Questions tagged [type-theory]
formal systems to specify properties of objects
446
questions
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1answer
44 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
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1answer
30 views
Example of Dependent Types?
Say you have 3 objects, a global MemoryStore, which has an array of MemorySlabCache objects, and each ...
0
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0answers
53 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
42 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
44 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
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1answer
341 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 ...
1
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1answer
94 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
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0answers
40 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
84 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
247 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
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0answers
78 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
71 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
47 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
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1answer
39 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
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1answer
703 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
19 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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0answers
24 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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0answers
21 views
2
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1answer
55 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
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1answer
50 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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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 ...
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0answers
21 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
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2answers
55 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
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1answer
47 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
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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
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1answer
67 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
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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
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1answer
40 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
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1answer
32 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
247 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
62 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
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0answers
36 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
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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
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2answers
47 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: Γ(x) = k. For example, ...
2
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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
577 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
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0answers
48 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
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2answers
53 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
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2answers
136 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.
...
2
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0answers
23 views
Is possible to construct a fixed set of typeclases as powerful as unconstrainde typeclasses?
We can construct a fixed set of combinators with a computational power equivalent to lambda calculus.
Can we do the same with typeclasses (ad-hoc polymorphism)?
For example, construct a finite set ...
6
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2answers
89 views
Why is the type ∀t.t un-inhabited in System F?
How do you prove that there exists no term with the type $\forall t. t$ in System F?
I tried searching through Pierce's TAPL and Reynold's ToPL, but could not find anything. I suspect that the proof ...
5
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2answers
2k views
What do the ∀ and ∃ symbols mean in the Axiom of Choice?
On the Wikipedia page for the Axiom of Choice the following statement is given:
$(\forall x^\sigma)(\exists y^\tau)R(x,y)\rightarrow(\exists f^{\sigma \rightarrow \tau})(\forall x^\sigma)R(x, f(x))$
...
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0answers
44 views
What is the difference between a Type and an Abstract Type?
In my data structures course we are given definitions for Type and Abstract Type but they confuse me.
A type is a set of values and the operations you can do on them. The set of operations is ...
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0answers
42 views
in the lambda calculus with products and sums is $f : [n] \to [n]$ $\beta\eta$ equivalent to $f^{n!}$?
$\eta$-reduction is often described as arising from the desire for functions which are point-wise equal to be syntactically equal. In a simply typed calculus with products it is sufficient, but when ...
1
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1answer
47 views
Dynamic testing of down casts as explained in TAPL
On page 195 of Pierce's TAPL book, he states that one can replace a down-cast operator by some sort of dynamic type test. Then he gives the following rules:
T-Typetest:
$\dfrac{\Gamma \vdash t_1:S \;...
1
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0answers
36 views
How does progress fail in system $F_{\omega}$ when types $T_1 \to T_2$ and $T_2 \to T_1$ are equivalent?
Pierce's TAPL book gives in exercise 30.3.17 the setting where $T_1 \to T_2 \equiv T_2 \to T_1$ (the function type are assumed to be equivalent). In the solutions, he claims that this assumption ...
1
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1answer
88 views
Confluence to show equivalent terms have one common reduct
In lemma 30.3.9, Pierce states a confluence property for $F_{\omega}$:
$S \to_* T \land S \to_* U \implies \exists V. T \to_* V \land U \to_* V$
He then states the following proposition:
$S \...
3
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0answers
31 views
On the diamond property for $F_{\omega}$ in TAPL
In page 455, of Pierce´s TAPL and page 560, the single-step diamond property of reduction:
$S \Rrightarrow S' \land T \Rrightarrow T' \implies \exists V. T \Rrightarrow V \land U \Rrightarrow V$
...
2
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0answers
57 views
Are the permutation or weakening lemmas needed for term substitution?
In TAPL book, page 453, Pierce discusses the following lemma:
$\Gamma , x:S , \Delta \vdash t:T \land \Gamma \vdash s:S \implies \Gamma, \Delta \vdash [x \mapsto s]t:T$
He claims that when $t$ is ...
3
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0answers
26 views
What the type of an overloaded function should be?
Anecdotally, Virgill III language forbids overloading since overloading resolution is at odds with the language support of functions as first-class citizen, when resolution can't happen at compile ...
