39
votes
Why colon to denote that a value belongs to a type?
The main reason to prefer the colon notation $t : T$ to the membership relation $t \in T$ is that the membership relation can be misleading because types are not (just) collections.
[Supplemental: I ...
- 29.4k
21
votes
Accepted
Subtypes as subsets of SML datatypes
These kinds of types -- where you define a subtype (basically) by giving a grammar of the acceptable values -- are called datasort refinements.
They were introduced by Tim Freeman and Frank Pfenning,...
- 32.8k
20
votes
Accepted
Relative consistency of PA and some type theories
The short answer to your question 1 is no, but for perhaps subtle reasons.
First of all, System $F$ and $F_\omega$ cannot express the first-order theory of arithmetic, and even less the consistency ...
- 14k
20
votes
Accepted
Swapping arguments of variables in higher-order pattern unification
I have developed this, but haven't yet published it in a more strucured/formal manner.
"Enhanced pattern unification" abstract here.
Demo implementation.
Video recording.
You're absolutely ...
- 1,639
16
votes
Do I have to give up the Law of the Excluded Middle in order to Learn $\lambda$-Calculus?
To me, your question seems analogous to saying "I've heard that non-Euclidean geometry requires me to give up Euclid's fifth axiom, which is very useful in many mathematical contexts." You don't have ...
- 261
16
votes
In the Hott book, are the most of the type formers redundant? And if so, why?
You are asking several questions which are similar but distinct.
Why doesn't the HoTT book use Church encodings for data types?
Church encodings do not work in Martin-Löf type theory, for two ...
- 32.8k
15
votes
Accepted
In the Hott book, are the most of the type formers redundant? And if so, why?
Let me explain why the suggested encoding of the empty type does not work. We need to be explicit about universe levels and not sweep them under the rug.
When people say "the empty type", they might ...
- 29.4k
15
votes
Accepted
What is the difference between System F and Hindley-Milner type system?
Yes, In Hindley-Milner universal quantifiers are allowed only at the outside of a type (and therefore omitted). For example, in HM you can have the type $\forall \alpha.(\alpha\to\alpha) \to (\alpha\...
- 643
15
votes
Accepted
Applications of algebraic geometry in type theory/programming language theory
To my knowledge (which is definitely incomplete), there has been relatively little work on this, presumably because it requires assimilating two relatively intricate bodies of knowledge. However, ...
- 32.8k
15
votes
Accepted
Why is regularity a problem in cubical type theory?
The difficulty is in making such a reduction compatible with all the other reductions involving transport/coe. From one perspective it is a “confluence” problem. It is unfortunate that in the ...
14
votes
Accepted
Do I have to give up the Law of the Excluded Middle in order to Learn $\lambda$-Calculus?
You seem to be confusing several things here.
First of all, like Alexis said in her answer, I don't see why you would need to accept/reject the principles of a given logical theory in order to study ...
- 5,598
14
votes
Accepted
Can Isorecursive types capture mutually recursive data types?
In general, for any type (or domain, or complete lattice) $X$ we can consider the least fixed-point operator $\mu_X : (X \to X) \to X$. For recursive types we take $X = \mathsf{Type}$, i.e., we apply $...
- 29.4k
14
votes
Accepted
Why colon to denote that a value belongs to a type?
Because what's on the right of the colon isn't necessarily a set and what's on the left of the colon isn't necessarily a member of that set.
Type theory started out in the early 20th century as an ...
14
votes
Accepted
Defining inductive types in intensional type theory purely in terms of type-theoretic data
It turns out that $W$ types plus identity types (eq/= in Coq) allow you to construct pretty much all the general inductive types ...
- 256
14
votes
Why is the Curry-Howard isomorphism?
I suspect someone is going to come along with a deep category-theoretic reason for the connection, but in the meantime, here is my insight.
Both logic and programming with functions are built around ...
- 2,952
14
votes
Accepted
Which universities in the U.S. are doing research in type theory?
Any such list is always subjective, but the best approach to answer this question is:
Look at journals/conferences in the area you're interested in. For type theory, I'd look at LICS, LMCS, POPL, ...
- 2,952
14
votes
Accepted
Where is the model theory in programming language theory?
The model theory of programming languages is called denotational semantics. You can google the term to find out more about it, I'll give an extreme synthesis of it.
Denotational semantics is a ...
- 5,598
14
votes
Where is the model theory in programming language theory?
Let me amend Damiano's answer with more specific comments.
Terms of type bool are not logical statements. Expressions like ...
- 29.4k
13
votes
Accepted
Does the Law of Excluded Middle imply the Axiom K in Martin-Löf's Intensional Type Theory?
Yes, LEM implies K. See HoTT book Theorem 7.2.5, known as Hedberg's theorem, which shows that any type with decidable equality satisfies axiom $K$. If we assume excluded middle, all types have ...
- 29.4k
12
votes
Do I have to give up the Law of the Excluded Middle in order to Learn $\lambda$-Calculus?
Type theory is a mathematical theory in which we can do very many different things (set theory is like that as well). We can use type theory for computability, or homotopy theory, or use it to express ...
- 29.4k
12
votes
Accepted
What kind of theoretical object corresponds to a C++ concept?
From a programming language theory perspective, as opposed to the computability perspective other answers and comments have offered, C++ templates combined with concepts correspond to bounded ...
- 16.7k
12
votes
Accepted
What is the intuition behind linear logic?
I'm not sure this question is ideal for CSTheory, but given that it's already gathering upvotes, here is an answer somebody might have given had the question been posted on cs.stackexchange.
In ...
- 11.5k
11
votes
Accepted
Can Elementary Affine Logic be used as the core type system of a practical programming language?
Something very similar, but using light affine logic (LAL) instead of EAL, was attempted a few years ago by Baillot, Gaboardi and Mogbil (you may find the paper here). I think their work may be ...
- 5,598
11
votes
Proof that the calculus of constructions extended with recursive types isn't strongly normalizing?
With general recursive types you can define the type
type T = T -> T
With that type you can type self-application -- and in fact, every term of the untyped ...
- 1,320
11
votes
Philosophy behind monotonicity requirement for inductive types
When you write an inductive, you are defining a type by an equation. For example, if we write $F_1(X)=X^X=X\to X$, bad should satisfy $F_1($...
- 556
11
votes
Accepted
How do continuations represent negations (under the Curry–Howard correspondence)?
When one associates negation with continuations, it is probably not ideal to think of it in terms of an 'empty' type. Continuation passing can be done with respect to any result type, and if that type ...
- 1,021
11
votes
Applications of Barendregt–Geuvers–Klop conjecture
I'm not (exactly) an expert, but my understanding is that there are very few practical applications of this conjecture, except possibly simplifying the decision procedure for type-checking in ...
- 14k
11
votes
$\eta$-reduction not locally confluent on well-typed terms
Yes, $\eta$ reduction for unit is terribly behaved. Suppose you are in a context $\Gamma \triangleq x:1, y:1$.
Then, the unit term $\Gamma \vdash \left\langle\right\rangle : 1$ has the following eta-...
- 32.8k
11
votes
Accepted
Stronger "induction" principles than induction-recursion
Anton Setzer has work on type theories that are stronger than standard induction-recursion. In some ways, though, it's still in terms of simultaneous inductive and recursive definitions. The ...
- 1,021
10
votes
Accepted
Strong normalization property of CoC inside CoC
I'll summarize the comments from chi, and sketch the proof that
There can be no proof of $\mathrm{False}:=\forall X:*.X$ in the CoC in head normal form. Furthermore, this fact can be proven in a weak ...
- 14k
Only top scored, non community-wiki answers of a minimum length are eligible