Questions tagged [typed-lambda-calculus]
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Can you always throw away the types when evaluating lambda expressions?
As I understand it, in the simply typed lambda calculus you can type-check your terms, then throw away all the types and do the evaluation exactly as if it was the untyped lambda calculus. Is that ...
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$\lambda$-definability and structure preserved by homomorphisms
I imagine there are some standard results that bear on this, but I'm having trouble finding a proof or refutation of it.
Some prelimary definitions.
A Henkin structure $A = (A^\cdot, ⟦\cdot⟧_A)$ for ...
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Constructive Strong Normalization of the Extended Calculus of Constructions
The extended calculus of constructions (ECC) is basically the calculus of constructions with cumulative universes. I use the definition which Zhaohui Luo used in his PhD theses which contained a proof ...
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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, ...
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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 ...
user61651
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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 ...
user61651
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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 ...
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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 ...
user61651
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$\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?
user61651
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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?
user61651
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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 ...
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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-...
user61651
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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 ...
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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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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
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Is there an efficient beta-equivalence algorithm?
Is there an efficient algorithm to determine if two terms are beta-equivalent? Specifically, I am curious about simply-typed-lambda-calculus, so you can assume both terms are strongly normalizing.
I ...
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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)\, ...
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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}....
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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 ...
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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 ...
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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-...
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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 ...
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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:
...
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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 ...
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Structural equality of Pi Types with heterogeneous equality?
I'm trying to implement a proof of the following type:
...
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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 ...
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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 ...
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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[...
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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 ...
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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 ...
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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 \...
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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 $\...
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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 ...
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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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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 ...
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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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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 ...
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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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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)....
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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-...
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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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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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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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 ...
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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}} = \...
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