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Questions tagged [type-theory]

formal systems to specify properties of objects

28
votes
1answer
2k views

Does there exist a Turing complete typed lambda calculus?

Do there exist any Turing complete typed lambda calculi? If so, what are a few examples?
4
votes
2answers
608 views

What is the type theory judgement symbol?

In type theory judgements are often presented with the following syntax: My question is what is that symbol in the middle called? All the papers I've found seem to use an image rather than a unicode ...
14
votes
1answer
550 views

Type inference with product types

I’m working on a compiler for a concatenative language and would like to add type inference support. I understand Hindley–Milner, but I’ve been learning the type theory as I go, so I’m unsure of how ...
22
votes
3answers
1k views

Categorisation of type systems (strong/weak, dynamic/static)

In short: how are type systems categorised in academic contexts; particularly, where can I find reputable sources that make the distinctions between different sorts of type system clear? In a sense ...
17
votes
3answers
2k views

How to read typing rules?

I started reading more and more language research papers. I find it very interesting and a good way to learn more about programming in general. However, there usually comes a section where I always ...
8
votes
3answers
374 views

Simple explanation as to why certain computable functions cannot be represented by a typed term?

Reading the paper An Introduction to the Lambda Calculus, I came across a paragraph I didn't really understand, on page 34 (my italics): Within each of the two paradigms there are several versions ...
25
votes
1answer
936 views

Is there a typed SKI calculus?

Most of us know the correspondence between combinatory logic and lambda calculus. But I've never seen (maybe I haven't looked deep enough) the equivalent of "typed combinators", corresponding to the ...
11
votes
1answer
400 views

Constraint-based Type Inference with Algebraic Data

I am working on an expression based language of ML genealogy, so it naturally needs type inference >:) Now, I am trying to extend a constraint-based solution to the problem of inferring types, based ...
20
votes
2answers
2k views

What is beta equivalence?

In the script I am currently reading on the lambda calculus, beta equivalence is defined as this: The $\beta$-equivalence $\equiv_\beta$ is the smallest equivalence that contains $\rightarrow_\beta$...
20
votes
2answers
2k views

Recursive definitions over an inductive type with nested components

Consider an inductive type which has some recursive occurrences in a nested, but strictly positive location. For example, trees with finite branching with nodes using a generic list data structure to ...
28
votes
2answers
527 views

Characterization of lambda-terms that have union types

Many textbooks cover intersection types in the lambda-calculus. The typing rules for intersection can be defined as follows (on top of the simply typed lambda-calculus with subtyping): $$ \dfrac{\...
19
votes
2answers
591 views

Are universal types a sub-type, or special case, of existential types?

I would like to know whether a universally-quantified type $T_a$: $$T_a = \forall X: \left\{ a\in X,f:X→\{T, F\} \right\}$$ is a sub-type, or special case, of an existentially-quantified type $T_e$ ...