[hhas01]

Curious: from what I can see, a type system is really just an embedded declarative DSL for doing set algebra.

So is there a technical reason why education and implementations always intertwine it with a larger client language, rather than treating it as a complete, self-contained entity in its own right? Or is that lack of decomposition just oversight?

[Drup]

A self-contained type-system is called "a logic". There are plenty of logics, many of which are not related to set algebra. Furthermore, many type systems are not directly related to set algebra (for instance, Rust's ownership). There are also more profound reasons (what is the type of a type, cf the notion of impredicativity and questions about decidability), but those are usually of little practical consequences for "mainstream" languages.

With only a logic, you can't show properties about the language. In particular, we like to show that programs that are well typed avoid some errors (like segfaults), or that if an expression ( `log2(length(s))`) has a type (`int`), when you evaluate it to a value, the value (`4`) has the same type. If you want to consider such (nice!) properties, you need to consider the type system and the language together.

[black_knight]

Some type systems have set theoretical semantics, but this does not usually correspond to the semantics of the program.

For instance the type ∀ A. (A → A) → A would be empty in the set theoretical semantics, but in many programming languages there are terms of this type – the fixed point operator.

A better framework for semantics of many programming languages is domain theory[0]. But it also has its limitations.

The type system types the terms of the language. Thus it has to be intertwined with them. One of the purposes is to allow only well-typed terms, which then have a better understood semantics.

[0]: https://en.wikipedia.org/wiki/Domain_theory

[tomp]

You can count the number of type systems that are "logically OK" (the formal terms are sound (doesn't admit wrong programs, unlike e.g. Java), complete (can admit all correct programs (for some definition of "correct"), unlike e.g. Scala), decidable (always finishes)) on one hand. The rest amount to various amounts of hacks - common ones include specifying function parameter types to allow type inference to function, various escape hatches (like Object in Java and interface {} in Go), generics incorrect or limited to various degrees. Subtyping is tremendously hard. The specific hacks implemented in different programming languages are often implementation-specific (and arbitrary), and often result in weird interactions between different language features.

Basically, type systems == science (figuring out new things), programming languages = engineering (making new findings useful).

[TheAsprngHacker]

> You can count the number of type systems that are "logically OK" (the formal terms are sound (doesn't admit wrong programs, unlike e.g. Java), complete (can admit all correct programs (for some definition of "correct"), unlike e.g. Scala)... on one hand. The rest amount to various amounts of hacks

Huh? Rice's Theorem states that any static semantic analysis must either reject valid programs or accept invalid programs. This says nothing about whether a type system is rigorous or hacky.

> common ones include specifying function parameter types to allow type inference to function

It's true that type inference becomes undecidable without annotations for things like dependent types, and there's nothing hacky about that.

[ImprobableTruth]

Rice's theorem only aplies to turing complete programs, so if youre "weak enough" there's no problem being both sound and complete. I think it's easy to see why people would think of such systems as "nicer" (even though the others are of course still rigorous).

>It's true that type inference becomes undecidable without annotations for things like dependent types, and there's nothing hacky about that.

But it's not just for more advanced type systems, type inference is undecidable for basically any system more complex than HM typing. So in practice virtually all programming languages with type systems that go beyond the absolue basics can't fullfill the "promise" of type inference, which is that you don't have to explicitly annotate types. I think if you take into consideration that even relatively "good" type systems like Haskell's occasionally require you to annotate not just functions, but 'random' terms, you might see why some people might consider it "hacky".

[chrisseaton]

> a type system is really just an embedded declarative DSL for doing set algebra

I vaguely remember that there is some maths that tells us that types aren't sets, but can't recall the details. Something to do with Russel's Paradox and function types that refer to themselves as an argument? (I'm not a mathematician.)

[_se]

Try it yourself. It's much more difficult than you seem to think it is.

[hhas01]

Well, isn’t that true of everything? Being a bear of very little brain, it’s why I’m always searching for ways to decompose the larger problem into smaller, simpler, self-contained chunks. Plus, a less common perspective sometimes yields insights that the mainstream may miss.

Thanks for the inputs, folks. Will cogitate further.