Questions tagged [typed-lambda-calculus]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
2answers
175 views

Are there strongly normalizing lambda terms that cannot be given a System F type?

I know that all well-typed System F terms are strongly normalizing, but is the converse true as well? In other words, does System F typeability precisely characterize program termination? (And if so, ...
2
votes
0answers
131 views

Set-theoretic encoding of functions in type theory

Functions usually get encoded in set theory as follows. A function $A\to B$ is a subset $f\subset A\times B$ such that $\pi_1:f\to A$ is a bijection. In type theory to give a function $A\to B$ is to ...
2
votes
1answer
97 views

Dependent eliminator for empty type in intensional Martin-Löf type theory

In calculus of inductive constructions you can just say that the empty type is the type with no constructors and it automatically builds the dependent eliminator. But let's say I'm setting up ...
-1
votes
1answer
92 views

How much type information do Hindley-Milner proof assistants need to remain sound?

A known benefit of the HM type system is that you can usually infer a term's most general type with no user-provided type annotations. For example, if my theory contains the standard axiom: $$\forall ...
1
vote
2answers
105 views

Explicit type system with infinite non-cumulative universe hierarchy

Is there an open-source proof assistant or at least an explicit set of rules written down somewhere for a type system with an infinite non-cumulative universe hierarchy and unique typing? I want to ...
0
votes
0answers
47 views

Complexity of finding typing derivation trees

In type theories where type checking is decidable do we have estimates for how much time/space it takes to find a typing derivation tree of a valid typing judgment? Do any published references do this ...
2
votes
2answers
127 views

$\mathbb{N}$ in intensional MLTT with judgmentally commutative $+$ and $\times$

Is there a way to implement natural numbers in intensional Martin-Löf type theory so that addition and multiplication is judgmentally commutative?
4
votes
1answer
128 views

Model of MLTT with $\eta$ rule where function extensionality fails

Consider intensional Martin-Löf type theory with judgmental $\eta$ rule for dependent product types. Is there a model of it where function extensionality fails?
2
votes
1answer
75 views

Normal term of double negation of W-type

Consider the intensional Martin-Löf type theory without axiom of choice or the law of excluded middle. Let $A:U_0$ be a type and $B:A\to U_0$ be a function such that $\Sigma_{a:A}(B(a)\to 0)$is ...
2
votes
1answer
90 views

Typing inference as a map on abstract syntax trees

Is there a reference that explains typing inference for Martin-Löf type theory as a computable map from abstract syntax trees of terms to abstract syntax trees of types? I don't want to identify non-...
5
votes
1answer
104 views

Is the Mendler-encoding in System-F adequate?

In the paper "Efficiency of Lambda-Encodings in Total Type Theory" it is mentioned that the Church-encoding is adequate and the Parigot encoding is not adequate. This means that any ...
2
votes
1answer
134 views

Complexity of type inference in the simply typed lambda calculus

A similar question was answered here: Is simply typed lambda calculus equivalent to primitive recursive functions What I conclude from the answers is that the complexity is that of the extended ...
6
votes
0answers
133 views

Category theory lambda cube?

If simply typed lambda calculus corresponds to cartesian closed categories, what types of categories do other calculi in the lambda cube correspond to? https://en.m.wikipedia.org/wiki/Lambda_cube
6
votes
1answer
212 views

An efficient Beta-Equivalence algorithm?

Is there an efficient algorithm to determine if two terms are beta-equivalent? I'm specifically curious about simply-typed-lambda-calculus, so you can assume both terms are strongly normalizing. I ...
-2
votes
1answer
93 views

Beta reduction and vacuous lambda abstraction [closed]

Suppose we have the following typed lambda term (where $s$ does not occur in E (which is of type $s \to p$) and $s$ and $s'$ have the same type), and want to apply $\beta$-reduction: $(\lambda s. E)\, ...
6
votes
1answer
134 views

What arithmetical theorems can plain $\lambda \Pi$ reason about?

I've read that System F cannot state or prove theorems of the First Order Theory of Arithmetic. I assume this is because we lack dependent types, so we cannot explicitly express $\forall n:\mathbb{N}....
5
votes
1answer
102 views

Decidability of rank-k polymorphism vs. System F

There's a paper by Kfoury from 1992, "Type Reconstruction in Finite Rank Fragments of the Second-Order $\lambda$-Calculus", that proves that type inference for Curry-style rank-$k$ polymorphic lambda ...
3
votes
1answer
128 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
votes
0answers
100 views

Type System Of $\lambda\mu$-Calculus

reading this paper on CPS-tranformation from the $\lambda\mu$-calculus, I'm a bit confused about the type system presented: Why second-order formulas in the types? Is this according to the Curry-...
5
votes
2answers
198 views

What's the point of stack judgement in CBPV?

Call-by-push-value (CBPV) introduces two main families of types, values and computations, and their corresponding judgements. However, in some extensions/variants/adaptation of CBPV, there is a third ...
3
votes
0answers
187 views

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: ...
6
votes
1answer
286 views

Fixed set of type constructors to simulate all intensional inductive families?

I'm wondering, are there small dependent calculi that can simulate a language with inductive families (that is, has a type isomorphic to each inductive family, at least as powerful of induction ...
6
votes
1answer
206 views

Structural equality of Pi Types with heterogeneous equality?

I'm trying to implement a proof of the following type: ...
7
votes
2answers
285 views

Termination checking for Scott-encodings in System F with positive-recursive types

Is there any research on termination analysis on Scott-encodings in System F with positive-recursive types. All papers I have found use languages with constructors and case analysis (for example ...
9
votes
1answer
277 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
votes
2answers
256 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[...
12
votes
1answer
614 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 ...
8
votes
1answer
240 views

What's the difference between Moggi's computational metalanguage and Moggi's lambda calculus?

This is a reference confusion. Sometimes I see people use the term "Moggi's computational metalanguage" to refer to the calculus presented by Moggi, and sometimes to "Moggi's computational lambda ...
2
votes
2answers
249 views

Moggi's computational metalanguage

In Notions of Computation and Monads Moggi models the notion of a computation of type $A$, $TA$, using a monad $T$. Among other things this ensures the $T\eta$ rule: $$\frac{x: A \vdash a:TB}{x:A \...
0
votes
1answer
50 views

Head variables of terms after application

We work in the Church-style simply typed lambda calculus. All terms shall be considered in long normal form. Any term of type $A_1\rightarrow A_2\ldots\rightarrow A_n \rightarrow 0$ is of the form $\...
10
votes
1answer
215 views

Program inversion algorithms for higher-order programs

The term program inversion has multiple shades of meaning, but probably got started with J. McCarthy's 1956 work The Inversion of Functions Defined by Turing Machines in the context of AI. By now ...
7
votes
1answer
326 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 ...
11
votes
3answers
284 views

Calculus of Constructions: compress expression to its smallest form

I'm aware that the Calculus of Constructions is strongly normalizing, meaning every expression has a normal for that cannot be beta,eta-reduced further. So in fact this is the most efficient ...
12
votes
1answer
250 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 ...
4
votes
1answer
744 views

Is simply typed lambda calculus equivalent to primitive recursive functions

It's well known that the computation models untyped lambda calculus and $\mu$-recursive function are equivalent in terms of computability (in fact they are both Turing complete.) It is also well known ...
4
votes
0answers
83 views

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 ...
5
votes
1answer
282 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)....
5
votes
1answer
109 views

Inference in typed lambda calculus theories

I'd like to do automated inference, say solving word problems or reducing to normal form, in an equational theory of the typed lambda calculus (with product and unit types). Equivalently, in category-...
7
votes
1answer
264 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). ...
3
votes
1answer
451 views

Does the simply typed lambda calculus have general iteration?

In more expressive calculi such as System F, the Church numerals, by virtue of their design, allow for iteration over an arbitrary type. Can this effect be replicated in the simply typed case? To be ...
5
votes
0answers
85 views

Characterisation of the BCIW definable functions

Given a full model of the simply typed lambda calculus, it's possible to characterise the lambda definable functions as those that are invariant under every "Kripke logical relation". (See here.) I ...
3
votes
1answer
320 views

Enumerating all simply typed lambda terms of a given type

How can I enumerate all simply typed lambda terms which have a specified type? More precisely, suppose we have the simply typed lambda calculus augmented with numerals and iteration, as described in ...
8
votes
0answers
283 views

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]...
22
votes
2answers
4k views

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 ...
2
votes
0answers
174 views

Heyting algebra in simply typed lambda calculus

The Emil Jeřábek's comment in Can boolean algebra be expressed in simply typed lambda caclulus? give rise to the following question: Can some non-trivial Heyting algebra be expressed in simply typed ...
3
votes
1answer
160 views

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 ...
1
vote
1answer
118 views

Observational Equivalence of open terms in PCF

The notion of observational equivalence is rather intuitive, but formally I'm having some doubts in the particular case of open terms. Lets consider the simple case where the terms ...
3
votes
0answers
52 views

Functionality of a hierarchy of definable functions over $\mathbb{N}$

Let $T$ be the complete hierarchy of functions over $\mathbb{N}$. That is: $T$ = $\bigcup T_{\tau}$ for all simple types $\tau$ built up from the basic type $\mathbb{N}$, with $T_{\mathbb{N}} = \...
4
votes
1answer
132 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 ...
20
votes
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 ...