Questions tagged [typed-lambda-calculus]
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55
questions
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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
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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
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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 ...
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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
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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
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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 ...
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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 ...
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...
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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]...
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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
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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 ...
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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
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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
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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 ...
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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 ...
