Questions tagged [type-theory]

Type structure is a syntactic discipline for enforcing levels of abstraction.

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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 ...
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245 views

How do continuations represent negations (under the Curry–Howard correspondence)?

Under the Curry–Howard correspondence, types can be thought of as propositions, and values inhabiting a type can be thought of as proofs that the corresponding proposition is true. (E.g., the ...
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246 views

Is it possible to verify a typechecker for a total dependently-typed language in that language's logic?

I understand the diagonalization argument against implementing an eval function in a total language, and that typechecking in a dependently typed language requires ...
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1answer
406 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 ...
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1answer
121 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 ...
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1answer
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Is the church-style affine calculus of constructions with unrestricted recursion consistent?

Suppose we take the church-style calculus of constructions, except with affine functions (variables must occur at most once) and mutual recursive definitions. For example: ...
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106 views

On the interpretation of coinduction in type theory

The notion of (co)datatype can be modeled satisfactorily in category theory as fixed-points of polynomial functors. Then, (co)induction principles are derived from initial algebras/terminal ...
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Implementations of Dependent Type Theory

I am trying to find a minimal implementation of dependent type theory that supports Pi Types (obviously) Modules containing records Inductive data types Universe Hierarchy A notion of equality ...
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170 views

Is my understanding regarding how to implement Quotient Types correct?

I was trying to understand Quotient Types, and determine if Self-Types can be used to implement them. From a Reddit post, Here is an example and explanation that may be more familiar to non-...
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Proof that CIC or Dybjer-style eliminators are strongly-normalizing?

Related to this question I'm wondering, what is the standard technique for showing that dependent types with eliminators are strongly normalizing? I'm thinking something like the Calculus of ...
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Hereditary Substitution with Inductives and Eliminators?

I'm wondering, is there any existing work on hereditary substitution with inductive type families and dependent eliminators? In particular, normalizing the application of an eliminator to an ...
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468 views

Is it possible to derive induction by extending CoC with recursion?

Suppose we extended the CoC with primitive recursion; that is, we added a term µ x . t such that equality allowed unrolling recursive terms: ...
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Completeness of realizability semantics for higher-order type theory

In this answer I mention a paper by Geuvers in which he describes a class of models for a type theory $\lambda P_2$ which is a sub-system of the CoC and roughly corresponds to 2nd order predicate ...
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Coherence spaces and full completeness for the implicative fragment of linear logic

Linear logic isn't complete for coherence space semantics since $1$ and $\top$ get identified. But it is, I believe, complete for the fragment of linear logic whose only connective is $\multimap$. I ...
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Modeling union types using sum types

It is trivial to model sum types using only union types and product types: simply add a discriminant. $A + B \cong (0 \times A) \cup (1 \times B)$. What I am wondering is whether or not there is a ...
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257 views

Subtyping rules for extension of System $F_\omega$ with subtyping and kind-level variance tracking

I need an extension of System $F_\omega$ with subtyping, and where the variance of type constructors is reflected in their kind. Unfortunately, System $F^\omega_{<:}$, as defined in chapter 31 of ...
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Category theory in plain MLTT

I want to define a category in simple MLTT (not in HoTT). I defined it with the help of setoids. I.e. category consists of: a type of objects with equivalence relation (Obj : Set) a type of arrows ...
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Type-and-effect systems, stochasticism and effect squelching: how about quicksort?

There's a feature of Haskell's type system which bugs me: you can't implement a randomized sorting algorithm without the use of randomness spilling out into all of its callers. That seems undesirable....
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299 views

Philosophy behind monotonicity requirement for inductive types

Is there a good philosophical reason for why inductive types with negative occurrences or non-monotonicity should not be considered valid constructions? According to my understanding of the Bishop/...
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338 views

Type a variable-argument function?

Is it possible to type a variable-argument function? EDIT: like those defined in Scheme.
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3answers
201 views

Is it reasonable to allow the type of a λ/∀-bound variable to refer to itself?

Usually, in Pure Type Systems, the type of a λ/∀-bound variable is only accessible on its body. That is, on λ (X : A) -> B, <...
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244 views

Extending Hindley-Milner to type mutable references

I have been trying to implement a programming language from scratch, and have gotten reasonably far. It reads just like Python, other than the fact that let is used ...
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4answers
307 views

Why do non-proper types have no terms?

I'm attending a course on Type Theory. The textbook is 'Types and Programming Languages' by Prof. Benjamin C. Pierce. In Chapter 29, Prof. Pierce introduces 'type operator', which can generate ...
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Typechecking liveness properties of coprograms

Clarification: in Total Functional Programming terminology, a program terminates with useful input, while a coprogram doesn't necessarily terminates, and repeatedly produces useful input. I am ...
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Confusing (to me) statement from “Type Classes in Haskell”

I'm reading up on type classes, and started looking at the paper Type Classes in Haskell. In Section 2.2 - Superclasses, the authors use the following example: ...
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1answer
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Verified type checkers

Most of the work on programming language metatheory mechanization focus on the declarative properties of the languages (e.g., type soundness), but fail to address the algorithmic side, i.e. the type ...
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1answer
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What is a canonical term of $\text{Id}_A(x,y)$ if $x$ is not jugdmentally identical to $y$?

In the context of constructive type theory, a term inhabiting some type is said to be in canonical form if it is explicitly built up using the constructors of that type. Particularly, the only ...
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Universe polymorphism: the inference of universes and their constraints

When making a universe polymorphic definition in Coq, universes and their constraints are automatically inferred. Are they somewhat the most general ones (in a sense similar to the principal type ...
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159 views

Can Curry-Howard prove a theorem from the types in your program, that has nothing to do with your program? [closed]

The following link states: Curry-Howard means that any type can be interpreted as a theorem in some logical system, and any term can be interpreted as a proof of its type. This does not mean that ...
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512 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

Monomorphic vs Polymorphic type theory

I am currently reading the book Programming in Martin-Löf type theory by Nordström et al. In the book they have two important parts, one about monomorphic set theory and the other about polymorphic ...
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1answer
309 views

Constructing terms of function types out of the empty type

If a function $f$ is understood as its graph, i.e. a set of pairs $\langle x,y\rangle$ where $x$ is input and $y$ is output, then the empty set $\emptyset$ is a valid function, and for any set $A$, we ...
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189 views

Decidability in Extensional Type Theory

What are the ways in which one can add a decidable equivalence relation in a type system with undecidable type checking/extensional equality?
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1answer
479 views

Generalization and instantiation of types in Hindley-Milner type inference

I’m currently reading Heeren, B., Hage, J., & Swiestra, D. (2002). Generalizing Hindley-Milner Type Inference Algorithms in an attempt to understand Hindley-Milner-style type inference. I'm ...
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111 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 (...
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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: ...
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131 views

Is it possible to check equality of equi-recursive types, or recursive λ-terms?

Can we determine if two λ-terms are equal? Given two lambda terms, let's say they are equal if their (possibly infinite) Bohm trees are. Under this definition, for example, ...
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What's the example of natural transformation in 'Type" that is not a parametric function?

Take a type theory of your choice (perhaps System Fω). Parametric functions are known to be natural transformations in 'Type' category. Yet not every natural transformation in 'Type' is a polymorphic ...
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197 views

Mildly dependently-typed metalanguage for mildly context-sensitive object languages

This is almost certainly not a new idea, but I haven't seen it elaborated or discussed elsewhere. A very natural way to represent the abstract syntax of an object language in a typeful metalanguage is ...
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168 views

Type theoretic equivalent of isomorphism class

How one defines the notion of isomorphism class in type theory? For concreteness I will describe what I mean with an example in Coq. Suppose I have a record ToyRec: ...
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1answer
404 views

Dependent Sums and Products

I'm trying to understand the connections between a few different concepts fundamental to dependent type theory. Dependent functions ($\Pi$-types) Including non-dependent functions ($A \rightarrow B$)...
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1answer
68 views

Does an initial algebra for a class have to belong to the class itself?

In the context of algebraic data types, a concept of initial algebras is usually defined, e.g., in the following way: An algebra $S$ is initial in a class $C$ of algebras iff for every $A\in C$ ...
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2answers
213 views

How to translate the axiom schema of induction by Curry-Howard?

I'm trying to understand the Curry-Howard correspondence. I am comfortable with it for propositional logic, but get confused when $\forall, \exists$ quantifiers come in the picture. The axiom schema ...
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1answer
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Representable functions on System T

In Proof and Types by Girard et alii. Section 7.4.2, I think that the authors want to show that: (1) The set of functions definable in System T coincides with the set of recursive functions whose ...
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1answer
226 views

Damas-Milner-like subset of the calculus of constructions with global type inference

Damas-Milner is a subset of System Fω that gives up expressivity (type-level computation) for usability (type inference). The experience with Haskell and ML attests to the practical value of this ...
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1answer
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Completeness of Constraint Typing (type inference) question regarding $\sigma'$

The theorem of completeness of type inference states the following: Suppose $\Gamma \vdash t:S| _{\mathcal{X}}C$, ...
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1answer
461 views

Wouldn't the calculus of constructions with linear types be a simple functional core that is consistent and expressive?

I have recently asked if there is a simple functional core that is consistent and expressive. In another question, cody pointed out that this is an open problem to have a language that is: Consistent/...
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2answers
269 views

How to translate general recursion into a set of $\mu$-recursive operator applications?

I'm trying to find a scheme to translate a functional language with let rec into a set of primitives called "generalized arrows", i.e. $\kappa$-calculus with ...
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1answer
88 views

Type of induction principle for fixpoint types

To the Calculus of Constructions we could add a general fixpoint type constructor (accepting inconsistencies or assuming F is a ...
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304 views

About the position of side conditions in an inference rule

Sometimes I see people put side conditions above the inference line as if they were premises of an inference rule. This feels strange. My understanding (which may be wrong) is that a side condition ...