Questions tagged [type-systems]
The type-systems tag has no usage guidance.
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What is the difference between a Top type and a Unit type [migrated]
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 ...
-3
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1answer
67 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 ...
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1answer
56 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
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2answers
153 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[...
4
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1answer
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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
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1answer
227 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 ...
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2answers
184 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 ...
4
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1answer
93 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
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1answer
93 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
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1answer
118 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
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0answers
41 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
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1answer
123 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.
8
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1answer
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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
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1answer
124 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
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0answers
60 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 ...
3
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1answer
80 views
Completeness of Constraint Typing (type inference) question regarding $\sigma'$
The theorem of completeness of type inference states the following:
Suppose $\Gamma \vdash t:S| _{\mathcal{X}}C$,
...
6
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1answer
314 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 $\pi$-...
6
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1answer
203 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
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2answers
219 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
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1answer
212 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
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1answer
128 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 ...
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0answers
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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]...
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2answers
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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 ...
3
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1answer
125 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 ...
6
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2answers
255 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 ...
7
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1answer
277 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 ...
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1answer
315 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 ...
5
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1answer
370 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 ...
8
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3answers
664 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. ...
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1answer
295 views
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1answer
113 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
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1answer
244 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 ...
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0answers
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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 ...
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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 \...
5
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1answer
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Can type inference be classified in two groups: unification-based and control-flow-based?
I recently came across the 1995 paper Safety analysis versus type inference (pdf link) by Palsberg and Schartzbach that contrasts unification-based type inference and static analysis methods based on ...
5
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1answer
196 views
Commutativity of addition in polymorphic lambda calculus
In the article "Extensional models of polymorphism" by Breazu-Tannen and Coquand, natural numbers are presented using a Church-like encoding:
$polyint = \forall t . (t \to t) \to t \to t$
Addition ...
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0answers
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Are the types that show monads are more powerful than continuations revealing something of fundamental importance?
In 1992 in the paper Imperative Functional Programming, Simon Peyton Jones and Philip Wadler write:
So monads are more powerful than continuations, but only because of the types! It is not clear ...
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Non-objected oriented type theories that can express the $\nu Obj$ calculus
Odersky et al.'s $\nu Obj$ calculus [1] adds just enough dependent typeness on top of object oriented programming to express interfaces that define types (and consequently module systems and other ...
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1answer
681 views
Practical implementation of Hindley–Milner with typeclasses — matching vs most general unifier
I'm trying to get a deep understanding of a (great) paper "Typing Haskell in Haskell". I'm having difficulties understanding the implementation of two methods there — the ...
5
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2answers
250 views
Isomorphism between algebraic data-types
I have two types of trees in Haskell, defined as the least solution of the following equations:
$T_1(A) \cong 1 + A + T_1(A) \times T_1(A)$
$T_2(A) \cong 1 + A \times T_2(A) + T_2(A) \times T_2(A)$
...
3
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1answer
436 views
Occurs check in type inference
I'm reading about type inference in chapter 30 of Programming Languages: Application and Interpretation and I'm trying to understand exactly how the occurs check works in an example I came up with.
...
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1answer
1k views
Homotopy type theory and Gödel's incompleteness theorems
Kurt Gödel's incompleteness theorems establish the "inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic".
Homotopy Type Theory provides an alternative ...
3
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1answer
358 views
Language with extensible type system?
Is there a practical programming language that has an extensible type system?
Or alternatively, an add-on type system that can be used with existing languages?
With extensible I mean that the typing ...
5
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1answer
117 views
Extending simple types to allow `fix`
I'm reading some lecture notes saying that “fix cannot be defined in the simply typed lambda-calculus” and that “no expression that can lead to non-terminating ...
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2answers
2k views
What parts of homotopy type theory are not possible in Agda or Coq?
When we look at the book, Homotopy Type Theory - we see the following topics:
...
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2answers
1k views
Is compiler for dependent type much harder than an intepreter?
I have been learning something about implementing dependent types, like this tutorial, but most of them is implementing interpreters. My question is, it seems that implementing a compiler for ...
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1answer
111 views
Type, operation and function, and their limits
First of all, sorry for my English.
I would like to know, when I want to define a new type (I'm currently developing a computer interpreted language), how can I determine which "functions" are ...
5
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1answer
186 views
Typing relations terminology – how do I read typing relations?
I am currently trying to read up on type theory and have some quick questions on terminology.
In the following rule,
$$
\frac{x:T_1 \vdash t_2 : T_2}{\vdash \lambda x:T_1.t_2:T_1\to T_2}
$$
How ...
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1answer
726 views
Constraint types (IBM/X10) compared to dependent types
Constraint types have been proposed by IBM in their X10 programming language (it's a commercial programming language, not open source software).
Nystrom, Nathaniel, et al. "Constrained types for ...
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0answers
83 views
Semantics of a programming language [duplicate]
A newbie question, if I may... Could you be so kind and explain to me in plain english meaning of 'denotational semantics' and 'operational semantics'? I'm familiar with the definitions and have read ...
