Short answer: “a type system centered on the use of set-theoretic types (unions, intersections, negations) that satisfy the commutativity and distributivity properties of the corresponding set-theoretic operations”.
replyLong answer, well, there are blog posts[0], the Design Principles of the Elixir Type System paper[1] and related presentations[2, 3, 4] that talk about it at length. Giuseppe Castagna’s site has many more related papers: https://www.irif.fr/~gc/topics.en.html
[0]: https://elixir-lang.org/blog/2022/10/05/my-future-with-elixi...
[1]: https://www.irif.fr/~gc/papers/elixir-type-design.pdf
[2]: https://www.youtube.com/watch?v=gJJH7a2J9O8
Sets and types are foundational mathematical concepts so I’m looking for how elixir’s types fit in that context. Union and intersection are not something that belongs only to sets.
replyIt means that the types are built on unions, intersections, and negations[1]. It's a polymorphic type system with inference at the function level. It also does some type narrowing with pattern matching.
reply[1] https://www.irif.fr/_media/users/gduboc/elixir-types.pdf
Unions, intersections and negations are available in types as well and are by no means exclusive to sets. The distinguishing feature of a set vs type is that a value belongs to just one type while it can belong to several sets.
replyTypes do not inherently have any such restrictions. A value can belong to several types. In fact, if you posit types to have union, that necessarily follows.
replyI think they do, and as you mentioned you can explicitly remove such a restriction. Sets and types are once again two different kinds of objects in mathematical theory, and a set-theoretic type doesn’t seem to be based either on set theory or type theory.
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