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.
29 questions with no upvoted or accepted answers
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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 ...
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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 ...
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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/...
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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 = \...
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Is there a notion analogous to separability but requiring definable rather than countable sets?
Among models of $\lambda$-calculus, some like the Bohm tree model have the property that every element is a directed sup of definable elements, whereas others like the $D_\infty$ and $P(\omega)$ ...
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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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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 ...
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$\omega$ 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 $N$ ...
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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 ...
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Difference between lambda-calculus with well-formed formulas vs properly-formed formulas
In S.C. Kleene's 1935 paper "$\lambda$-definability and recursiveness," he proves that all $\lambda$-definable functions are general recursive in the Herbrand-Godel sense and vice-versa. However, the ...
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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)(\...
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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 ...
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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 ...
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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 ...
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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 $A$ ...
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Is it possible to implement η-reduction in interaction nets?
There are several ways to encode λ-terms in interaction nets; for instance, using the original optimal algorithm by Lamping, or compiling λ-calculus into interaction combinators. However, all the ...
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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. ...
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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 \...
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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 ...
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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 ...
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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 ...
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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 $\lambda$...
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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 ...
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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 ...
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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 ...
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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 ...
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Schönhage's SMM with only one instruction
It is possible to implement $\lambda$-calculus in Schönhage's storage modification machine using an infinite set of nodes and one single program consisted exclusively of (about hundred) instructions ...
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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 ...
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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 ...
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