January 28, 2024

After my last post, a few people seemed interested in the papers I read, so I thought I'd write a (necessarily long) post mapping out my experience of programming language papers, with tons of recommendations for programming language developers to start exploring. At the end I'll have a list of all the papers I mention throughout the post. Note that this is just my own journey through the academia, and will be missing plenty of important work. Hopefully what I have here will serve as good jumping off points for your own PL research.

If you aren't familiar reading academic papers, don't worry! It's not as hard as you think. My best advice for getting started is patience. Papers will often use terms you (or I) are not familiar with, but then they'll define the terms later in the paper. I don't know why people write this way, but it seems to be a convention. Also, when you read a paper, start with the abstract, introduction, and conclusion. Leave the actual technical details of the work to the end.

For terms that they don't define, there are two youtube playlists I HAVE to recommend, by Bartozs Milewski and Richard Southwell. I especially recommend the latter one, for getting me up to speed on a ton of important concepts in an accessible way. The things that it taught me that you should probably know: intuitionistic logic, sequent calculus, dependent types, the Curry-Howard isomorphism. If you don't like youtube videos, many blog posts across the internet explain pieces of these ideas. Exercise your Google-Fu! (Or whatever search engine you use.) Self learning requires a lot of googling.

The prerequisites for this post are all over the place: if you don't understand something, just skip it! There may be other things in other sections that are more digestable. I hope this post will have something for people who are fairly advanced too, since some of these things are pretty niche, even in PL.

When your code doesn't compile because of a type error, it means the language is doing something called *static analysis.*
"Static" is an adjective referring to everything at compile-time.
"Dynamic" is the opposite, referring to everything at runtime.
Thus, "dynamic analysis" means doing some runtime checks, like guaranteeing that indexing into an array is in bounds
by crashing (or producing `undefined`

) if it's not.

Type theory is the bread and butter of static analysis.
If you say `int x = 5;`

then you're doing type theory, because `x`

is given the *type* `int`

.
If you then try to assign `x`

the value `"hello!"`

then that's a type error,
because any value of `x`

must have the type `int`

, but `"hello!"`

has the type `string`

.
Types are very familiar to programmers.

In a "statically typed" language, types are checked at compile-time.
In a "dynamically typed" language, types are checked at runtime.
In a "strongly typed" language, incompatible types leads to a crash or compile error.
In a "weakly typed" language, incompatible types are *coerced,*
meaning one of the incompatible values is modified to make the types match.
JavaScript is the classic example of a weakly typed language: `parseInt`

needs its argument to be a string,
so if you give it something else then that thing will be implicitly turned into a string before `parseInt`

starts working with it.

Dynamic typing is still absolutely a part of type theory, and I've found some of the papers on it to be fascinating. When you combine static and dynamic typing, you get a system that can express anything correct and rule out some incorrect things, whereas type systems usually try to rule out all incorrect things and still allow some correct things. They're inverse approaches. Quasi-static typing (Satish Thatte) is my personal favorite paper on this, related to a much more famous paper with a godsquad of four researchers you should pay attention to. Other interesting work on combining static and dynamic types (often called "gradual typing") are blame (Wadler) and hybrid typing (Flanagan), both from researchers that are worth keeping an eye on. Wadler did much of the theoretical work behind Haskell, and Flanagan did much of the work behind ANF compilation. I'm sure I'm forgetting important work here but that should be enough to give a good jumping off point for your own research.

That's all I'm going to say about dynamic types now,
since type theory generally tries to remove runtime checks as much as possible.
The farthest you can go with this is something called *dependent types.*
Dependent type theory is enormous but some names to know are Per Martin-Löf
(a logician and philosopher who came up with many of the original ideas)
and Thierry Coquand.

Dependent type theory can be quite difficult to grasp the first ten times you try, but eventually the ideas will start to click, no matter your background. Earlier I mentioned Richard Southwell's wonderful videos for learning dependent type theory, and I've also written a post that tries to gently introduce the ideas.

The main thing you need to know about dependent types for now is that they allow you to express any
facts you want about your programs (more or less) and offer you a way to prove those facts.
So you could write a function that gets things from an array, but the given array has to be
big enough for the given index to make sense.
Then to use this function, you just write a proof that the array is big enough for the index.
It's quite a bit of effort compared to using types in traditional languages,
but it's very useful when program correctness really matters.
If you want the language to *figure out the proofs for you,*
check out Liquid Haskell's refinement types.
Note that in general dependently typed languages use intuitionistic logic for their proofs;
if you're interested in expressing classical logic, check out the continuation stuff I mention later in the post.
Lean notably allows classical logic with its dependent types.

The classic dependently typed system is called the Calculus of Constructions, or CoC. There's no way to express your typical functional-language ADTs in it, so it's often extended to the Calculus of Inductive Constructions, or CIC, which is really just CoC + (G)ADTs. Cedille (which I discuss further later in the post) has another approach to adding ADTs using pure lambda calculus, the details of which I won't discuss in this post.

Since we can do arbitrary computation in the type system in a dependently typed language,
we suddenly need types *for our types,* which we call *kinds.*
A very simple and beautiful formalization of this structure is
pure type systems
or "PTS"s, by a very famous researcher named Henk Barendregt.
The paper I just linked calls them "generalized type systems" but that's not what anyone calls them anymore.
His "\(\lambda\)-cube" in that paper is extremely famous, with CoC being the most powerful system present.
So there aren't ADTs in any corner of the cube.
(And every corner of it is a system that definitely halts, so none of it is even turing complete!)

If you want you can have computation on the kind level, the heirarchy of types-of-types-of-types-of... can go on forever (in which case they're called "universes") and the "Extended CIC" has such an infinite heirarchy. The Coq (maybe renamed to Rocq?) dependently typed language is based on the extended CIC.

Dependent types allow you to write functions that produce types,
and call those functions at compile-time to decide the types of your values.
That means you can have arbitrary code running at compile time,
so most dependently typed languages aren't allowed to loop forever.
(There are other reasons infinite loops are bad, which have to do with how dependent types express propositions and proofs.)
This segues to a super fascinating field about *termination checking.*
Think of it as "how to get around the halting problem" or
"how bad is it, *really,* to be turing incomplete?"
The goal is to be able to express normal programs while also knowing that none of those programs loop forever.

A very important part of termination checking is *rewrite theory.*
It's a very fun theory about computation where you have a list of symbols
(a "term") and a set of rules (a "term-rewriting system" or TRS)
and you apply those rules in any order to the term until none of them apply anymore.
A rule might look like \(t\&f\rightarrow f\), which shows a computation rule for logic.

Term rewriting represents computation as a bunch of terms with a "rewrite-to" relation between them; that is, computation is done in set theory, and function composition is just relation transitivity. It's quite elegant, and gives a nice language for talking about computation. In particular, if there are no infinite sequence of steps in your rewrite relation, then the rewrite system halts on every term you give it. Often you show that the sequences are finite by assigning each term a positive integer in such a way as to guarantee that each step in the relation always reduces the positive integer. Since every term's integer is positive, you know that those subtractions can't go on forever.

A fun paper combining functional programming and rewrite theory is Decidable Type Inference for the Polymorphic Rewriting Calculus. A great book on rewrite theory and termination checking is Term Rewriting and All That by Franz Baader and Tobias Nipkow (who's a top researcher in rewrite theory).

Most dependently typed languages assign each term its "depth," which is meant like the depth of a binary tree. For a linked list, its depth is its length. This is a useful positive integer to assign because recursion (the only way to loop forever in a functional language) usually is applied to the subcomponents of a term, that is, terms whose depth are 1 less. To support this and other reasoning, they'll often encode positive integers themselves as something like linked lists, and do math using the lengths/depths. The typical paper for termination checking a dependently typed language is foetus. Another concept to know that encodes the depth idea more explicitly is sized typing.

Another very important angle of termination checking is *recursion schemes.*
As far as I know this came about in the very famous paper on
bananas, lenses, envelopes, and barbed wire.
They divide recursion into two types, called *catamorphisms* and *anamorphisms,* which are dual in category theory (look up "F-Algebras").
Catamorphic recursion is when the size or depth is going down, as I discussed earlier;
computation is trickling down a tree into smaller and smaller subtrees.
Catamorphisms are guaranteed to halt.
Anamorphic recursion is when the size is going up, because you're building a tree and recursing on it.
For example, you might recursively push something onto a linked list until its length is 10 or more.
Anamorphisms (or corecursion) might not halt, in general,
so much of the recent research into termination checking focuses on them,
with terms like "coinduction" (anamorphism) and "corecursion" (apomorphism).
There are many other recursion schemes (hylo-, apo-, para-, etc.-morphisms, the list goes on).

One important recursion scheme is called "Mendler-style" recursion. It's a fancy way of doing catamorphisms that the Cedille dependently typed language uses for all of its termination checking. Cedille is interesting for many reasons, which I'll get into in the next section.

In general when we think of types and type theory, we're thinking in terms of only *intrinsic* type theory,
or "Church-style" type theory, or "type theory a la Church." (I swear it's called that.)
This means that the meaning of an expression is dependent on its type.
`fn foo(i: int) { return i; }`

does nothing but return what you give it,
but we might say that if it was for `float`

instead of `int`

then it would be a different function.
*Extrinsic* typing argues that the `int`

version and the `float`

version are *the exact some function,*
namely its *untyped* behavior, and we just have various methods for *assigning* types to untyped expressions.
Much of the end result is the same, it's just a different way of looking at things.
Extrinsic type theory is also called "Curry-style" type theory or "type theory a la Curry."

The most famous consequence of this way of thinking is *intersection types.*
These mean assigning one expression *multiple* types (so it's in the "intersection" of these types).
The most famous language that does this is TypeScript, by far.

Some intersection type systems can well-type all the expressions that halt, and none of the expressions that don't, thereby characterizing halting in the type system. Obviously, the typechecking for these systems is undecidable.

Within academia another very famous extrinsic system is Cedille.
Cedille is a dependently typed language with *dependent* intersection types,
meaning the second type in the intersection can refer to the value with that has the first type
(which is the second type's own value too, meaning the terms type can talk about the term itself).
It also has an implicit products
which are like an infinitary intersection type.
(Here's a paper.)
Cedille also has an extrinsic equality type to let you really say that a no-op function for integers is equal to a no-op function for floats.
It does all of this to be able to express inductive types (ADTs) with a much smaller extension to the CoC than the CIC requires.
One of the Cedille papers is this but many of them are very interesting.
If you like podcasts, Type Theory Forall has a couple wonderful episodes on Cedille.

Extrinsic type theory is very deep and has their own versions of many intrinsic things, like the \(\lambda\)-cube. My favorite is the "\(\Delta\)-chair", discussed here.

Much of my favorite work on making functional languages fast is connected to Greg Morrisett, a researcher I really admire. Work that he was a part of is a big inspiration for SaberVM, as well as the Rust programming language. I recommend the papers on Cyclone, Morrisett's PhD thesis, intensional polymorphism (Robert Harper is a researcher to be aware of; also I think Sixten uses something like this for its polymorphism), and the calculus of capabilities (which I have a post on).

A few of those suggestions use *regions,* a notion typically connected to
Tofte and Talpin's region calculus.
It's a super famous paper about basically using arenas for functional programming, with lifetime inference.

The Gibbon compiler takes a subset of Haskell and improves cache locality and parallelism by turning the datastructures into arrays. (Without the language user having to change anything in their code!)

Continuations are very very useful for compiling functional programs, as they let the functions kind of disappear. The idea of a continuation is that instead of a function returning, it takes a function as its argument and calls that function with what it would have returned. Thus the caller says "when you're done with your work, don't give it to me. Instead, send it over to this other function." It's a transformation of the program into something that can just be expressed with jumps in machine code. SaberVM uses "Continuation-Passing Style" (CPS) because it's such a useful structure for functional language compilers. Compiling With Continuations is the go-to reference on this, and I found it pretty accessible and helpful when I read it. CPS is also really good for optimizations. Since then ANF has become a popular alternative, for being simpler in certain ways while still allowing many of the CPS optimizations. The ANF researchers are yet more names you'll see more often.

There's a lot of interesting work on parallelism in functional languages. Par Means Parallel is a paper using linear logic (everything used exactly once) for parallelism. Choreographic programming is a really neat way of architecting distributed computation. LVars use monotonicity to guarantee deadlock freedom.

Linear logic leads to good resource usage too. Linear Types Can Change The World is a super famous paper using linear types for safe in-place updates of immutable values. this paper also uses linear types, for much more of the memory management. Alias Types use linear logic but separate the references from the "proof things" so that the references can still be used nonlinearly.

The last section I'll touch on is Information Flow Control, or IFC. This is a big part of "language-based security," where information security is achieved through programming language features. DCC is the classic reference, which is able to track and restrict the flow of information through programs. FLAC extends this with trust relations that can change during the runtime of the program, and some very elegant security and lattice theory. DCC and FLAC use monads, and this paper brings in comonads as well for more interesting features, in particular related to modal logic. This paper offers a potential simplification of DCC.

Amal Ahmed and William Bowman have done a lot of work on something called "secure compilation" or "full abstraction," where the compiler produces something that is itself typed and preserves all the security guarantees of the source language. Because of the expressivity of System F (the polymorphic typed lambda calculus), it's seen as a useful target for compilation, as it can express many of the features of source languages. For example, Haskell extends System F only a little bit, and has syntax sugar the compiles to it for everything else. A really interesting paper relating this to information flow is Noninterference For Free. Noninterference is the property that IFC systems try to have (think of it as "secrets don't get leaked") and "theorems for free" is how we refer to the power of polymorphism in System F, referencing this classic paper.

One final piece of advice I have for self-learning is talking to people. Part of that means joining discord servers and subreddits and whatnot that focus on programming languages, but another important part is talking to researchers. When you find a paper with ideas that you really like, collect a list of questions you have about it and send an email to the authors. Cold emailing has been very successful for me, and many wonderful papers I discovered were sent to me as responses from researchers I cold-emailed. I discovered intersection types from emailing the authors of a paper using rewrite theory. I discovered an dependent type paper about Haskell from emailing one of the guys behind Cedille. You have no idea what papers people will recommend to you, but you can rest assured that the PL research community is generally quite friendly, patient, and thoughtful.

Finally, here's the promised list of papers:

Quasi-static typing (1990) by Satish R. Thatte.

Dynamic typing in a statically-typed language (1989) by Martin Abadi, Luca Cardelli, Benjamin Pierce, and Gordon Plotkin.

Well-Typed Programs Can’t Be Blamed (2009) by Philip Wadler and Robert Bruce Findler.

Hybrid Type Checking (2006) by Cormac Flanagan.

Refinement Types: A Tutorial (2020) by Ranjit Jhala and Niki Vazou.

Introduction to Generalized Type Systems (1991) by Henk Barendregt.

Decidable Type Inference for the Polymorphic Rewriting Calculus (2006) by Horatiu Cirstea, Claude Kirchner, Luigi Liquori, and Benjamin Wack.

Term Rewriting and All That (1998) by Franz Baader and Tobias Nipkow.

foetus – Termination Checker for Simple Functional Programs (2002) by Andreas Abel.

Calculating sized types (1999) by Wei-Ngan Chin and Siau-Cheng Khoo.

The Implicit Calculus of Constructions as a Programming Language with Dependent Types (2008) by Bruno Barras and Bruno Bernardo.

Syntax and Semantics of Cedille (2018) by Aaron Stump.

Experience With Safe Manual Memory-Management in Cyclone (2004) by Michael Hicks, Greg Morrisett, Dan Grossman, and Trevor Jim.

Compiling With Types (1995) by Greg Morrisett.

Compiling Polymorphism Using Intensional Type Analysis (1995) by Robert Harper and Greg Morrisett.

Typed Memory Management in a Calculus of Capabilities (1999) by Karl Crary, David Walker, and Greg Morrisett.

Region-Based Memory Management (1997) by Mads Tofte and Jean-Pierre Talpin.

Efficient Tree-Traversals: Reconciling Parallelism and Dense Data Representations (2021) by Chaitanya Koparkar, Mike Rainey, Michael Vollmer, Milind Kulkarni, and Ryan R. Newton.

The Essence of Compiling with Continuations (1993) by Cormac Flanagan, Amr Sabry, Bruce F. Duba, and Matthias Felleisen.

Par means parallel: multiplicative linear logic proofs as concurrent functional programs (2019) by Federico Aschieri and Francesco A. Genco.

Choreographic Programming (2013) by Fabrizio Montesi.

LVars: Lattice-based Data Structures for Deterministic Parallelism (2013) by Lindsey Kuper and Ryan R. Newton.

Linear Types can Change The World! (1990) by Philip Wadler.

Linear Regions Are All You Need (2006) by Matthew Fluet, Greg Morrisett, and Amal Ahmed.

Alias Types (2000) by Frederick Smith, David Walker, and Greg Morrisett.

A Core Calculus of Dependency (1999) by Martín Abadi, Anindya Banerjee, Nevin Heintze, and Jon G. Riecke.

A Calculus for Flow-Limited Authorization (2021) by Owen Arden, Anitha Gollamudi, Ethan Cecchetti, Stephen Chong, and Andrew C. Myers.

Monadic and Comonadic Aspects of Dependency Analysis (2022) by Pritam Choudhury.

A Perspective on the Dependency Core Calculus (2018) by Maximilian Algehed.

Noninterference For Free (2015) by William Bowman and Amal Ahmed.

Theorems for free! (1989) by Philip Wadler.

The Delta-calculus: syntax and types (2018) by Luigi Liquori and Claude Stolze.

Dependently-Typed Programming with Logical Equality Reflection (2023) by Yiyan Liu and Stephanie Weirich.