Questions tagged [type-theory]

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

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9
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1answer
787 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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3answers
3k 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 ...
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1answer
1k 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. ...
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5answers
6k 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 ...
43
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4answers
4k views

How would I go about learning the underlying theory of the Coq proof assistant?

I'm going over the course notes at CIS 500: Software Foundations and the exercises are a lot of fun. I'm only at the third exercise set but I would like to know more about what's happening when I use ...
48
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3answers
3k views

How do 'tactics' work in proof assistants?

Question: How do 'tactics' work in proof assistants? They seem to be ways of specifying how to rewrite a term into an equivalent term (for some definition of 'equivalent'). Presumably there are formal ...
46
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2answers
6k views

Explaining Applicative functor in categorical terms - monoidal functors

I'd like to understand Applicative in terms of category theory. The documentation for Applicative says that it's a strong lax ...
34
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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 ...
20
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2answers
2k views

Is there a typed lambda calculus which is consistent and Turing complete?

Is there a typed lambda calculus where the corresponding logic under the Curry-Howard correspondence is consistent, and where there are typeable lambda expressions for every computable function? This ...
28
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6answers
2k views

Why naturals instead of integers?

I'm interested in why natural numbers are so beloved by the authors of books on programming languages theory and type theory (e.g. J. Mitchell, Foundations for programming languages and B. Pierce, ...
26
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1answer
1k views

Are types propositions? (What are types exactly?)

I've been reading a lot on type systems and such and I understand roughly why they were introduced (in order to resolve Russel's paradox). I also understand roughly their practical relevance in ...
24
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2answers
2k views

Why do Agda and Coq disagree on strict positivity?

I've stumbled across a confusing disagreement between Agda and Coq that is not obviously related to the most well known distinctions between their type theories (e.g., (im)predicativity, induction-...
19
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1answer
1k 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: ...
16
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1answer
600 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. ...
15
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1answer
398 views

Logical Reations for an Impredicative System in a Predicative MetaTheory

Logical Relations for Impredicative languages like System F seem to rely critically on impredicativity of the ambient logic. Specifically, the interpretation for the forall-type will be defined in ...
26
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2answers
2k views

Context Sensitive Grammars and Types

1) What, if any, is the relationship between static typing and formal grammars? 2) In particular, would it be possible for a linear bounded automaton to check whether, say, a C++ or SML program was ...
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3answers
2k views

Type inference for imperative statements other than assignment

In my search for research papers about type systems for imperative languages, I only find solutions for a language with mutable references but without genuine imperative control structures such as ...
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2answers
2k views

Implicit vs explicit subtyping

This page asserts that many languages do not use implicit subtyping (structural equivalence), prefering explicit/declared subtyping (declaration equivalence) I've mostly used programming ...
16
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3answers
993 views

Can we prove weak normalization for System F by induction on a transfinite ordinal

Weak normalization for the simple typed lambda calculus can be proved (Turing) by induction on $\omega^2$. An extended lambda calculus with recursors on natural numbers (Gentzen) has a weak ...
14
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0answers
366 views

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, ...
11
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2answers
772 views

Formalizing the theory of finite sets in type theory

Most proof assistants have a formalization of the concept of "finite set". These formalizations, however, differ wildly (although one hopes that they are all essentially equivalent!). What I don't ...
6
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1answer
664 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 ...
15
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1answer
463 views

Minimal specification of Martin-Löf type theory

I'm reading the formal presentation of Martin-Löfs type theory (appendix of the HoTT book). The authors introduce a hierarchy of universes, then $\Pi, \Sigma,+, {\bf 0}, {\bf 1}$ and also $W$-types as ...
12
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3answers
810 views

For what languages is there already a theory of observational equivalence?

For a correctness proof, I'm looking for a usable notion of program equivalence $\cong$ for Barendregt's pure type systems (PTSs); missing that, for enough specific type systems. My goal is simply to ...
7
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1answer
482 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 ...
6
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1answer
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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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1answer
466 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 ($A \rightarrow B$)...
11
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2answers
379 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 ...
10
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2answers
281 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 ...
10
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1answer
816 views

Algorithm to determine function equality on the simply typed lambda calculus?

We know that beta-equality of simply typed lambda-terms is decidable. Given M,N:σ→τ, is it decidable whether for all X:σ, MX $≃_β$ NX?
9
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1answer
463 views

In System F à la Church, can we automatize type inference for the for-all elimination?

The question is the following. Generally when one have a term like $\Lambda X.t$, we can eliminate the forall by applying this term to a type, as instance $(\Lambda X.t)[T]\to t[X:=T]$. Now, suppose ...
7
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2answers
512 views

"Correctness" of type theory

How to "proof" that type theory is correct? Or at least explain that it's meaningful in some sense. In what extent is this a mathematical question and in what is a philosophical one? When type ...
5
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1answer
413 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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498 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: ...
4
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0answers
143 views

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 ...
11
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1answer
703 views

Can affine lambda calculus solve every problem in P?

In Advanced Topics in Types and Programming Languages it is mentioned, in the chapter on sub-structural type systems, that a "carefully crafted" affine lambda calculus with a recursion combinator for ...
9
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1answer
323 views

Example of where violation of strict positivity condition in inductive types leads to inconsistency

Most dependent typed systems have a strict positivity conditions for inductive types. Does anybody know an example where violation of the condition leads to inconsistency in the system?
9
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1answer
859 views

Will Martin-Löf Type Theory lead to a greater ability to write provably correct code?

This post refers to the Curry-Howard isomorphism and the Martin-Löf Type Theory. The post makes the claim of a future 'unification' between the the describing language of math, and the operation ...
9
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0answers
200 views

reference request: deciding validity of higher-order quantified boolean formulas is not Kalmar-elementary

$\newcommand\iddots{⋰}$In "A simple proof of a theorem of Statman" (TCS 1992), Harry Mairson gives a simple proof of Statman's result that deciding $\beta\eta$-equality of terms in simply typed lambda ...
7
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2answers
794 views

An example of a totally computable function that is not definable in system T?

Could you give me an example of a totally computable function of type N × N → N that is not definable in System T? Thanks.
7
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1answer
302 views

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
234 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
190 views

From Church-encoding to induction principle

I am looking for an algorithm to go from a Church-encoded datatype to their induction principle in the Calculus of Constructions. For example: ...
3
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1answer
133 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 ...
3
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0answers
220 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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0answers
265 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: <...
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1answer
188 views

Extensional type theory and function extensionality

Is the principle of function extensionality $ (\forall x. f(x) = g(x)) \implies f = g$, derivable from ETT? Most notably is this derivable in Agda with axiom K?
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2answers
222 views

Why is plus-comm a b $:\equiv$ refl (plus a b) not a proof of the commutativity of addition?

I've been curious about the 'geometric situation' that one has when considering the type $\prod_{(n,m:\textrm{Nat})}(\textrm{plus}\ n\ m) = (\textrm{plus}\ m\ n)$. Here, addition is defined in the ...