Questions tagged [type-systems]
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82
questions
1
vote
2answers
102 views
Explicit type system with infinite non-cumulative universe hierarchy
Is there an open-source proof assistant or at least an explicit set of rules written down somewhere for a type system with an infinite non-cumulative universe hierarchy and unique typing?
I want to ...
3
votes
1answer
141 views
Defining binary natural numbers without quotient types
Let's say we work in a dependent type theory with W-types and we want to have a type for binary natural numbers. We don't want to add quotient types or higher inductive types to the system.
How to ...
5
votes
1answer
103 views
Is the Mendler-encoding in System-F adequate?
In the paper "Efficiency of Lambda-Encodings in Total Type Theory" it is mentioned that the Church-encoding is adequate and the Parigot encoding is not adequate. This means that any ...
0
votes
1answer
58 views
Pi-calculus (or session types) - proof for weakening lemma
I'm writing a thesis about session types and am currently writing a section concerning type soundness for the system. I started to proof weakening lemma, which states, that
$$
\text{If } \Gamma \vdash ...
7
votes
2answers
199 views
Does the Hindley-Milner type system (i.e. STLC with prenex polymorphism) have a category-theoretic model?
It is well known that any CCC (cartesian closed category) is a model of the simply-typed $\lambda$-calculus. It is less well known that System F admits a categorical model, but it is also well studied ...
3
votes
1answer
193 views
Why would the term “dynamically typed” be considered a misnomer?
In the book "Types and Programming Languages", the author writes:
The word "static" is sometimes added explicitly - we speak of a "statically typed programming language",...
2
votes
0answers
138 views
Types as abstract interpretations
In Types as Abstract Interpretations, Cousot seems to propose a method to derive various type systems by succesive abstract interpretation
of the denotational semantics of an untyped lambda-calculus. ...
6
votes
0answers
108 views
Programmatic higher inductive/inductive-inductive types with equalities between equalities
I can think of practical HITs in verified software that capture some form of alpha equivalence, context equivalence or the example which defines well-typed syntax of type theory from Signatures and ...
3
votes
0answers
143 views
Set:Set or Negative Inductives in a Total Language?
In total dependently typed languages, general recursion is forbidden, since this can allow for non-terimination. However, dependently typed language can still describe Turing-complete computations (...
8
votes
1answer
421 views
PHOAS with extrinsic typing?
Parameterized Higher Order Abstract Syntax (PHOAS) is a representation of syntax trees that allows the host language's binding to be used to represent binding in the language being modelled, while ...
5
votes
2answers
198 views
What's the point of stack judgement in CBPV?
Call-by-push-value (CBPV) introduces two main families of types, values and computations, and their corresponding judgements. However, in some extensions/variants/adaptation of CBPV, there is a third ...
3
votes
0answers
180 views
Is System-F with higher-kinded newtypes equivalent in computational power to System-F omega?
If we have System-F with higher-kinded types and newtypes, then we can express everything (I think) of System-F omega, except we have to manually (un)pack.
For example:
...
7
votes
1answer
172 views
Example of a function that you can write in Calculus of Constructions but not in System-F
It's been suggested in an answer to this question that the Calculus of Constructions has more computational strength than System-F. What are examples of functions that you express in CoC that you ...
2
votes
1answer
275 views
Difference between statically and dynamically typed languages
When writing a course on computer science where students get an introduction to both Python and OCaml, I was on the verge of saying that Python is dynamically typed and OCaml is statically typed. I ...
7
votes
2answers
285 views
Termination checking for Scott-encodings in System F with positive-recursive types
Is there any research on termination analysis on Scott-encodings in System F with positive-recursive types.
All papers I have found use languages with constructors and case analysis (for example ...
11
votes
2answers
400 views
Intuition Behind Strict Positivity?
I'm wondering if someone can give me the intuition behind why strict positivity of inductive data types guarantees strong normalization.
To be clear, I see how having negative occurrences leads to ...
1
vote
1answer
128 views
Dependent C-style types with subtyping rule
I'm looking for previous work regarding an extension of a C-style type system in which types may have constraints and have a defined subtyping rule. In particular, I'm interested in defining algebra-...
-3
votes
1answer
92 views
Soundness of type (systems)
For someone without strong background in theoretical computer science: can soundness be a property of a type (given a type system), or a property of type systems only? In other words, can we say that ...
0
votes
1answer
67 views
Is a reference on T a subtype of T?
If I take the book Practical Foundations for Programming Languages by Robert Harper, the following definition is given for subtyping:
A subtype relation is a pre-order on types that validates the ...
5
votes
2answers
256 views
Typing of substitution in a bidirectional type system
In most typed lambda calculi, we have the following lemma:
If $\Gamma \vdash t_1 : \tau_1$ and $\Gamma, x : \tau_1, \Delta \vdash t_2 : \tau_2$ then $\Gamma,\Delta[t_1/x] \vdash t_2[t_1/x] : \tau_2[...
5
votes
1answer
213 views
Complexity of type-checking in relation to complexity of normalization
In order to verify that a terminating program terminates, one thing that can be done is to actually run the program. That may take a lot of time. If the program is typed in a total type-system, we can ...
7
votes
1answer
448 views
Fixed points in dependent type theories
Most dependent type theories aim for some notion of correctness in two respects:
The type system must be decidable.
The type system must be consistent. e.g. $\forall \tau. \tau$ should not be ...
1
vote
2answers
424 views
What are the limitations of dependent typing?
It seems dependent types can provide lots of desirable guarantees about program behavior. What kinds of program properties can they NOT guarantee (besides what's computationally infeasible)? Is there ...
5
votes
1answer
151 views
What's the difference between proving weak normalization and implementing evaluator?
Implementing an normalization (cut elimination) procedure for a type system A in a language with a total type system B, automatically proves cut elimination for type system A since the implementation ...
2
votes
1answer
159 views
Is there a formalization of normalization of impredicative system F?
In particular Agda seems not strong enough to prove that.
Is the predicative Calculus of Inductive Constructions universes (Coq without Prop) sufficient?
How about with the impredicative Prop?
3
votes
1answer
136 views
Is there a 'very fast growing' hierarchy that would capture System F?
Particular ordinals in slow-growing and fast growing hierarchies can capture the expressiveness of many predicative type systems.
Is there a hierarchy of function that could possibly capture ...
1
vote
0answers
54 views
Effect handlers, arrows and applicatives
After reading Lindley's paper on effect handlers for arrows and applicatives, I got the gist about dynamic and static flow and that it was added to the effect system and so on. However, I do not ...
9
votes
1answer
176 views
System F and System T names
Does anyone know where do the names System "F" and System "T" comes from? I am not asking who introduced those names (Girard System F, and Gödel System T), but what the "F" and the "T" means.
7
votes
1answer
323 views
Avoiding Cycles with Unification and Subtyping
Context
I realize that subtyping often doesn't admit principle types, and that inference in the presence of subtypes is undecidable. I'm working in a context where typechecking should simply fail ...
2
votes
1answer
364 views
programming language with type-level functions
Is there any programming language out there that allows the same set of tools it offers, to be used at the type level as well? I know, Haskell and some other ML family languages allow parametric types ...
4
votes
0answers
63 views
Finite intersection property of polymorphic type families
Let $\Phi$ be a type functor definable in polymorphic lambda calculus:
$$ \alpha : * \vdash \Phi(\alpha) : * $$
$$ f : A \to B \vdash \mathsf{Map}^{A,B}_\Phi(f) : \Phi(A) \to \Phi(B)$$
Suppose further ...
2
votes
1answer
100 views
Completeness of Constraint Typing (type inference) question regarding $\sigma'$
The theorem of completeness of type inference states the following:
Suppose $\Gamma \vdash t:S| _{\mathcal{X}}C$,
...
7
votes
1answer
546 views
Determinism and pi-calculus
Milner embedded $\lambda$-calculus into $\pi$-calculus, showing that the $\pi$-calculus is capable of Turing-complete, deterministic calculation. Since parallel compositions of processes in the $\pi$-...
5
votes
1answer
281 views
Recursive types and the empty type
In John Mitchell's book "The Foundations of Programming Languages", he considers a typed lambda calculus with unit, exponential, product, (binary) coproduct types, and arbitrary recursive types (p126)....
7
votes
2answers
299 views
Efficiently ordering typed programs
Sometimes it is useful to enumerate in increasing order
programs that have a given type. A
simple example is test
generation for compilers: we want to test a new optimising phase and
are ...
6
votes
1answer
253 views
Type checking, Hypothetical judgments, meaning explanations and computational type theory
We say that a system is a computational type theory if it is a type theory defined by not a bunch of inference rules, but some sort of Martin-Löfian meaning explanations (e.g. in the sense of NuPRL). ...
8
votes
1answer
140 views
How to prove relations between “classes” of types?
After reading Effects as Sessions, Sessions as Effects, I was wondering how would a proof of equivalence between both take place, or even, a proof of Sessions types being a Type and Effect System.
In ...
8
votes
0answers
283 views
Denotational semantics of System $F_\omega$ with recursive types and general recursion
Is there a denotational semantics for System $F_\omega$ in literature that supports both recursive types and general recursion?
I'm looking for a model of Ralf Hinze's variant of System $F_\omega$ [4]...
22
votes
2answers
4k views
How do you get the Calculus of Constructions from the other points in the Lambda Cube?
The CoC is said to be the culmination of all three dimensions of the Lambda Cube. This isn't apparent to me at all. I think I understand the individual dimensions, and the combination of any two seems ...
4
votes
1answer
142 views
Is there any work relating type systems and Cook-Reckhow proof systems?
An important subfield of computational complexity is proof complexity, mostly due to Cook and Reckhow. E.g. one notable result is that a proof system that has efficient proofs (i.e., in length ...
5
votes
2answers
343 views
Is there a “lambda cube” for interaction nets?
The lambda calculus is an untyped language that is often extended with logical frameworks such as the vertices of the λ-cube. Is there something similar to it, but for interaction nets? What about ...
6
votes
1answer
319 views
What type system fits the subclass of λ-terms that can be reduced optimally?
There is a subset of λ-calculus terms that can be reduced by Lamping's Abstract Algorithm without using the Oracle. That is an interesting subset, because only for those terms it is proven that ...
11
votes
1answer
456 views
Dependent types over Church-encoded type in PTS/CoC
I'm experimenting with pure type systems in Barendregt's lambda cube, specifically with the most powerfull one, the Calculus of Constructions. This system has sorts ...
4
votes
1answer
422 views
Featherweight Generic Java formalization in Coq
I've been searching for some nice formalization of FGJ (Featherweight Generic Java) in Coq. I am going to develop an extension of FGJ in Coq, so I hope there is an appropriate Coq implementation which ...
9
votes
3answers
829 views
Universal and existential types
I'm trying to wrap my head around the concepts of universal and existential types but everywhere I look, I see either logical or operational intuitions (or implementations) (e.g. TAPL book by B. ...
10
votes
1answer
364 views
4
votes
1answer
132 views
Rendering of type-level computation
Programming languages with dependent types and/or higher-kinded types feature what might be called compile-time computation at the type-level. This is usually defined as follows (I'm omitting some ...
5
votes
1answer
471 views
Can all linear lambda calculi be linearity checked syntactically?
Given a lambda calculus with explicit linearity and usual application and abstraction, can the linearity check be done on an untyped syntax tree if we keep track of the structural types? Are the ...
1
vote
0answers
122 views
Has there been work on formal Semantics for linear algebra?
Could I get some references on formal semantics for a calculus on linear algebra that helps you study matrix or tensor based programming languages? I am looking for anything that encompasses linear or ...
8
votes
0answers
83 views
Decidability of equality between higher-kinded equirecursive types (or: between nonregular Böhm trees)
In §3 of Polytypic values possess polykinded types, Ralf Hinze described a calculus of types with higher-kinded recursive types. There is a fixed-point combinator $$\mu_\kappa : (\kappa \rightarrow \...
