Questions tagged [lambda-calculus]

Church's formal system used in computatability, programming languages and proof theory to represent effective functions, programs and their computation, and proofs.

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153 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, ...
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1answer
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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-...
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CCCs, computational calculi and point-surjectivity

The models of some computational calculi are in a correspondence with Cartesian Closed Categories with an object $U$ that has some relationship to its exponential object $U^U$ e.g. a retraction ...
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1answer
124 views

Structural normalization algorithm for the simply typed lambda calculus

I would like to know if there is a (piecewise) structural normalization algorithm for the simply typed lambda calculus. By structural I mean a recursive function that only calls itself on subterms of ...
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Nominal Tree Languages i.e. with Binders and Infinite Symbols?

I'm wondering if there has been any research done into automata that accept languages of trees that can bind arbitrary variables, and are considered equal under alpha equivalence. I've found so far: ...
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1answer
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Phonology and lambda calculus

I wonder whether there is any relationship between lambda calculus and phonology (study of phonemes). Specifically, how one would use the concepts of lambda calculus (typed or untyped) in the study of ...
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Applications of solvability in the λ-calculus

What are applications of solvability / unsolvability, and of operational characterizations of solvability? Solvability In the (untyped pure call-by-name) $\lambda$-calculus, a closed term is said to ...
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1answer
671 views

Can we derive Cubical Type Theory from Self-Types?

Self Types are known for being a simple extension to the Calculus of Constructions that allow it to derive all inductive datatypes of a proof assistant like Coq and Agda, without a "hardcoded&...
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What is the time complexity of substitution algorithms(normalization by evaluation, explicit subtitution)?

I'm studying the substitution algorithms of lambda calculus. I think now I understand how they work, but I couldn't find any materials about their time complexity yet. This is what I've thought about ...
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Forming ordered pairs using monads and doing without the Kuratowski encoding of ordered pairs

Suppose we have a set $S$ of constants of the Simply-Typed Lambda Calculus (STLC) various types, and the operation of union $\cup$ which takes two constants and forms their union. For example, $S$ ...
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Is this a reader monad?

I'm unsure whether the following three equations constitute a valid instance of a reader/environment monad on the simply-typed lambda calculus, where $\alpha$ is any type (I subscript some terms with ...
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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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Categorical equivalent of higher order logic

From Simply typed lambda calculus and higher order logic, I get the impression that HOL is STLC + equality + equality axioms. I was wondering if there is a particular kind of category modelling this.
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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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1answer
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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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Would it be possible to derive `transp` natively from Path, Interval and typecase?

Assume for a moment that we extended Agda with an Interval and a Path type, but not transp (which is a primitive currently). I'm ...
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A substitution to add variables in the context

I'm doing type-inference in a dependently typed language, using (as is commonly done) a λ-calculus with explicit substitutions like that of Abadi (with a representation based on debruijn indices) in ...
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143 views

Set:Set or Negative Inductives in a Total Language?

In total dependently typed languages, general recursion is forbidden, since this can allow for non-terimination. However, dependently typed language can still describe Turing-complete computations (...
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1answer
286 views

Applications of Barendregt–Geuvers–Klop conjecture

I was learning about type systems from Benjamin C. Pierce's Types and Programming Languages and came across the Lambda cube in the chapter on Higher-Order Polymorphism. After reading up more about it ...
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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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Other popular lambda calculus encodings besides Church's?

I know once could start from the popular Church encodings of booleans, ints, etc and arbitrarily obfuscate and complicate them to obtain new representations of the same concepts, but are there other ...
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PHOAS with extrinsic typing?

Parameterized Higher Order Abstract Syntax (PHOAS) is a representation of syntax trees that allows the host language's binding to be used to represent binding in the language being modelled, while ...
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Metrics for modelling convergence in the lambda-calculus

I wonder if there have been efforts to reconcile the measure approach to termination and Scott's domain theory or other topological models of computation. In other words, can we translate this measure ...
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Church numerals and Kleene numerals

Church numerals $\overline{0} = \lambda fx. x$ and $\overline{n} = \lambda f x. f^n x$ are provisions for applying a function $n$ times to an argument. An alternate system of numerals, possibly ...
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Semantic read-back of sharing graphs

A "sharing graph" is a representation of a $\lambda$-term that modifies an abstract syntax tree by adding edges connecting each variable use to the place where that variable is bound. They are used ...
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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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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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1answer
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Infinite $\beta \eta$-reduction sequence implies infinite $\beta$-reduction sequence

In Sorensen and Urzyczyn's book there is a lemma (1.3.11) which I am having a hard time proving. 1.3.11 Lemma: If there is an infinite $\beta \eta$-reduction sequence starting with a term $M$ then ...
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Is Combinatory Logic (CL) still relevant for programming language theory?

I've been reading up on R. Smullyan's "To Mock a Mockingbird" and Hindley's "Lambda-Calculus and Combinators: An Introduction". I've even read Schonfinkel's 1924 paper introducing the idea of ...
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What is the “question” that programming language theory is trying to answer?

I've been interested in various topics like Combinatory Logic, Lambda Calculus, Functional Programming for a while and have been studying them. However, unlike the "Theory of Computation" which ...
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Extending EAL with recursion makes it incompatible with the abstract algorithm?

A few years ago, I've asked if Elementary Affine Logic can be used as the core type system of a practical programming language. The accepted answer argues that, yes, although such language would be ...
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Proof that CIC or Dybjer-style eliminators are strongly-normalizing?

Related to this question I'm wondering, what is the standard technique for showing that dependent types with eliminators are strongly normalizing? I'm thinking something like the Calculus of ...
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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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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 ...
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When a type is a value?

In functional programming and in the theoretical setting of the $\lambda$-calculus it is standard to consider a lambda abstraction $\lambda x.M$ as a value. In my understanding, the intuitive reason ...
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Busy Beaver Equivalent for the Untyped Lambda Calculus

In the same way that the Busy Beaver function is defined for Turing Machines, we could define a similar function for the untyped lambda calculus: Over all terms in the ULC composed of ...
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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 ...
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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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Preservation under Substitution with Telescopes

In the simply typed lambda calculus, one can show the following result, known as "preservation under substitution": If $\Gamma \vdash v : \tau_1$ and $(x : \tau_1) \vdash t : \tau_2$, then $\Gamma \...
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Locally-nameless representation: normal order & opening with a bound variable

This question concerns the representation used in Arthur Charguéraud's paper “The locally nameless representation” and is somehow a follow-up on this question, where it is asked about the ...
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Equivalent formulation of complexity theory in Lambda Calculus?

In complexity theory the definition of time and space complexity both reference a universal Turing machine: resp. the number of steps before halting, and the number of cells on the tape touched. ...
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Is the church-style affine calculus of constructions with unrestricted recursion consistent?

Suppose we take the church-style calculus of constructions, except with affine functions (variables must occur at most once) and mutual recursive definitions. For example: ...
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Relationship between lambda-definability, specification and definability in model theory

I am new to lambda calculus and definability theory, and I am trying to clarify my understanding of the relationship among the following concepts: An element $a$ in the domain of a type $A_\sigma$ is ...
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Representing bound variables with a function from uses to binders

The problem of representing bound variables in syntax, and in particular that of capture-avoiding substitution, is well-known and has a number of solutions: named variables with alpha-equivalence, de ...
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1answer
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Why isn't it “enough” to prove induction with one extra “INat” argument?

It is well known that it is impossible to prove the induction principle for Natural numbers on the Calculus of Constructions. That is, ...
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1answer
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Is a CEK machine an implementation of a CESK machine?

We know that a CESK machine can be defined as: a state-machine in which each state has four components: a (C)ontrol component, an (E)nvironment, a (S)tore and a (K)ontinuation. One might imagine ...
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If the untyped language is terminating, can we still derive a contradiction from `Type : Type`?

Question If a pure type system has a terminating proof language, can we have Type : Type at the logic level without causing paradoxes (i.e., without causing ...
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Is it possible to create a “quote” function that, given a native λ-term, returns its λ-encoded representation?

Suppose we implement the λ-calculus inside the λ-calculus itself with λ-encodings and Bruijn indices: ...
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1answer
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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 ...

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