Questions tagged [type-theory]
formal systems to specify properties of objects
4
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1answer
165 views
What should we return when pattern matching on a Higher Indutive Type and the case is a Path?
Context: Cubical Type Theory
Consider a simple HIT, say, an HitInt:
...
3
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1answer
32 views
T&PL: Language grammar with terms
I'm autodidacting Pierce's Types and Programming Languages. On page 27 he states a definition for "terms, concretely" in constructing a language of terms, thus:
For each natural number $i$, define a ...
1
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1answer
25 views
What is the difference between a Top type and a Unit type
Wikipedia defines a Top type: (edited for readability)
The Top type [...] is the universal supertype, as all other types in any given type system are subtypes of Top
However, the article goes on ...
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0answers
16 views
The underlying type theory of HOL/Isabelle
Is there a good source on the type theory of HOL/Isabelle/other HOL-based LCF-style theorem provers?
0
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0answers
19 views
Computational type theorists: how do you compare terms for equality here?
I am attempting to implement Simple Type Theory in the language D. How do you compare subterms to a term $R$ for the sake of computing the covering abstractors of $R$ in $M$?
By reference (class ...
3
votes
1answer
168 views
Drawbacks of adding type equality to 1ML
In the 1ML – Core and Modules United (F-ing First-Class Modules) paper, the author gives the following example for why module types do not form a lattice under subtyping:
...
4
votes
1answer
138 views
Typing rule for binding groups
In "Typing Haskell in Haskell", by Mark P. Jones, is provided a sort of haskell-like specification for typing Haskell. As stated in this paper, binding groups is a area "neglected in most theoretical ...
0
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0answers
41 views
Mathematical resource material accompanying TAPL
I'm currently reading Types and Programming Languages by Benjamin C. Pierce and just arrived at chapter 21 Metatheory of Recursive Types.
Prior to this chapter I found the book challenging but ...
3
votes
1answer
55 views
Is there a generally accepted name for creating types that select a subset of other types?
Tl;Dr;
Given:
type A = { int: foo, int: bar }
type B = select foo from A
What is the name of the typing relationship between A and B?
What is the name of the ...
1
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0answers
22 views
Summary of types of equivalence and equality in type theory, with notations and examples
Coming from non-computer science background, I am trying to understand the different types of equivalence and equality usually used in type theory. Ideally, I am looking for clear definitions and ...
3
votes
1answer
63 views
Roadmap to formal verification
I would like to learn about different approaches to formal verification of software programs that goes beyond what Wikipedia has to offer. Ideally one would not only get an overview but also ...
5
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2answers
110 views
LET REC recursive expression static typing rule
I'm taking a programming languages course and had a question regarding the typing rules for a recursive let rec expression in a static typing system.
To be more ...
1
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3answers
92 views
Soundness and completeness w.r.t. programming languages
I'm studying programming languages (more specifically type systems) and came across a concept I couldn't quite wrap my head around: soundness and completeness.
I'm taking a class, and according to my ...
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0answers
24 views
Composition of compostion as a functor
"Composition of Composition" (i.e., (.) . (.)) in Haskell), has type ...
1
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1answer
37 views
PL: What solves the type isomorphism $X \cong (X \rightarrow \mathbf{2})$?
In Practical Foundations for Programming
Languages, on page 138 (page 156 of the pdf), it says:
Requiring solutions to all type equations may seem suspicious, because
we know by Cantor’s Theorem ...
1
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0answers
16 views
Propositional extentionality in the lean theorem prover?
Propositional extentionality in the lean theorem prover is stated as the following axiom:
axiom proptext {a b : Prop} : (a $\iff$ b) \to a = b
My confusion about this is as follows: Previously I’...
1
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0answers
28 views
Type theory based automated theorem prover?
I know that there exist type theory based proof-checker, and I know that there are logic/sequent-calculus based automated theorem provers.
But I haven’t heard of a type-theory based automated theorem ...
1
vote
1answer
36 views
Curry-howard isomorphism in object oriented programming languages
I want to get a better intuition for the curry howard isomorphism, and my intuition is mainly based on object oriented programming languages like JavaScript.
So as an example, I am going to formalize ...
1
vote
2answers
56 views
What is the type signature of a Turing Machine?
Maybe my question is a bad question, but if it is, I want to know eactly how it is a bad question.
Suppose we have some Turing machien $M$ that takes as input a natural number $n$ in the form of a ...
0
votes
1answer
46 views
Bounded Quantification: Full F<: intuition
I'm currently looking into Chapter 26 of Types and Programming Languages and am a bit confused by the "intuition" for Full F<: (p. 395):
A type T = ∀X<:T1.T2 describes a collection of ...
2
votes
1answer
55 views
Does co-inductive and co-recursive types also have their recursors?
I'm new to type theory, and recently read introductory materials where dependent type are discussed. One of my friend asked me, "Those dependent types are having recursors & 'inductors'(dependent ...
3
votes
1answer
42 views
Multiplicative Pure Type Systems
All the references about Pure Type Systems I know consider only systems that allow to recover natural deduction systems with additive rules. Is there any variant that allows it to recover natural ...
3
votes
2answers
82 views
Are Bad churches inhabited?
In type theory, some inductively defined data types allow you to prove absurdity. For example
...
1
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2answers
44 views
How can MLTT etc encode computability?
I am recently thinking about proving the undecidability of some problem. This problem has been formalized in Coq and by staring at it, people including me think "for sure" this is undecidable. "For ...
2
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2answers
82 views
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0answers
29 views
What is the difference between ADTs and ASDLs?
ASDL stands for Abstract Syntax Description Language (ASDL), whereby ADT stands for Algebraic data type.
By looking at Python.asdl it appears to me to be the same thingy, just with different names, ...
3
votes
0answers
37 views
Type system for Query DSL with only simple GADTs: what typing judgments are needed?
Background
I have several F# codebases with reasonably high level of complexity of code. In order to convince myself that the code is solid I do whatever I can to write as much of it as possible type-...
1
vote
1answer
58 views
Is there any sort of formal definition of terms like 'data type', 'abstract data type', etc?
If not (because I assume not) is there some kind of reference or book that provides some theoretical foundation to these concepts? I've been learning about data structures and abstract data types for ...
12
votes
4answers
2k views
What does the leading turnstile operator mean?
I know that different authors use different notation to represent programming language semantics. As a matter of fact Guy Steele addresses this problem in an interesting video.
I'd like to know if ...
3
votes
2answers
381 views
What makes a proof assistant a proof assistant?
You open a code editor, define a syntax with lambdas, a few primitives. Then you invent some nice computation rules, some cool typing rules, and write a corresponding interpreter and "type checker". ...
1
vote
1answer
53 views
how to deduce a function subtype rule from a given function type definition
This question relates to liskov substitution principle seems to have two conventional meanings but is really a different question, so I'm posing it as a new question.
I'm doing a bit of research into ...
2
votes
1answer
44 views
liskov substitution principle seems to have two conventional meanings
This is a question about the semantics of the name, rather than about the principle itself. What is the Liskov Substitution Principle (LSP)?
LSP seems to have two meanings in the literature I've ...
1
vote
1answer
35 views
How to define the natural numbers as a W-type?
I'm having trouble understanding the rules for W-types in type theory as defined here: https://ncatlab.org/nlab/show/W-type#wtypes_in_type_theory
Can someone give an example of how these rules could ...
4
votes
0answers
37 views
Isomorfism between inductive and coinductive types (through double negation)
The paper "CPS Translating Inductive and Coinductive Types" mentions that there is an isomorphism between inductive (mu) and coinductive (nu) types, which they use for their translation. It states ...
3
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0answers
46 views
PL: How can I prove the type of something using “Inversion for Typing”?
I'm currently going through this book about programming languages, and in section 4.2, Lemma 4.2 it says this:
Lemma 4.2 (Inversion for Typing). Suppose that $\Gamma \vdash e : \tau$. If $e = \...
7
votes
1answer
65 views
Flawed argument in the proof of function extensionality in cubical type theory?
I am reading the lectures about cubical type theory in this github repo. In lecture 1 the author defines function extensionality the following way:
...
19
votes
0answers
427 views
Can a calculus have incremental copying and closed scopes?
A few days ago, I proposed the Abstract Calculus, a minimal untyped language that is very similar to the Lambda Calculus, except for the main difference that substitutions are ...
1
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2answers
49 views
What is the use case of multi-type-parameters generic interface?
Previously, I ask a similar question, but the answer given only demonstrates the usage of multiple-type-parameters(MTP) data structure, but not MTP generic interface.
Based on my experience, generic ...
4
votes
0answers
153 views
Row polymorphism extended to modules
One common observation in type systems is that having subtyping makes type inference hard [1]. Consequently, for records, many modern functional languages shun subtyping (OO style) in favor of row ...
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0answers
43 views
What's are the consequences of subject expansion property?
Subject reduction is a well and widely used property of typed rewriting systems.
Subject expansion is much less known.
What are the applications of this property and which systems enjoy it?
3
votes
1answer
42 views
Are all terms of type forall a. a operationally bottom?
Is there a proof that all terms of type $\forall{a}. a$ are operationally $\bot$, in a non-weakly-normalising version of System F?
If you ask a free theorem calculator such as this one for the free ...
5
votes
1answer
60 views
Hindley-Milner type inference for language with implicit type casting
I've only implemented the HM algorithm on a small academic language with a few primitive types and functions. In that case, the unification algorithm would return a type error if two different ...
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0answers
51 views
On the termination of mutually recursive functions
In Finding Lexicographic Orders for Termination Proofs in Isabelle/Holl the authors construct a method for proving termination of functions based on constructing a matrix that registers for each row ...
2
votes
1answer
56 views
Induction on typing derivation in refinement types system
From the text Principles of Type Refinement page 14:
The author introduces in definition 2.2.7 the rule:
$$ \dfrac{\Pi \vdash t : R \qquad R \le S}{\Pi \vdash t : S} $$
and gives the following ...
0
votes
1answer
52 views
Can you automatically generate a parser for a type using type theory some how?
Was wondering since all the types are spelled out constructively, and the constructions can all be reflected symbolically on a computer, if you can automatically parse expressions in a type?
6
votes
1answer
116 views
Relationship between Higher Kinded Polymorphism, type inference, and Currying
On Hacker News there is an interesting exchange about the async\await RFC for Rust.
The author of the proposal withoutboats is responding to a comment about the ...
4
votes
1answer
714 views
What is the use case for multi-type-parameter generics?
In C#, one can define a class/method/function with multiple type parameters. For example,
...
1
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3answers
76 views
What are the concequences of the unit type and the unit value being the same?
What are the practical and theoretical implications of the unit type and the unit value being the same or different entities?
For example, in Haskell the unit type and unit value are both spelled <...
2
votes
1answer
91 views
What are the implications of Lean not having the type `Set`?
In Coq we have an impredicative base type, called Prop, and a predicative base type, called Set, both of type ...
5
votes
2answers
91 views
Definition of “properly partial” versus “total” value types
In the Foundations chapter of Elements of Programming (Stepanov and McJones, 2009), this paragraph appears:
A value type is properly partial if its values represent a proper subset of the abstract ...
