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Questions tagged [lambda-calculus]

For questions on the formal system in mathematical logic for expressing effective functions, programs and computation, and proofs, using abstract notions of functions and combining them through binding and substitution.

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Does substitution on named terms correspond to substitution on de Bruijn terms?

Altenkirch wrote (in the unpublished draft α-conversion is easy): I leave it to the reader to show that (some natural translation function) preserves substitution, i.e. it maps substitutions on named ...
Pavel Shuhray's user avatar
1 vote
1 answer
87 views

Selection terms in the untyped lambda calculus

In the untyped $\lambda$-calculus, are there terms $S$ and $T$ such that for any $n$ and any terms $t_1, \dotsc, t_n$, $$S(T(t_1)\dots(t_n)) \twoheadrightarrow_{\beta} t_1$$ Of course, if $n$ is fixed ...
provocateur's user avatar
0 votes
0 answers
23 views

For which categories can the Cartesian closedness of a category be described as a family of covariant functors?

The paper Functorial Polymorphism describes how parametric polymorphism can be described as dinatural transformations. It involves second order lambda calculus but my question is restricted to simply ...
Johan Thiborg-Ericson's user avatar
6 votes
1 answer
183 views

Consistency in pure type systems

Summary My question is about how (i) a certain presentation of pure type systems in the $\lambda$-cube, bears on (ii) a standard definition of consistency in pure type systems. In short, I'm ...
Spaceka13's user avatar
3 votes
1 answer
118 views

Where can I learn about Cartesian closed functors between categories of simply typed lambda calculus?

I'll try to describe the subject I am looking for literature on, or concept names that I can Google. For each $n \geq 1$, let $\mathbf{STLC}_n$ be the category where the objects are all simply typed ...
Johan Thiborg-Ericson's user avatar
1 vote
0 answers
98 views

Can application in untyped lambda calculus be seen as the uncurried unit of some monad?

Simply typed lambda calculus in one type variable in a Cartesian closed category $\mathbf{C}$ can be interpreted as a family of Cartesian closed functors (described below, do they have a name?) from ...
Johan Thiborg-Ericson's user avatar
1 vote
0 answers
66 views

Does lambda polymorphism have some universal property?

To evaluate some typed lambda calculus applications, the type of the function might have to be "lifted" in order to match the type of the value it is applied to. For example, in the ...
Johan Thiborg-Ericson's user avatar
7 votes
1 answer
156 views

Is lambda calculus polymorphism a type of generalized monad?

Let $\mathbf{C}$ be a Cartesian closed category. Then simply typed lambda calculus in $\mathbf{C}$ in one type variable can be interpreted as a category $\mathbf{STLC}_{\mathbf C}$ where the objects ...
Johan Thiborg-Ericson's user avatar
12 votes
2 answers
833 views

An overview of mathematical-logical approaches in formalizing natural languages

Crossposted on Mathematics SE I am an undergraduate mathematics student with a keen interest in pursuing research in the formalization of natural languages (from a more mathematical-logical approach),...
Heleyrine Brookvinth's user avatar
1 vote
0 answers
65 views

Second order lambda calculus as dinatural transformations in some category of CCCs

Let $\textbf{CART}$ be a category where the objects are all Cartesian closed categories (henceforth shortened as CCC). Is there any way to define the arrows so that $\textbf{CART}$ itself becomes ...
Johan Thiborg-Ericson's user avatar
2 votes
1 answer
159 views

Semiring axioms which almost implement inverse, searching for domains other than lambda calculus

I'm working with an idempotent semiring which contains elements $C_i, \hat{C_i}$ with the following properties: $$ {C}_i \hat{C_j} = 0 \quad\text{where}\quad i \neq j \quad\quad\quad\quad(\beta_0)$$ $$...
Łukasz Lew's user avatar
1 vote
0 answers
56 views

The set of closed untyped $\lambda$-terms is not context-free?

The set of untyped $\lambda$-terms is obviously context-free. But, according to Barendregt's paper Discriminating coded lambda terms (six lines before Theorem 1.5), the set of closed untyped $\lambda$...
Paul Blain Levy's user avatar
0 votes
0 answers
84 views

Univalence and higher inductive types in the lambda calculus model of type theory

In appendix A1 of the homotopy type theory book by the Univalent Foundations Project, the authors give a formal presentation of Martin-Löf type theory in lambda calculus. However, they did not give ...
Madeleine Birchfield's user avatar
3 votes
0 answers
102 views

When does weak normalization imply strong normalization?

Is there a possibility to get strong normalization for some kind of $\lambda$-calculus out of weak normalization with some other assumptions? For example: The term $(\lambda_y z)((\lambda_x xx)(\...
Zermelo-Fraenkel's user avatar
12 votes
2 answers
980 views

Is there a proof of strong normalisation that uses ordinal numbers?

I am currently trying to find a proof for strong normalisation of an extension of $\lambda$-calculus. I've tried several approaches and one would be to assign an ordinal number $\operatorname{cs}(t)$ ...
Zermelo-Fraenkel's user avatar
4 votes
0 answers
90 views

$\omega$ incompleteness of $\lambda$ calculus

In Plotkin's 'The $\lambda$-Calculus is $\omega$-Incomplete' (The Journal of Symbolic Logic Vol. 39, No. 2 (Jun., 1974), pp. 313-317), an example is given of two (untyped) $\lambda$-terms $M$ and $N$ ...
provocateur's user avatar
19 votes
2 answers
1k views

Do combinatory logic bases need a function of 3 variables?

All the known bases of combinatory logic, such as $\{S,K\}$, or $\{K,W,B,C\}$, have one or more combinators using 3 variables: \begin{align*} S ={} & \lambda x\lambda y\lambda z. x z(y z), \\ B ={}...
John Tromp's user avatar
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5 votes
1 answer
241 views

Internal language proof of Lawvere's fixed point theorem for cartesian closed categories

This proof of Lawvere's fixed point theorem suggests (since it uses $\lambda$ notation) that it is written in the internal language of cartesian closed categories (which is the $\lambda$-calculus, as ...
user1005113's user avatar
1 vote
0 answers
94 views

Is Set complete for the free CCC/lambda calculus over a monoidal signature?

To be precise, given a monoidal signature $S$ (i.e, a set of generating objects $O$ and morphisms with source and target taken in the free monoid over $O$) , we can generate the free Cartesian closed ...
FeralX's user avatar
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3 votes
0 answers
193 views

What is the connection between these proofs of strong normalization in $\lambda^\to$ and LK?

In Ralph Loader's lecture notes on lambda calculus (section 3.3), he states that a combinatorial proof of the SN of simply typed lambda calculus uses a technique that is "in essence that used by ...
Trebor's user avatar
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7 votes
1 answer
338 views

Criterion for the consistency of pure type systems

Pure type systems are characterized in an almost combinatorial way: a set of axioms $\star_i : \star_j$, and a set of triples $(\star_i, \star_j, \star_k)$ saying when the dependent product $\prod_{x :...
Trebor's user avatar
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1 vote
0 answers
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What is the proof theoretic strength of PCF?

Godel's system $T$ means different, although equivalent, things to different people. To people working in the traditon of mathematical logic, $T$ is a quantifier-free equational theory of arithmetic ...
Not_Here's user avatar
  • 482
7 votes
0 answers
175 views

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 ...
alessio-b-zak's user avatar
0 votes
0 answers
110 views

Expressing a model transformation by using monads in the simply-typed lambda calculus

In https://link.springer.com/content/pdf/10.1007/s10670-019-00128-z.pdf , page 16, the following clause is given for a modal operator $\langle R_k \rangle$ (see definition 4.2 for the definition of a ...
user65526's user avatar
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9 votes
0 answers
309 views

The geometry of lambda calculus?

I stumbled upon "the geometry of quantum computation" --- to quote the abstract: Determining the quantum circuit complexity of a unitary operation is closely related to the problem of finding ...
Siddharth Bhat's user avatar
23 votes
0 answers
3k views

What's the smallest $\lambda$-calculus term not known to have a normal form?

For Turing Machines, the question of halting behavior of small TMs has been well studied in the context of the Busy Beaver function, which maps n to the longest output or running time of any halting n ...
John Tromp's user avatar
  • 1,699
3 votes
0 answers
132 views

When can all elements of $[A\to B]$ can be represented as computable functions?

(crosspost from math stackexchange) While working through Barendreght's book on the Lambda Calculus, and Abramsky's notes on Domain Theory, I had the following realization: It's often stated that ...
Alex Appel's user avatar
4 votes
0 answers
104 views

Reflexive object and infinite products

The category CPO of cpos and continuous functions has a reflexive object, i.e. an object $A$ such that $A\times A\simeq A$ and $A\simeq A^A$. Since CPO has countable products, my question is whether ...
Steve K.'s user avatar
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1 vote
0 answers
66 views

Posets with two partial (self-)distributive operations

Let $(X, {\sqsubset}, {\circ}, {\ast})$ be a set $X$ with a strict partial order $\sqsubset$ and two partial binary operations $\circ$ and $\ast$ such that for any $a, b, c \in X$: $a \circ b$ and $a ...
Anton Salikhmetov's user avatar
2 votes
0 answers
279 views

Lambda calculus as set-theoretic operations

It is possible to interpret typed lambda calculus a-la Church as logical operations (because of Curry-Howard correspondence). Also, there is a isomorphism between logical and set-theoretic operations. ...
семен антонов's user avatar
11 votes
2 answers
3k views

How can the simply typed lambda calculus be Turing-incomplete, yet stronger than second-order logic?

It is well-known that the simply typed lambda calculus is strongly normalizing (for instance, Wikipedia). Hence, it is not strong enough to be Turing-complete, as also mentioned on the Wikipedia page ...
Mike Battaglia's user avatar
3 votes
0 answers
254 views

Upward confluence in the interaction calculus

The lambda calculus is not upward confluent, counterexamples being known for a long time. Now, what about the interaction calculus? Specifically, I am looking for configurations $c_1$ and $c_2$ such ...
Anton Salikhmetov's user avatar
20 votes
1 answer
2k views

What is Chemlambda? In which ways could it be interesting for a mathematician?

I${}^{*}$ have randomly come across a couple of websites (Chemlambda project, chorasimilarity) that seem to be about a certain "thing" (a computer program, I think) called Chemlambda that does "stuff" ...
Qfwfq's user avatar
  • 22.8k
23 votes
1 answer
969 views

What, mathematically speaking, does it mean to say that the continuation monad can simulate all monads?

In various places it is stated that the continuation monad can simulate all monads in some sense (see for example http://lambda1.jimpryor.net/manipulating_trees_with_monads/)) In particular, in http://...
user65526's user avatar
  • 629
3 votes
0 answers
147 views

One strong fixed-point property

Each topological space $A$ with fixed-point property is connected (all clopen subsets are trivial). This is an analog of Rice theorem (all decidable subsets are trivial). Suppose, we have a space $A$ ...
George Cherevichenko's user avatar
9 votes
0 answers
521 views

The Curry Howard Isomorphism and models for an intuitionistic modal logic and its bimodal translation

My question regards the Curry Howard Isomorphism and how it constrains models in the case of a particular logic. Consider quantified Lax Logic $QLL$. https://pdfs.semanticscholar.org/468e/...
user65526's user avatar
  • 629
2 votes
0 answers
182 views

Notation in 'The lambda calculus, its syntax and semantics' by H.P. Barendregt

I'm reading the book 'The lambda calculus its syntax and semantics'. In part 5, chapter 19: Local structure of Models, more specifically 19.2 Local structure of $D_\infty$, the notation $D_\infty \...
tpsp_lcs's user avatar
8 votes
0 answers
151 views

Is every total computable function definable by a strongly total lambda term?

Every computable (total) function $f : \mathbb{N} \to \mathbb{N}$ is definable in untyped pure lambda calculus in the sense that there is a term $F$ such that, for every Church's numeral $c_n = \...
Valery Isaev's user avatar
  • 4,410
8 votes
1 answer
310 views

Is every total computable function definable by a normalizing lambda term?

$\newcommand{\nat}{\mathbb{N}}$ $\newcommand{\then}{\ \Longrightarrow\ }$ A partial function $f : \mathbb{N} \to \mathbb{N}$ is said to be $\lambda$-definable if there is a term $F \in \Lambda$ such ...
Andrew Polonsky's user avatar
8 votes
3 answers
990 views

How to handle sums in Tait's reducibility proof of strong normalisation?

I've been reading Girard et al's 'Proofs and Types', which in Chapter 6 presents a proof of strong normalisation for the simply typed lambda calculus with products and base types. The proof is based ...
RAC's user avatar
  • 83
2 votes
1 answer
313 views

Substructural types, the lambda calculus, and CCCs

It's well known that the simply-typed lambda calculus corresponds to a cartesian closed category. How would substructural type systems be characterized in category theory? For example, linear type ...
C. Bednarz's user avatar
9 votes
2 answers
2k views

Is simply typed lambda calculus with fixed-point combinator Turing-complete?

There are many sources cite that simply typed lambda calculus extended with fixed-point combinator is Turing complete. For example, Does there exist a Turing complete typed lambda calculus? or the ...
kittyphon's user avatar
  • 101
7 votes
3 answers
1k views

Relationship of lambda calculus to the rest of math

I just started reading "The calculi of lambda conversion" by Church. Church defines functions like: id x = x, and says the domain and range are understood to be as permissible as possible. ...
Polymer's user avatar
  • 181
5 votes
0 answers
226 views

Proper full submodels of full models of type theory

Let $N$ be the standard full model of the simply typed lambda calculus with infinite base type $o$ and let $X$ be an infinite and coinfinite subset of $N(o)$. I want to know if there's a full ...
Andrew Bacon's user avatar
3 votes
0 answers
261 views

Understanding Strong Normalization for Identity Types in Martin-Löf Intensional Type Theory [closed]

Roughly, the strong normalization property for Martin-Löf Intensional Type Theory (MITT) tells us that every closed term $t$ of type $M$ will eventually reach a canonical normal form $t’$ such that it ...
StudentType's user avatar
8 votes
1 answer
741 views

Easier Girard's paradox in a circular pure type system (PTS)

System U is an inconsistent PTS in that one has a term of type $\bot = \forall p\colon \ast \ldotp p$, and such a term is explicitly constructed in Hurkens' A Simplification of Girard's Paradox. One-...
H Koba's user avatar
  • 369
14 votes
1 answer
620 views

Why the reflection rule trivializes higher paths in Martin-Löf Extensional Type theory?

Martin-Löf Extensional Type theory differs from its intensional counterpart in that it contains the so-called reflection rule that says that if $p : x = y$, then actually $x \equiv y$ (i.e. $x$ and $y$...
StudentType's user avatar
3 votes
2 answers
322 views

Comparing really big numbers

Is there an intractability theorem that says that in any sufficiently rich system for defining really big numbers, there will be two numbers for which it's very, very, ... very difficult to decide ...
James Propp's user avatar
  • 19.5k
1 vote
0 answers
124 views

Optimal reduction using token-passing nets

I am looking for implementation of optimal reduction for λ-calculus based on interaction nets (McCarthy's amb allowed) in the spirit of "Token-Passing Nets: Call-by-Need for Free" by François-Régis ...
Anton Salikhmetov's user avatar
6 votes
1 answer
351 views

On an automatic translation of typed lambda calculus in untyped lambda calculus

I have a question regarding the "compilation" of typed lambda calculus in untyped lambda calculus. Take for example the inductive definition of lists, with introduction rules: and: We can ...
meditans's user avatar
  • 243