Questions tagged [type-theory]

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

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1answer
1k 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 + ...
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1answer
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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 ...
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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 ...
9
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1answer
359 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 / ...
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4answers
350 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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1answer
148 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 ...
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1answer
214 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 ...
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500 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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1answer
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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?
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2answers
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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, ...
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1answer
149 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, ...
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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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2answers
379 views

Explicit set of types and terms in MLTT

Whenever I read a presentation of MLTT, especially in the context of the correspondence of MLTT with LCCCs (eg. Seely's paper), they say "the type constructors/formation rules are..." and then list a ...
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1answer
335 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 ...
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Equality Theorems with Type Theoretic Proof

I am investigating how I might be able to translate even commonplace equalities/ inequalities via the so-called Curry-Howard Correspondance - from a generic, set theoretic plus AOC foundation - into a ...
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1answer
166 views

Type for "ways values can be different"

I am looking for a concept in type theory that I am sure has probably been explored, but do not know the name behind. Let's consider a ML-like language with product and sum types and a Hindley-Milner ...
12
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1answer
287 views

Eta expansion in the pattern lambda calculus

Klop, van Oostrom, and de Vrijer have a paper on the lambda calculus with patterns. http://www.sciencedirect.com/science/article/pii/S0304397508000571 In some sense, a pattern is a tree of variables ...
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3answers
555 views

Proving running time upper bounds for algorithms in dependent type theory

Proof assistants are a valuable tool for verifying the correctness of proofs of mathematical theorems. When dealing with proofs of correctness of algorithms, one is not only interested on showing ...
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1answer
607 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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1answer
577 views

What is the intuition behind linear logic?

I'm trying to understand linear logic to understand linear type systems better. However, when I read the rules, I fail to get an intuition behind it as I've done in modal logic - $\Box A$ means $A$ is ...
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What kind of theoretical object corresponds to a C++ concept?

I am lacking a background in theoretical computer science but I would have liked to understand to what kind of theoretical objects C++ concepts corresponds to. Basically, C++ concepts allow to define ...
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1answer
105 views

Completeness of Constraint Typing (type inference) question regarding $\sigma'$

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

Relationship between Pataraia's theorem and inductive-recursive definitions?

Pataraia's fixed point theorem gives a constructive proof of the fact that if you have a monotone function $f$ on a DCPO, then it has a least fixed point. I've frequently used this fixed point theorem ...
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1answer
599 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 $\pi$-...
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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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2answers
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What's the definition of join on iso-recursive types?

In languages with subtyping, there is often a "join" operation defined to compute the least upper bound of two types. It's used in type-checking, for example to find the smallest type that covers both ...
6
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1answer
267 views

Can a totality checker be used to guarantee a proof on the calculus of constructions + inductive types is correct?

If we extend the Calculus of Constructions with Fix, we gain a lot of expressivity for barely no added complexity. That includes being able to derive induction, perform large eliminations, prove ...
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Are there simple core languages which are consistent and expressive?

The Calculus of Constructions is a very simple core functional language with dependent types. Per curry-howard isomorphism, it could, potentially, be very useful for writing programs and proofs. It, ...
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2answers
196 views

Proof that the calculus of constructions extended with recursive types isn't strongly normalizing?

What is the proof that the calculus of constructions, extended with recursive types (i.e., Fix at the type-level) isn't strongly normalizing?
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1answer
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How to mechanically derive the recursor of a type from its constructors?

In Martin-Löf Dependent Type Theory a type is commonly prescribed by how to construct its canonical terms and how to show that its canonical terms are definitionally equal. This means that the ...
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1answer
845 views

Does the Law of Excluded Middle imply the Axiom K in Martin-Löf's Intensional Type Theory?

So I've been wondering if the Law of Excluded Middle (LEM) implies the so-called Axiom K in Martin-Löf's Intensional Type Theory. The Axiom K states that $$\Pi_{A : Type} \Pi_{x : A} \Pi_{p : \text{Id}...
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2answers
405 views

Higher-rank polymorphism over unboxed types

I have a language in which types are unboxed by default, with type inference based on Hindley–Milner. I’d like to add higher-rank polymorphism, mainly for working with existential types. I think I ...
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465 views

How to do Type Inference using an SMT solver?

I understand that Hindley-Milner Type Inference can be implemented using an off-the-shelf SMT (Satisfiability Modulo Theories) solver? How would this work, for example for a very simple type system (...
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261 views

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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2answers
201 views

Given a type system T, and a type $A$ in that type system, is there an (effective) surjection from $\mathbb{N}$ onto the set of terms of that type?

I assume that in hoping for an effective bijection, we run into undecidability issues, but intuitively, it seems like there should at least be a surjection from $\mathbb{N}$ to the terms of any type, ...
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1answer
149 views

What is the proof-theoretic significance of the existence of a Brown-Palsberg self-interpreter for system $F_\omega$?

In A Self-Interpreter for F-omega, Brown and Palsberg construct for each term of each type $x:T$ a representation $\bar x : \Box T$ (a metatheoretical function which can't be represented in the ...
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1answer
624 views

Relative consistency of PA and some type theories

For a type theory, by consistency, I mean that it has a type which is not inhabited. From the strong normalization of the lambda cube, it follows that system $F$ and system $F_\omega$ are consistent. ...
5
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1answer
437 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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1answer
819 views

Can Elementary Affine Logic be used as the core type system of a practical programming language?

Elementary Affine Logic is a type system which captures the class of λ-terms that can be reduced in elementary time. Moreover, EAL-typeable terms can be reduced using the abstract fragment of Lamping'...
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1answer
283 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). ...
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0answers
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Connection between nonmonotonic logic and type theory (lambda calculus)

There is known connection between classical and modal logics and type theory (lambda calculus), but are there connections between nonmonotonic logics (e.g. defeasible logic) and type theory (lambda ...
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Typed Lambda Calculus models and denotations

I'm trying to draw a general mental picture about the models and the denotational semantics of the typed lambda calculus, in its different variants. I'm particularly interested in how the semantics ...
8
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1answer
141 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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1answer
266 views

Proof of decidability of type checking of calculus of (co)inductive constructions?

I often see it asserted that type checking is decidable for CIC, but I haven't seen it proven. Is there a good paper (or simple demonstration) of this?
4
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1answer
264 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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4answers
927 views

Do I have to give up the Law of the Excluded Middle in order to Learn $\lambda$-Calculus?

I know very little about what I am talking about in what follows, so I appreciate any all help in pointing out all of my mistakes -- otherwise I won't be able to learn more and advance in my knowledge ...
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1answer
260 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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205 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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2answers
840 views

Subtypes as subsets of SML datatypes

One of the few things that I dislike about Okasaki's book on purely functional data structures is that his code is littered with inexhaustive pattern matching. As an example, I'll give his ...
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2answers
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"Impredicative" in type theory

I am confused. I think I've read two usages of the word "impredicative" in type theory: When people talk about the "impredicative" version of Martin-Löf's type theory, which they say it is ...

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