Questions tagged [type-theory]
formal systems to specify properties of objects
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questions
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Is there a relationship between visitor pattern and DeMorgan's Law?
Visitor Pattern enables mimicking sum types with product types. Where does the "sum"-iness come from?
For example, in OCaml one could define ...
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0answers
30 views
Do any industry programming languages use Martin-Löf style identity types?
Most programming languages have some kind of type systems but are there any programming languages widely used outside of academia (in consumer-oriented tech, finance etc.) that have intensional ...
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0answers
26 views
What's the relationship between “semantic type soundness” and “functional correctness”?
In the Milner Award lecture "The Type Soundness Theorem That You Really Want to Prove (and now you can)" and related Sigplan blog post (with collaborators), Derek Dreyer argues that semantic ...
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22 views
Interpreting Minimal STLC using a $\lambda 1$ Category
On page 139, example 2.4.5 of "Categorical Logic and Type Theory" by Bart Jacobs demonstrates the interpretation of the abstraction typing rule with respect to a $\lambda 1$ category. ...
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1answer
64 views
Resources for implementing dependent type theory
I want to implement Martin Löf's intuitionistic type theory in a functional language such as Haskell, preferably also implementing a lexer/parser for the language. How should I start approaching it? ...
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16 views
Is there a uniform way of giving for any mathematical formula a hypercomputer that computes it?
Some mathematical formulas directly suggest an algorithm for computing it (even if sometimes an inefficient one).
For example, if we recursively define $\sum_{i=1}^nx_i=x_n+\sum_{i=1}^{n-1}x_i$, then ...
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1answer
35 views
Can we somehow get functoriality from purely type-theoretic reasoning?
In this question, I asked about how to prove naturality from parametric polymorphism, using parametricity. The current answer to that question simply assumes that the functors in question satisfy the ...
6
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2answers
165 views
Rigorous proof that parametric polymorphism implies naturality using parametricity?
This question asks for an informal explanation of why all polymorphic functions between functors are natural transformations (This is a claim made by Bartosz Milewski). One answer to that question ...
2
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1answer
37 views
What does $\text{dom}(\Gamma)$ mean in the context of an inference rule?
In the wikipedia page on pure type systems, it gives the following inference rule:
$\frac{\Gamma \vdash A : s \quad x \notin \text{dom}(\Gamma)}{\Gamma, x : A \vdash x : A }\quad \text{(start)}$
...
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55 views
Resources for connections between dependent type theory and LCCC
Can someone recommend introductory articles/papers on the connections between dependent type theory and locally cartesian closed category? Many Thanks!
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1answer
58 views
Is a system of equations derived from mutually recursive ADTs always uniquely solvable?
After looking at Can a computer determine whether a mathematical statement is true or not? for a while, I worry we may be into incompleteness/halting problem territory with this question, so an answer ...
2
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1answer
61 views
Curry–Howard correspondence and functional programming “reliability”
The first time I heard about functional programming, someone told me "it's more reliable to code in a functional style because your type system is like a proof of correctness".
I recently ...
2
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2answers
105 views
Does any language need to have a bottom type?
From wikipedia:
In type theory, a theory within mathematical logic, the bottom type is the type that has no values. It is also called the zero or empty type, and is sometimes denoted with the up tack ...
3
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1answer
74 views
What is an uninhabited type?
https://en.wikipedia.org/wiki/Type_inhabitation
The wiki article above says,
To be sound, such a system must have uninhabited types.
What is the definition an uninhabitated type? Do all programming ...
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0answers
27 views
Generalised letrec semantics for mutual recursion
I'm new to system types and I was wondering how mutual recursion is defined through generalized
e::= ..|let rec x1=e1 ,...., xn=en in e .What has to be added in the "simple" let rec ...
2
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0answers
27 views
How to specify mutated types mathematically?
Say I have an object which I pass to a method, and the method returns that same object, just mutated.
So it goes like this:
...
2
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1answer
56 views
Greatest fixpoint of the type of lists
I'm working through Samuel Mimram's book Program = Proof. In the first chapter, he discusses recursive types in OCaml, and inductive types. An exercise he provides on the topic has me a little bit ...
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45 views
What is the difference between type theory and logic programming (in terms of declarative programming and specification)
How is does type theory (coq, lean, agda), and logic programming (prolog, datalog) differ from each other.
Logic programming is a way of declarative specifying an algorithm, using classical 1st order ...
5
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1answer
284 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
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1answer
70 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
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1answer
69 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
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1answer
116 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
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2answers
149 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 ...
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29 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
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1answer
73 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 ...
3
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2answers
115 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
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4answers
437 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 ...
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87 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 ...
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2answers
79 views
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1answer
122 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 ...
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1answer
39 views
Example of Dependent Types?
Say you have 3 objects, a global MemoryStore, which has an array of MemorySlabCache objects, and each ...
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0answers
65 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
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1answer
76 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
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0answers
53 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 ...
4
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1answer
367 views
Why do we need a separate notation for П-types?
Main
I am confused about the motivation behind the need for a separate notation for П-types, that you can find in type systems from λ2 on. The answer usually goes like so - think about how one can ...
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1answer
117 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, \, ...
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0answers
45 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
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0answers
91 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
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1answer
256 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 ...
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0answers
92 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....
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2answers
86 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
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0answers
51 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:
...
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1answer
52 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 ...
5
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2answers
754 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 ...
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0answers
23 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 ...
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34 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 ...
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2
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1answer
101 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,...
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1answer
59 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 \...
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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 ...
