Questions tagged [type-theory]
formal systems to specify properties of objects
455
questions
5
votes
1answer
202 views
Relationship between inductive families and type-returning functions
Dependently typed languages such as Agda support inductive families, also called indexed datatypes, which allow type parameters to vary between constructors. This can be used to define a set of ...
3
votes
1answer
57 views
Which language is used to construct a type system?
Typically, OCaml and Scala seem to be used for designing any programming languages tool. But what features offer them an edge over other languages.
A related question, is a type system for a language ...
2
votes
1answer
53 views
What are the most used statements in programming (ranked)?
I was wondering if there are any resources for a study/ranking of the most frequently used statements (by statements I mean assigning, invoking, instantiating etc, like in C#) in programming overall (...
3
votes
1answer
91 views
How to prove that the Church encoding, forall r. (F r -> r) -> r, gives an initial algebra of the functor F?
The well-known Church encoding of natural numbers can be generalized to use an arbitrary (covariant) functor F. The result is the type, call it ...
4
votes
2answers
113 views
What are the implications of Homotopy Type Theory?
I've recently come across the topic of homotopy type theory and I'm interested to learn more. I have a very limited background in type theory.
Can anyone tell me, in functional programming terms or ...
0
votes
0answers
22 views
Is the borrow checker mechanism in Rust based on quantitative types?
I was trying to understand if the borrow checking mechanism for references is actually a quantitaive type in disguise because it does look very similar. In case these are just similar but unrelated ...
2
votes
1answer
55 views
How does the undecidability of Extensional Martin-Löf Type Theory apply to real type-checking compilers?
It is claimed in many sources (for example, here) that adding a rule like "if Id(X,Y) then X really equals ...
0
votes
1answer
39 views
How do you have a type typed “Type” when implementing a programming language?
I am working on the base of a language model, and am wondering how to represent the base type, which is a type Type. I have heard of an "infinite chain of ...
2
votes
4answers
422 views
Is there such thing as “sequential types”?
I am wondering how you could possibly define the implementation of an imperative function as a type. Is it possible? Currently I only see the input parameters and output result used in the definition ...
0
votes
0answers
78 views
Can we think of a non-symmetric product type in Haskell?
Meta note: I asked this question here a while ago. It got an answer:
type a /\!! b = (a, ((b -> Void) -> Void))
Unfortunately, I do not reckon it to be ...
1
vote
2answers
72 views
1
vote
1answer
65 views
What's the difference between Row Polymorphism and Structural Typing?
The definitions I've stumbled across seem to indicate they express the same idea. That's that the relationship between record types is determined by their fields (or properties) rather than their ...
1
vote
1answer
35 views
Example of Dependent Types?
Say you have 3 objects, a global MemoryStore, which has an array of MemorySlabCache objects, and each ...
0
votes
0answers
64 views
Consistency of a set of bidirectional typing rules
Main
Is there any way to algorithmically check the consistency of a set of bidirectional typing rules, e.g. the absence of cycles and the uniqueness of the derivation tree? This problem is naturally ...
2
votes
1answer
50 views
Does canonicity imply weak normalization?
Context: type theory.
My understanding of:
WN: every term can rewrite to NF.
Canonicity: every term rewrites into canonical form.
Then it leads to an intuition where if canonicity holds, then we get ...
3
votes
0answers
50 views
Semantics of “write-once” variables for complex data structures
Question
My use case for what is described below is not a language or compiler implementation, but finding a reasonable semantics for this feature in a an abstract calculus.
Ideally, you give me a ...
3
votes
1answer
352 views
Why do we need a separate notation for Š-types?
Main
I am confused about the motivation behind the need for a separate notation for Š-types, that you can find in type systems from Ī»2 on. The answer usually goes like so - think about how one can ...
1
vote
1answer
101 views
Creating a large tuple from smaller tuples via a monad or applicative
Suppose I have a term $a :\alpha$ of the Simply-Typed Lambda Calculus (in the following, $\alpha, \beta, \gamma$ stand for arbitrary types) and I want to lift it to a term
$\lambda x_{\beta}. \;(x, \, ...
2
votes
0answers
44 views
Difference between the logic and the type system of a proof assistant?
In Comparing Mathematical Provers (section 4.1), Wiedijk classifies logics and type systems of different proof assistants? I do not see what he means by type system of the assistant. He only says:
A ...
3
votes
0answers
89 views
(Co)-monads and terminating implementations
The bounty above should read 'I would like to know whether the example I discuss is a com-monad and why (why not).'
Suppose we set $\mathbb{M} \alpha := r \to \alpha$, where $r$ is some fixed type, ...
3
votes
1answer
255 views
The meaning and relevance of the locution ''no terminating implementation'' in type theory
In the context of a discussion of Haskell https://stackoverflow.com/questions/62509788/the-intuition-behind-the-definition-of-the-co-reader-monad, I was told that
There is no terminating ...
1
vote
0answers
89 views
How do type classes make ad-hoc polymorphism less ad hoc?
The title of the paper that introduced type classes is "How to make ad-hoc polymorphism less ad hoc".
It seems the type classes approach is being compared to how OOP does ad-hoc polymorphism....
1
vote
2answers
79 views
How to express a type that represents an associative array whose keys determine the type of the value?
I'm fairly new to type systems and theory, so I would appreciate some guidance in a problem that sparked my interest.
I would like to understand what type system features are required so a compiler ...
2
votes
0answers
48 views
Inhabitation of STLC is in PSPACE
Urzyczyn: Inhabitation in Typed Lambda-Calculi (A syntactic approach) gives a proof that STLC inhabitation problem is in PSPACE (section 2, lemma 1). I don't understand certain aspects of the proof:
...
1
vote
1answer
43 views
Terms for different models of sum types
There seem to be at least a couple different possible ways of modeling sum types in a type system, but I haven't been able to find consistent terms for referring to them:
A sum type is formed from a ...
4
votes
1answer
711 views
Question on the “Tutorial implementation of dependently typed lambda calculus”
I have a slight technical struggle with this marvelous tutorial. On page 5 the tutorial talks about typing rules for Simply Typed Lambdas and presents following judgement as derivable via rules on ...
1
vote
0answers
21 views
Does Quantitative Type Theory make the Prop universe obsolete?
Coq (and other type theories such as Setoid Type Theory) have a Prop universe for propositions. As far as I understand this universe is needed to be sure that the propositions can be erased. In ...
0
votes
0answers
29 views
Is Observational Equality better than intensional equality?
The Observational Equality from Epigram 2 seems to be intensional equality (like Coq and Agda have), but it also supports function extensionality. In that sense it seems that Observational Equality is ...
1
vote
0answers
24 views
2
votes
1answer
67 views
Difference between assignment, binding, and substitution?
I am trying to understand the difference of assignment, binding, and substitution. I know the three things are related, but to me it's not exactly clear what word refers to what. Example, illustration,...
1
vote
1answer
52 views
Substitution lemma for types
TAPL (page 549) proposes the following lemma in order to prove soundness of System F type system:
Substitution lemma for types:
$E, X, \Delta \vdash t: T \implies E, [X \mapsto S] \Delta \vdash [X \...
1
vote
0answers
39 views
The abstract interpretation corresponding to the pure simply typed lambda calculus
In Types as Abstract Interpretation, Patrick Cousot sketched how different type systems could be constructed from the collecting semantics of a language. However, the notation of the paper is very old ...
0
votes
0answers
26 views
Cayley diagram for Frieze group
In Type Theory there is Rule: Every action is reversible.
There are 7 groups for 1d repeating pattern (Frieze groups).
Group 1: only translations.
Group 2: only glide reflection.
Why Cayley ...
0
votes
2answers
66 views
What is the single type in a dynamic typing language?
Regarding static typing and dynamic typing, Practical Foundation of Programming Languages by Harper says:
There have been many attempts by advocates of dynamic typing to
distinguish dynamic from ...
2
votes
1answer
49 views
Do dynamic/static languages associate types or classes to values or variables?
In Practical Foundation of Programming Languages by Harper
There have been many attempts by advocates of dynamic typing to
distinguish dynamic from static languages. It is useful to consider
...
0
votes
1answer
39 views
Doubts on the behavior of Unit Type in a type system
I have a doubt about the Unit Type in the context of Type Theory and its use in different case scenarios.
To start with, a Unit Type can be seen as a nullary Product Type, namely Unit, with one only ...
3
votes
1answer
71 views
What are Contexts in Lambda Calculus?
What is a Context? Is it like a scope in C? Does it have a start and an end? Can contexts contain other contexts?
I see Contexts being used in lambda calculi type system rules, but I don't understand ...
3
votes
1answer
72 views
What is the formalism to prove statements about uniqueness of functions with certain signatures
Suppose I want a function like
f: ((A, B) -> C) -> A -> B -> C
A statement I've often seen made is that f has just one implementation, namely the '...
1
vote
1answer
45 views
Why F-bounded polymorphism and F-bounded quantification are called, well, F-bounded
It's claimed in Wikipedia that:
F-bounded quantification or recursively bounded quantification,
introduced in 1989, allows for more precise typing of functions that
are applied on recursive ...
1
vote
1answer
34 views
Type inference with overloading
I am working on a type system supporting overloading. I have a rough idea of how type inference is usually implemented in such a scenario, but I am wondering how - after type inference is completed - ...
3
votes
1answer
251 views
How to get an element from an existential proposition in Type theory proof assistant (Lean prover)
I am trying to implement set theory in type theory from scratch, just for self pedagogical purposes. Specifically, I'm using the Lean Prover, and defining the element-of relation from scratch using ...
2
votes
2answers
63 views
Tightening application rules for STLC
The syntax STLC is usually written:
$e ::= x |\lambda x : \tau . e|(e \space e)|c$
However, the application rule appears to accept all expressions on the left hand side. Shouldn't the application ...
1
vote
0answers
40 views
Proof of Subject-Expansion Theorem in Type Theory
I am beginning to study type theory, using Hindley's book "Basic Simple Type Theory", and I would like your help in the proof of a theorem. I would like to know whether my idea for how to prove the ...
0
votes
1answer
39 views
Is there any correspondence between SUM type in type theory and arithmetical summation?
Is there any correspondence between the coproduct(sum) type in type theory and arithmetical summation?
For example what does 3+4 or x+6 mean in type theory?
3
votes
2answers
48 views
Parentheses after Typing Environment
I've been reading about System F Omega lately, and I keep stumbling across a construct in typing rules that I cannot find an explanation of: Ī(x) = k. For example, ...
2
votes
1answer
32 views
Type inference with imports
I understand how a type inference algorithm infers types within a single file by building on top of already inferred types and identified constraints (e.g. in the Hindley-Milner type system).
I am ...
5
votes
2answers
583 views
Does the underlying computational calculus in type theories affect decidability?
I'm looking for a high-level explanation although if that isn't possible or difficult, I'd prefer references to books/papers.
I understand that modern type theory is inspired by Curry-Howard ...
1
vote
0answers
51 views
Does Type:Type lead to inconsistency without general inductive types?
In e.g. Agda , it is possible to derive an element of the empty type by enabling the "type in type" option.
Every proof I have seen (and come up with) involves making a special inductive type ...
2
votes
2answers
56 views
What is the differences and similarities between refinement type and liquid types?
Looking at the examples here and here both refinement type and liquid types look very similar. What are the differences and similarities?
3
votes
2answers
148 views
When is cumulative type universes useful?
AFAIK, a hierarchy of type universe(Type^0: Type^1: Type^2: ...) was introduced to avoid inconsistency caused by Type: Type.
...
