Questions tagged [type-theory]
formal systems to specify properties of objects
411
questions
3
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0answers
17 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
votes
2answers
1k 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:
$(\forall x^\sigma)(\exists y^\tau)R(x,y)\rightarrow(\exists f^{\sigma \rightarrow \tau})(\forall x^\sigma)R(x, f(x))$
...
0
votes
0answers
36 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 ...
0
votes
0answers
34 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
vote
1answer
44 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
vote
0answers
31 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
vote
1answer
84 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
votes
0answers
29 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
votes
0answers
54 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
votes
0answers
22 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 ...
3
votes
0answers
41 views
Derivation of product type eliminator in type theory
In HoTT book, section 1.5 (Product Types) in order to define the eliminators for the product type it assumes a function of type $g:A \rightarrow B \rightarrow C$ and then goes on to define the ...
2
votes
0answers
55 views
How could 'Complete and Easy Bidirectional Type Checking' handle invariant parameters on type constructors
The paper Complete and Easy Bidirectional Typechecking for Higher-Rank Polymorphism provides examples for checking if one function type is a subtype of another, which I think demonstrates checking ...
5
votes
2answers
238 views
Are “logarithm types” a thing?
I'm attempting to formalize some thoughts I've had about paths into data structures. For example, a path into a list of Ts might be a pair of an index with a path ...
2
votes
0answers
41 views
Simulating extensible sums with dependent types?
ML-style languages have a concept of "extensible" or "open" sum types, where variants can be declared at any point, and there's not a fixed number of constructors for the type. They're usually used to ...
5
votes
0answers
41 views
Rules for consistency with mutual inductive families?
I'm trying to use a proof assistant to define a type and a relation that are mutually dependent on each other:
...
0
votes
1answer
106 views
Is this statement of P = NP in Agda correct?
Looking for a self-contained statement of P = NP in type theory, I stumbled upon this short Agda formalization (a cleaned up version is reproduced below).
The statement here does seem to express the ...
0
votes
0answers
35 views
How is substitution in type theory the composition of classifying morphisms in category theory?
In the article at nlab about relation between category theory and type theory, it is said that substitution in type theory is the same as composition of classifying morphisms in category theory.
...
1
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0answers
25 views
Is, beta reduction in type theory being considered as counit for hom-tensor adjunction in category theory, a denotational or operational semantic?
In the article at nlab about the relation between type theory and category theory, it is said that "beta reduction" in type theory corresponds to "counit for hom-tensor adjunction" in category theory ...
0
votes
1answer
76 views
What is the language feature which allows a variable to be associated with values of different types?
In Python, I can change the types of values associated with a variable:
>>> x=1
>>> x="abc"
In C, I can't do the same.
What is the name of ...
-1
votes
1answer
47 views
Is the choice of static and dynamic typing not visible to the programmers of the languages? [closed]
From Design Concepts in Programming Languages by Turbak
Although some dynamically typed languages have simple type markers (e.g., Perl variable names begin with a character that indicates the type ...
2
votes
2answers
41 views
Are type variables really only used in mathematical conversation about types?
Are type variables really only used in mathematical conversation about types? i.e. are type variables (meta-variables that only contain the type classification label) only exist in proofs for types ...
0
votes
1answer
87 views
What is the difference between $ \alpha \to \alpha $ vs $ \forall \alpha. \alpha \to \alpha$?
I was studying polymorphic types and I was finding the distinction with monomorphic types difficult to pin down (context CS 421). From the course I linked the have the following (vague attempt) at a ...
5
votes
1answer
79 views
What does $ \forall \alpha_1, \dots , \alpha_n . \tau $ mean formally as a type?
I was learning about polymorphic types but I couldn't understand the notation, can someone explain it means (context cs421 UIUC):
$$ \forall \alpha_1, \dots , \alpha_n . \tau $$
its supposed to be a ...
0
votes
1answer
42 views
How do you define and parse variables (free or bound) from user-entered strings?
I'm writing an application in which the user might enter expressions such as $\text{lim}_{i \in I} \beta(i)$ where $\beta$ is a functor. That's just an example, the expressions, which contain ...
2
votes
1answer
57 views
How is β-reduction a 2-morphism in Category theory?
According to Categorifying CCCs: Computation as a Process, computation or β-reduction process in untyped-lambda calculus is in fact a 2-morphism in category theory.
Can someone please describe me ...
1
vote
0answers
22 views
Do Rank-1 (prenex) polymorphism and Predicative polymorphism mean the same?
https://en.wikipedia.org/wiki/Parametric_polymorphism says:
Rank-1 (prenex) polymorphism
In a prenex polymorphic system, type variables may not be instantiated with polymorphic types.[4] ...
1
vote
0answers
26 views
What are the relation and differences between reification and type passing semantics?
https://en.wikipedia.org/wiki/Type_erasure says
type erasure refers to the load-time process by which explicit type annotations are removed from a program, before it is executed at run-time. ...
0
votes
0answers
20 views
What are the relations between these two descriptions of let polymorphism?
In Types and Programming Languages by Pierce, there are two descriptions of let-polymorphism.
Sec23.8 Fragments of SystemF on p359 says
This has led to various proposals for restricted fragments ...
0
votes
1answer
29 views
What are the difference and relation between type checking and type reconstruction?
In Types and Programming Languages by Pierce,
ML-style let-polymorphism was first described by Milner (1978). A num-
ber of type reconstruction algorithms have been proposed, notably the clas-
...
-1
votes
1answer
49 views
What is “Hindley-Milner (i.e., unification-based) polymorphism”?
In Types and Programming Languages by Pierce, Ch11 Simple Extensions extends the typed lambda calculus.
Section 11.5 Let Bindings says:
In Chapter 22 we will see another reason not to treat let as ...
2
votes
0answers
22 views
Does Hindley-Milner refer to the unification algorithm for type reconstruction, a type system, or a form of polymorphism?
What does Hindely-Milner refer to?
In Types and Programming Languages by Pierce, I only find that Section 22.4 Unification mentions "Hindley" and "Milner", when introducing the unification algorithm.
...
0
votes
0answers
24 views
Is it possible to deduce type from the lambda form?
I was continuing the exploration of lambda world this summer. When I take a look at the simply typed lambda calculus, it looks like there is no use for usual chuch numerals and boolean forms anymore. ...
1
vote
1answer
50 views
Types and Programming Languages - proof for theorem about principles of induction of terms
Types and Programming Languages book introduces a theorem about principles of induction on term (p. 31, theorem 3.3.4):
Suppose P is a predicate on terms.
Induction on depth:
If, for each ...
0
votes
1answer
27 views
What does valid method overriding mean?
In Types and Programming Languages by Pierce, from p257 to p258, about featherweight Java,
The predicate override(m, D, C→C0) judges whether a method
...
1
vote
2answers
54 views
Which is a type of objects in mainstream OO languages: a class, an interface, an abstract class, a metaclass?
In Types and Programming Languages by Pierce, Section 18.6 Simple Classes in Chapter 18 Imperative Objects says:
We should emphasize that these classes are values, not types. Also we can,
if we ...
9
votes
1answer
1k views
Why does Coq include let-expressions in its core language
Coq includes let-expressions in its core language.
We can translate let-expressions to applications like this:
let x : t = v in b ~> (\(x:t). b) v
I understand ...
1
vote
1answer
33 views
Does an ADT have multiple or only one representations/implementations?
Section 24.2 in Types and Programming Languages by Pierce defines ADTs in existential types:
A conventional abstract data type (or ADT) consists of (1) a type name A, (2) a concrete representation ...
0
votes
1answer
20 views
Why can System F1 a.k.a. λ → have kind `*`, but no quantification `∀`?
In Types and Programming Languages by Pierce, on p461 in Section 30.4 Fragments of Fω
30.4.1 Definition: In System F1 , the only kind is ...
-1
votes
1answer
47 views
What order logic does a system correspond to under Curry–Howard correspondence?
In Types and Programming Languages by Pierce,
Section 9.4 Curry–Howard correspondence on p109 has a table
Does the table mean that the simply typed lambda calculus ...
0
votes
0answers
10 views
Can two type expressions in different kinds have subtyping relation and equivalence relation?
In Higher-order bounded quantification ($F^ω_{<:}$), introduced in Ch31 in Types and Programming Langauges by Pierce, its subtyping and equivalence rules are:
Does subtyping relation only exist ...
0
votes
0answers
35 views
Is `→` a type operator?
In Types and Programming Languages by Pierce,
The level of types contains two sorts of expressions.
First, there are proper
types like Nat, Nat→Nat, Pair Nat Bool, and ∀X.X→X, which are ...
0
votes
0answers
25 views
Does λ→ have type operators?
In Types and Programming Languages by Pierce, Ch11 Simple Extensions introduces λ→ as the simply
typed lambda calculus with simple extensions, and introduces <...
0
votes
0answers
25 views
Is the language “untyped arithmetic expressions” in Types and Programming Languages not Turing complete?
In Types and Programming Languages by Pierce, is it correct that
the language introduced in Chapter 3 Untyped Arithmetic Expressions is not Turing complete? Because it doesn't provide recursion.
the ...
0
votes
0answers
16 views
How can we directly reaching inside and looking at the state of a partially existential object?
Types and Programming Languages by Pierce says in Section 26.5 Bounded Existential Types, about partial existential objects implemented in terms of bounded existential types:
We can make a similar ...
0
votes
0answers
26 views
What are the differences and relations between a type constructor and a type operator?
What are the definitions of a type constructor and a type operator?
What are their differences and relations?
I think a type operator is a function whose parameters are n types and return is a type. A ...
1
vote
1answer
54 views
How can an existential type be defined in terms of universal type?
In Types and Programming Languages by Pierce, how does the following achieve the definition of an existential type in terms of universal type, by polymorphic version of Church encoding of pairs?
...
0
votes
1answer
60 views
Why are ADT packages opened immediately after they are built, while existential objects opened as late as possible?
Section 24.2 in Types and Programming Languages by Pierce defines ADTs in existential types:
A conventional abstract data type (or ADT) consists of (1) a type name A, (2) a concrete representation ...
0
votes
1answer
70 views
How can mainstream OO languages support strong binary operations by classes?
Section 24.2 in Types and Programming Languages by Pierce compares ADT and existential objects,in terms of how well they support strong binary operations:
Other binary operations cannot be ...
0
votes
1answer
39 views
Are type abstraction values and universal types not for non functions, but only for functions?
In Types and Programming Languages by Pierce, Chapter 23 Universal Types has a summary of System F in the following figure, in particular, "type abstraction values" and their types "universal types".
...
0
votes
1answer
37 views
How does this allow list operations to be applied to lists with elements of any type?
In Types and Programming Languages by Pierce, Chapter 11 is simple extensions of the simply typed lambda calculus with any simple base types. Section 11.12 introduces Lists.
How does the section ...
