# Introduction

Welcome to Counterexamples in Type Systems, a compendium of horrible programs that crash, segfault or otherwise explode.

The "counterexamples" here are programs that go wrong in ways that should be impossible: corrupt memory in Rust, produce a ClassCastException in cast-free Java, segfault in Haskell, and so on. This book is a collection of such counterexamples, each with some explanation of what went wrong and references to the languages or systems in which the problem occurred.

It's intended as a resource for researchers, designers and implementors of static type systems, as well as programmers interested in how type systems fit together (or don't).

## Type systems

For the purposes of this book, a "type system" is a system that does local checks of a program's source code, in order to establish some global property about the program's behaviour. The local checks usually assign a type to each variable and verify that they are used only in contexts appropriate to their type, while the global property varies but is generally some sort of safety property: the absence of memory corruption, use-after-free errors, and the like.

This is intentionally a fairly narrow definition. Some examples of things that are not "type systems":

• Linters: Linters do local checks of a programs's source code, but don't try to establish any global property. Just because a program passes a linter's checks does not imply anything definite about its behaviour.

• Python's type system: Dynamic languages like Python do local type checks to establish global safety properties. However, the type checks are not done purely on the program's source code before running it, but examine the actual runtime inputs as well.

• C++'s type system: C++ (among other unsafe languages) has a type system that does many local checks of a programs's source code. However, passing these checks does not establish anything definite about a program's behaviour: many C++ programs pass all of the compiler's type checking, and yet exhibit use-after-free errors, memory corruption, and arbitrary behaviour when run.

• Some program analysis tools: Some static program analysis tools do only global checks (requiring the whole program's source code to be available at once), while others detect certain issues but do not attempt to establish any global property when no issues are found. Still others do fit the definition of "type system", though: the line here is blurry.

This is not to say that there's something wrong with the above tools, but they're not what this book is about. Some examples of things that do fit the definition are the type systems of languages like Java, Haskell and Rust.

## Soundness and counterexamples

Type systems make the claim that any program that passes the local type checks will definitely have the global safety properties. A type system is sound if this claim is true, and to say that a system is unsound is to say that there exists some program which passes all type checks, and yet whose behaviour violates the global safety properties (corrupts memory, crashes unexpectedly, etc.). A counterexample is such a program, exhibiting a soundness bug.

Different languages enforce different properties, so exactly what "soundness" means varies somewhat. For instance, a global property established by the Coq type system is that all programs halt. So, a program that looped infinitely would be a counterexample in Coq, but not in Java, which makes no such claim.

The goal of this book is to collect such counterexamples, especially those that crop up in similar forms across multiple languages. These fall into a couple of broad categories:

• Missing checks: The simplest sort of soundness bug is when some important type check is missing from the type checker. Usually, these are straightforward bugs, easily fixed once discovered, highly specific to a particular compiler, and not very interesting. However, certain missing checks are common mistakes that keep coming up across many different type checkers and languages, and a few of these are included here: for instance, missing variance checks or scope checks.

• Feature interactions: Often, two type system features which are perfectly sound when used in isolation have some tricky interaction when used together. Examples of this include mixtures of polymorphism and mutability or polymorphism and overloading or intersections and mutability.

• Subtle differences: If two parts of a type system use different but similar versions of the same concept, the gap between them can often cause soundness bugs. For instance, does "private" mean private to an object or private to a class? Does "unequal" mean "not provably equal" or "provably distinct"?

## Organisation (or lack thereof)

Since so many of the counterexamples here depend on interactions between disparate parts of a complex type system, it would be tricky to separate them all into distinct coherent topics, and no attempt to do so has been made.

Instead, each entry is tagged with the type system features involved, and in lieu of a table of contents there is an index and glossary listing the features and the counterexamples in which they appear.

The entries themselves are mostly independent, and it should be possible to read them in any order. However, a few of them do refer to previous entries, and some attempt was made to put the simpler entries earlier, so if you do intend to read the whole thing then going start to finish is best.

## A note on sources

While much type system research appears in formal academic publications, it is relatively rare (but not unheard of!) to publish a paper about an unsound system.

So, as well as academic papers, many of the counterexamples here are drawn from unpublished notes, the archives of the TYPES forum, blogs by programming language researchers, designers and implementors, and language-specific bugtrackers, forums and mailing lists. A major reason for the existence of this book is to collect these various sources together.

In particular, compiler bugtrackers are a major source of the counterexamples reproduced here. Try not to read too much into how often different bugtrackers show up: while it is tempting to assume that the more often a compiler appears, the buggier it is, this would be wrong for several reasons:

• the author works professionally on OCaml, so it is somewhat over-represented due to familiarity,

• the same applies to other languages that the author likes or at least vaguely understands, and

• languages whose community does a good job of publicly tracking, explaining, and writing up issues that arise are also over-represented.