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 ...
- 28.3k
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,...
- 32.4k
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 ...
- 13.7k
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 ...
- 1,444
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 ...
- 32.4k
15
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 ...
- 251
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 ...
- 28.3k
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
14
votes
Accepted
Logical Reations for an Impredicative System in a Predicative MetaTheory
In general, what we usually call the logical relations argument isn't really linked to impredicativity: the main idea is simply to interpret terms in some abstract algebra $\cal A$, and to represent ...
- 13.7k
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 ...
- 5,273
14
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, ...
- 32.4k
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
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
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 ...
- 2,830
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, ...
- 2,830
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 ...
- 28.3k
13
votes
Accepted
Contradiction between Gödel's Second Incompleteness Theorem and the Church-Rosser's property of CIC?
First, you are confusing consistency of CIC as an equational theory with consistency of CIC as a logical theory. The first means that not all terms of CIC (of the same type) are $\beta\eta$-equivalent....
- 5,273
13
votes
Accepted
Is there a good notion of non-termination and halting proofs in type theory?
Because one of the principal applications of Type Theory in formalizations has been to study programing languages and computation in general, a lot of thought has gone into ways of representing ...
- 13.7k
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 ...
- 28.3k
13
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 $...
- 28.3k
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
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 ...
- 5,273
11
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 ...
- 28.3k
11
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 ...
- 11.4k
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
$\eta$-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-...
- 32.4k
10
votes
Accepted
Church-Rosser property for dependently typed lambda calculus?
It might be useful to quickly give the counter-example to CR in typed calculi with $\beta$ and $\eta$:
$$ t=\lambda x:A.(\lambda y:B.\ y)\ x$$
And we have
$$ t\rightarrow_\beta \lambda x: A.x$$
and
$...
- 13.7k
10
votes
Accepted
Dependent types over Church-encoded type in PTS/CoC
You can't do this using the traditional Church encoding for Bool:
#Bool = ∀(Bool : *) → ∀(True : Bool) → ∀(False : Bool) → Bool
...
10
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 ...
- 5,273
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