Questions tagged [type-theory]

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

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6
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1answer
164 views

General Induction Principle

Let us suppose that we want to provide for each inductive type an axiom describing the associated elimination/induction principle. For example, given a definition for the naturals: ...
11
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2answers
324 views

Hereditary substitution with a universe hierarchy

I've read about hereditary substitution for the Simple Lambda Calculus and for The Logical Framework with distinct terms and types. I'm wondering, are there any examples of hereditary substitution in ...
18
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3answers
2k views

Classification of Typed/Untyped Lambda Calculi

Can anyone explain briefly (if thats possible!) or refer me to a reference, summarizing the differences between untyped lambda calculus and the more common typed lambda calculi? I'm particularly ...
9
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1answer
199 views

Understanding the Proof of Strong Normalization of the Calculus of Constructions

I have difficulties in understanding the proof of strong normalization for the calculus of constructions. I try to follow the proof in the paper of Herman Geuvers "A short and flexible proof of Strong ...
5
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2answers
215 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
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2answers
321 views

Preservation under Substitution with Telescopes

In the simply typed lambda calculus, one can show the following result, known as "preservation under substitution": If $\Gamma \vdash v : \tau_1$ and $(x : \tau_1) \vdash t : \tau_2$, then $\Gamma \...
8
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1answer
187 views

Type theory and computational complexity

Is there a type system, which restricts the lambda terms to the terms which fall inside a complexity class? Like the typable terms in the theory are strictly inside the complexity class ? Or is it not ...
7
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1answer
321 views

Finding a common factor in $\lambda$-terms that agree under certain substitutions

Suppose that $\mathcal{L}$ is the language of a simply typed lambda calculus of two base types, $e$ and $t$, with infinitely many constants at each type. A substitution $j$ is a mapping from ...
10
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1answer
451 views

Proof techniques for showing that dependent type checking is decidable

I'm in a situation where I need to show that typechecking is decidable for a dependently-typed calculus I'm working on. So far, I've been able to prove that the system is strongly normalizing, and ...
46
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5answers
5k views

What is the most intuitive dependent type theory I could learn?

I am interested in getting a really solid grasp on dependent typing. I've read most of TaPL and read (if not fully absorbed) 'Dependent Types' in ATTaPL. I've also read and skimmed a bunch of articles ...
3
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3answers
200 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, <...
15
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1answer
954 views

Can boolean algebra be expressed in simply typed lambda caclulus?

Boolean algebra can be expressed in untyped lambda calculus in (for example) this way. ...
3
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1answer
475 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 ...
4
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1answer
108 views

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: ...
7
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1answer
557 views

Is Church-pentation implementable in Agda?

Inspired by suggestion in this question, I've implemented predicative Church encoding of Peano arithmetic. Exponentiation works fine, unfortunately tetration requires the level of one of the arguments ...
9
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2answers
183 views

Decidability of type inference and type checking in MLTT

In Martin-Löf's An Intuitionistic Theory of Types: Predicative Part it is proved that type checking $a \colon A$ is decidable subject to $a$ being typeable in the first place, by proving a ...
4
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1answer
197 views

Why isn't it “enough” to prove induction with one extra “INat” argument?

It is well known that it is impossible to prove the induction principle for Natural numbers on the Calculus of Constructions. That is, ...
4
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1answer
133 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 ...
2
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1answer
94 views

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 ...
7
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1answer
172 views

If the untyped language is terminating, can we still derive a contradiction from `Type : Type`?

Question If a pure type system has a terminating proof language, can we have Type : Type at the logic level without causing paradoxes (i.e., without causing ...
7
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1answer
319 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 ...
17
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1answer
964 views

Why it's impossible to declare an induction principle for Church numerals

Imagine, we defined natural numbers in dependently typed lambda calculus as Church numerals. They might be defined in the following way: ...
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2answers
321 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 ...
6
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2answers
142 views

Is it possible to create a “quote” function that, given a native λ-term, returns its λ-encoded representation?

Suppose we implement the λ-calculus inside the λ-calculus itself with λ-encodings and Bruijn indices: ...
5
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2answers
119 views

Does Standard ML validate (CBV) eta equivalence?

$\eta$ equality of functions is fundamental in their Category-theoretic semantics but in practice even "functional" languages include "impure" features that violate it. Note that this is not an issue ...
33
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3answers
2k views

Type classes vs object interfaces

I don't think I understand type classes. I'd read somewhere that thinking of type classes as "interfaces" (from OO) that a type implements is wrong and misleading. The problem is, I'm having a problem ...
2
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0answers
192 views

What is the proof for the inconsistency of impredicaitivity + excluded middle + large elimination in type theory

Why is the combination of impredicativity + excluded middle + large elimination inconsistent in dependent type theory? My understanding of large elimination is I am doing large elimination if I am ...
11
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1answer
368 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
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1answer
113 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 ...
4
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1answer
325 views

Strong normalization property of CoC inside CoC

Wikipedia says that The CoC is strongly normalizing, although, by Gödel's incompleteness theorem, it is impossible to prove this property within the CoC since it implies inconsistency. Why is ...
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0answers
226 views

Is it possible to type Ackermann function with (stratified variant of) System F?

I was surprised to find no open-source implementation of Ackermann function in pure System F as an illustrative example. I finally managed to implement it myself in Haskell using Church encoding: <...
2
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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 $A\in C$ ...
12
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1answer
262 views

What makes a language (and its type-system) capable of proving theorems about its own terms?

I've recently attempted to implement Aaron's Cedille-Core, a minimalist programming language capable of proving mathematical theorems about its own terms. I've also proven induction for λ-encoded ...
7
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1answer
247 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 ...
5
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2answers
295 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 ...
16
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0answers
203 views

Using Dependent Type Theory for Categories that are not LCCC

I have recently been working with polynomial functors and monads based mostly on Gambino-Kock. There they define polynomial functors in a Locally Cartesian Closed Category (LCCC) and extensively use ...
5
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1answer
602 views

What exactly is “large elimination”?

I'm studying type theory (mostly Coq) and often encounter the term "large elimination", usually when talking about type universes hierarchy consistency, for example: impredicative polymorphism + ...
2
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2answers
266 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 ...
9
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1answer
258 views

Cantor's theorem in type theory

Cantor's theorem states that For any set A, the set of all subsets of A has a strictly greater cardinality than A itself. Is it possible to encode something like this using only types / ...
3
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4answers
297 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/...
0
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1answer
114 views

Language/type system closest to Haskell without general recursion

I've implemented a completely functional DSL, and now I'd like to reason about it. It would be helpful to be able to compare it to existing languages. The type system is parametric polymorphic with ...
1
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1answer
172 views

What is the coproduct: A + A? [closed]

In the HoTT book, it is said The type of booleans 2 : U is intended to have exactly two elements. It is clear that we could construct this type out of coproduct and unit types as 1 + 1. I don't ...
25
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1answer
3k views

What's the relation and difference between Calculus of Inductive Constructions and Intuitionistic Type Theory?

As stated in the title, I wonder any relation and difference between CIC and ITT. Could someone explain or point to me some literature that compares these two systems? Thanks.
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0answers
464 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: ...
7
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1answer
1k views

What is the difference between System F and Hindley-Milner type system?

Hindley-Milner type system is some restriction of System F, which allows type inference. But from simple descriptions I cannot see, what is the difference between them?
14
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2answers
653 views

In the Hott book, are the most of the type formers redundant? And if so, why?

In chapter 1 and Appendix A of the Hott book, several primitive type families are presented (universe types, dependent function types, dependent pair types, Coproduct types, Empty Type, Unit type, ...
9
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1answer
252 views

Where is relational parametricity in hyperdoctrine or topos models explored?

Reynolds originally proposed a relational semantics for the second order polymorphic lambda calculus[1]. However he later showed[2] that this approach was inconsistent with classical set theory. ...
9
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1answer
457 views

What is the role of the Bicolored Calculus of Constructions?

So, I'm reading a bit about elaboration, particularly, algorithms based on the Bicolored Calculus of Construction, and I'm a bit confused. I don't understand what exactly the purpose of the $CC^{bi}$ ...
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1answer
133 views

How could one define a language based on the Calculus of Constructions, but with fixed points and EAL-style duplication restrictions?

Suppose that we take the Calculus of Constructions as a basis, but take away exponential functions (allowing only linear functions), and add the controlled duplication rules of EAL. That'd, I believe, ...
4
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0answers
123 views

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 ...