskell) and doing classic PL stuff like type soundness, type inference, and proofs of contextual equivalence (by logical relations or bisimulation, on paper or in Coq).
A slide from a Keynote deck. The title is "semantics engineering". The left-hand side illustrates systems challenges:

 - a "C" monster
 - complicated specs
 - a dog in front of a laptop (programming is hard!)

The right-hand side illustrates PL formalism: inference rules, helper functions, grammars, etc.
I’ve been calling it this combination of executable systems and PL formalism “semantics engineering”, with inspiration from the PLT folks (though I don’t really use Redex).

You can check out a list of all my papers. Are any of these papers the sort of thing you’d like to write? Come join me for a PhD!

Who will you work with?

Stevens has about thirty research faculty, and we’re growing fast. We have a great group of people interested in PL, security, and systems: Eduardo Bonelli, Tegan Brennan, Dominic Duggan, Eric Koskinen, Philippe Meunier, David Naumann, Georgios Portokalidis, Susanne Wetzel, and Jun Xu. And there are of course many other fantastic researchers in other topics to learn from in class and collaborate with on research. And beyond all that, I got a lot out of my internships (AT&T Shannon Labs; MSR Cambridge), and I encourage my students to find stimulating opportunities.

Where is Hoboken, again?

Hoboken, NJ is directly across the Hudson River from Manhattan a/k/a New York City. There’s 24-hour train service and frequent ferries to and from New York. Hoboken is a vision zero city, where it’s safe and comfortable to bike and walk. There are other cool cities nearby, like Jersey City.

How do you apply?

You can learn more about the CS PhD program at Stevens and apply online. If you have questions, please don’t hesitate to get in touch.

Heaven, Hell, or Hoboken!

After six years at Pomona College, I’ve moved to Stevens Institute of Technology as an assistant professor in the computer science department. I miss my lovely Pomona colleagues—they’re hiring!—but I’m excited to be on the East Coast and to be doing more research with a new set of lovely colleagues.

A photo of my office nameplate. The Stevens logo in red, with the following text:

Michael Greenberg
Assistant Professor
Department of Computer Science

447 (in Braille)

I’ve got a new webpage, but the old webpage should stay up.

We’ll be spinning up the Stevens PL/systems/security seminar soon, and I’m hopeful we can involve lots of interesting people, as speakers and attendees. If you’re in the New York area, come by and say hi!

Also… I’ll be looking to hire PhD students for the coming year! More info on that soon.

Pomona College is hiring!

Pomona College’s computer science department is hiring Fall of 2021 for Fall 2022. I used to work at Pomona, and there is a lot to recommend it. Pomona College is a small liberal arts college (SLAC) in LA County, 35mi/45-240min outside DTLA. It’s a 2:2 teaching load.

Steps on campus, with a view of the mountains behind.

First and foremost, you’ll have excellent colleagues. They are friendly, collegial, supportive, and hardworking. There’s a sense of shared purpose and responsibility. Disagreements are resolved amicably, because everyone is on the same team. Nobody shirks. They’re great people!

Second, the students are bright. They’re motivated, broad-minded, and often interested in social justice. Pomona’s student body overall is quite diverse, along a variety of axes (income, ethnicity, national origin), and the CS enjoys that diversity. Pomona is a very wealthy institution, and it’s putting its wealth to work helping many students who have very little money.

Third, Pomona offers a great deal of research freedom. I felt zero pressure to get grants when I was there, which allowed me to pursue whatever research interests felt worthwhile.

I’ve written in the past about what I loved (and didn’t) about Pomona College. I’ve left that document up, since it provides more detail on what I think is really good about working at a SLAC in general and Pomona in particular.

Joining Pomona’s CS department will let you join a community of lovely colleagues. You’ll have the opportunity to shape the culture and trajectory of a department, work closely with smart and interesting students, do the research you want… all while enjoying the mountains, high desert, city, and coast. It could be right for you — if you’re not sure, feel free to get in touch and we can chat about it.

A student asked why STLC programs always terminate… so I showed them! Pomona offers the opportunity to teach interesting things to interested students. That student and I later worked through the Homotopy Type Theory book together.

SIGPLAN Blog: Making PL Ideas Accessible

I have a new post up on the SIGPLAN blog: “Making PL Ideas Accessible: An Open-Source, Open-Access, Interactive Journal. Inspired by Distill, I propose an open-access, open-source, interactive journal for disseminating clear presentations of current ideas and methods in programming languages.

It’s a particularly good moment to consider our research’s reach and impact: CORE has just downgraded many PL conferences in its rankings. Just because you don’t take an interest in rankings doesn’t mean rankings won’t take an interest in you. Let this spur a new wave of beautiful and enlightening explanations of PL ideas that can reach a a broad audience.

POPL Cocktail Hour

POPL 2021 is open for business on Clowdr! The synchronous band is in the afternoon and evening in Central European Time (CET = UTC+1). I live outside LA, which is UTC-8… so the POPL happy hours at 10:30am start a little early even for me.

So far as I know, this is the first POPL with a paper named after a cocktail. Accordingly, I’ve decide to host a POPL Cocktail Hour on Wednesday, January 20th at 5pm Pacific Time (PT = UTC-8). We’ll be meeting in the Clowdr break room. (You need to be registered to attend, but it’s not too late!)

I’ll be making the official POPL Cocktail, “Nordic Summer”. I got the recipe from the Moody Mixologist, but here it is:

  • 2oz aquavit
  • 1oz Aperol
  • 1oz lime juice (fresh, natch)

Add ice to a shaker, shake ingredients until it’s quite cold (i.e., it hurts to hold a metal shake), and then strain into a chilled coupe. SkÃ¥l!

I think it’d be great with a variety of substitutions—Capelletti or Campari or even Punt e Mes would do well instead of Aperol, and you can sub lemon for lime. I bet it’d be good long (i.e., with soda on top).

Here’s another one I came up with, which I’m calling the “Copenhagen Sour”:

  • 3/4oz lemon juice
  • 1 egg white
  • 1 1/4oz aquavit (okay, mine is made in Pasadena)
  • 1/4oz Cherry Heering (made in Copenhagen)
  • 3/4oz simple syrup

Dry shake the juice and egg white (i.e., no ice). Add ice, aquavit, Cherry Heering, and simple syrup and shake hard. Double strain into a frosty coupe.

Please join me for a tipple if you can—and bring your own recipes to share!

Cast notation: a case study

I recently wrote on the SIGPLAN blog about how PL notation is a barrier to entry. While the arguments are mine, I acknowledge the many folks who helped me write it in the post. Ideas from an interesting conversation with Neel Krishnaswami came up again in a comment from Jeremy Gibbons. The universe has spoken: here’s what I think about cast notation.

First, here’s what Neel said:

I think that the argument in this note is a bit under theorized. As I see it, there are three fundamental forces at work:

1. Infix operators are hard to parse without extra side information (eg, precedence).

2. Center-embedding is hard to parse for psycholinguistic reasons.

3. Semantics is about relating things, and it is easier for people to see connections when the notational differences between the two things are small.

So when you compare cast(e, S, T) to e <S => T>, the functional notation wins in point 1 and loses on point 2. This is because cast(cast(e, S, T), T, U) has nested structure in a way that e <S => T> <T => U> does not—the second one can parse as exp cast*.

I don’t know the work on gradual typing well enough to say with any confidence, but I would have expected the second notation to be a bit better for 3. The semantics of the term e is a map Γ -> S, and if the meaning of a cast is an embedding function S -> T, then [[ e <S => T> ]] = [[e]]; [[<S => T>]] — i.e., the parts of the term can be interpreted using composition in a diagrammatic order without changing the relative position of the subterms.

My guess in this last part is also an example of the dynamic that leads to bad notation — we pick our notation at the outset, based on a guess about which semantic properties are important, and we can get stuck with bad notation if the properties that actually are important differ from the ones we guessed when designing the notation. (Well, people can also just have bad taste, I guess.)

—Neel Krishnaswami, personal communication.

Both Neel and Jeremy zeroed in on a feature I really like about the e <S => T> or e :: S => T notations: they’re neatly diagrammatic. Jeremy goes further, noting that these notation suggess that cast might compose, as in e <S=>T> <T=>U> ~= e <S=>U>.

If I were forced to choose a notation, I do think these two are the best… with a slight preference for e :: S => T. (I think Phil Wadler introduced this notation, but I’m too tired to run it down just now. Let me know in the comments? Edit: thanks to James McKinna for pointing out the source—“Blame for all” by Ahmed et al.—in a comment, below!)

So why do I prefer a function? In short: the notations suggest identities which don’t actually hold.

Casts don’t compose

Whether you’re casting between simple gradual types or dependent refinements… casts don’t actually compose! Consider a term f : int -> ?, i.e., a function f that takes an int and returns… something.

We can cast f to be purely dynamic, writing f' = f :: (int -> ?) => (? -> ?). These types are compatible, i.e., they differ only by having or not having a ?. Now eventually f' may flow out of the dynamic part of our code and arrive at some context that thinks f' ought to be a bool -> bool function, casting it. So we get:

f' :: (? -> ?) => (bool -> bool) =
(f :: (int -> ?) => (? -> ?)) :: (? -> ?) => (bool -> bool) =
f :: (int -> ?) => (? -> ?) => (bool -> bool) 

Now, composition would say that we ought to be able to convert the above to f :: (int -> ?) => (bool -> bool), but such a cast is statically forbidden—these types aren’t compatible because their domains are incompatible, i.e., you can never cast from int to bool! (If you’re surprised that compatibility isn’t transitive in this way, a whole literature awaits. I recommend Abstracting Gradual Typing by Garcia, Clark, and Tanter as a starting point: transitivity can fail.)

In the contracts world, e :: S => T => U ~= e :: S => U is just as bad. What if S = U = {x:Int|true}, but T = {x:Int|x>0}. By eliminating the cast in the middle, we’ve forgotten to check that e is positive! Such a forgetful semantics comes up as a possible space-efficient semantics, though you can do better.

Cast congruence

Where Jeremy talked about composition of casts, Neel talked about compositional semantics: that is, the postfix notation directly suggested a diagrammatic denotation, as in [[ e :: S => T ]] = [[ e ]] ; [[ S => T]]. My experience with casts suggests that this intuition isn’t a helpful one, for two reasons.

First: the key ingredient for space-efficient evaluation is not using conventional composition for casts, but rather treating casts in your continuation specially. That’s no ordinary semi-colon! A “cast congruence” lemma lets you recover conventional reasoning, but it takes quite some work to get there.

Second, treating casts as first class (i.e., utterable without being directly applied) forces you to think about very strange terms, like <S1->S2 => T1->T2> <S1 => S2>. (Just… don’t. For why?) Just as for primitive operations, it’s simplest to force casts to be fully applied.

Use coercions

I don’t like these notations for casts because they offer bad suggestions. A textual notation is modestly more cumbersome here, but it’s worth it for clarity to newcomers. It’s particularly worth skipping fancy notation in this setting, because casts are merely a technical device for a core calculus, not a part of the surface language.

But the real truth is: if you’re interested in higher-order runtime checks, you really should be using Henglein’s coercions anyway. (And check out Henglein and Rehof’s Scheme implemented on those principles while you’re at it.) Coercions are much clearer than casts, compose naturally, and are the source of space efficiency and fast implementations. What’s more, blame safety for coercions is straightforward… it’s syntactic!

Postscript: the beam in my eye

You might say, “Well, Michael, it seems like all your papers use the <S => T> e notation. Who are you to judge?”

I used that notation first for POPL 2010, when we (me, Benjamin Pierce, and Stephanie Weirich) tried to figure out whether or not contracts and hybrid types were the same. (They are at simple types. They aren’t if you have dependency—they treat “abusive”, self-contradictory contracts differently.) I just stole Cormac Flanagan’s notation from Hybrid Type Checking, changing an \rhd to a \Rightarrow. (Ugh, why change it at all? My guess: I could draw \Rightarrow better on the board.)

I’ve stuck with that notation… and that’s the heart of the problem. People get used to a board notation, and then they decide that their preferred shorthand is what should go in the paper. What’s good for you when you’re doing the work may not be helpful or even clear to new folks.

I like Neel’s final diagnosis. Don’t invent a notation in the first paper. Use whatever you want on the board, but publish with text first. Like a good stew, your notation will be better on the second day, once the flavors have had time to marry. Later on, if you decide you do need a notation, you’ll know exactly which identities you want your notation to suggest… and which it should not suggest!

Racist Bullshit in Mathematics

Robin Gandy’s “On the Axiom of Extensionality–Part 1”, Journal of Symbolic Logic, Vol. 21, No. 1 (Mar., 1956) quotes Alan Turing using a racist phrase.

A screengrab if the bottommatter of the first page of Gandy's paper.

Received July 24, 1955.
1 Indeed A. M. Turing once told me that he had done this, and that the proof was fairly difficult. I have found among his manuscripts two versions of the proof: one is rather short and contains a fallacy which could not, I think, easily be put right; the other (perhaps a second draft) is unfinished and only a beginning. He may therefore have discovered and surmounted the fallacy. On the other hand, he always spoke of the axiom of extensionality as being 'the nigger in the woodpile', which suggests that he did not think his consistency was transcendental enough to accord with Gödel's theorem; but, by the results of this paper, it would have to be just that.
[Turing] always spoke of the axiom of extensionality as being ‘the nigger in the woodpile’, which suggests that he did not think his consistency proof was transcendental enough to accord Gödel’s theorem.

Yikes. Those unfamiliar with this particular racist phrase will be disappointed to learn that it’s still current enough in the UK to be used “totally unintentional[ly]”… whatever that means.

Gandy’s paper isn’t the first time I’ve been pulled out of my mathematical/logical/philosophical reverie by racist bullshit. When I was reading Ronald Clark’s The Life of Bertrand Russell, I posted a thread on Twitter of Russell’s many racist utterances, with a selection of three racist Bertrand Russell quotes; two anti-racist quotes repudiate his earlier statements, offering some modest redemption.

I found these episodes of casual, by-the-way racism jarring: they pulled me out of my investment with the material and my ability or even desire to identify with the author.

What’s galling is that Turing “always” spoke of the axiom of extensionality this way; Gandy thought the phrase worth repeating verbatim; the reviewers and editors and publishers thought the phrase acceptable; and those who cite the paper don’t seem to find this footnote worth remarking on. Gandy’s paper is important and widely cited—a foundational resource on extensionality in general, and functional extensionality in particular—but if I refer someone to it, I’m going to let them know to expect a disappointingly racist quote from Turing.