Tootfinder

Homogeneous Linear Calculus of Order 1 and a $\lambda$-Taylor Formula
Ronald Orozco L\'opez
https://arxiv.org/abs/2506.22976 https://

Homogeneous Linear Calculus of Order 1 and a $λ$-Taylor Formula
In this paper, a new calculus on sequences is defined. Also, the $λ$-derivative and the $λ$-integration are investigated. The fundamental theorem of $λ$-calculus is included. A suitable function basis for the $λ$-derivative and the $λ$-integral is provided, and various properties of this basis are given. A $λ$-Taylor formula for functions is given.

Positive Sharing and Abstract Machines
Beniamino Accattoli, Claudio Sacerdoti Coen, Jui-Hsuan Wu
https://arxiv.org/abs/2506.14131 https://

Positive Sharing and Abstract Machines
Wu's positive $λ$-calculus is a recent call-by-value $λ$-calculus with sharing coming from Miller and Wu's study of the proof-theoretical concept of focalization. Accattoli and Wu showed that it simplifies a technical aspect of the study of sharing; namely it rules out the recurrent issue of renaming chains, that often causes a quadratic time slowdown. In this paper, we define the natural abstract machine for the positive $λ$-calculus and show that it suffers from an inefficiency: the quad…

Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains
Nick Lindemulder, Emiel Lorist, Floris Roodenburg, Mark Veraar
https://arxiv.org/abs/2507.13478

Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains
We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded $H^{\infty}$-functional calculus on weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Our analysis applies to bounded $C^{1,λ}$-domains with $λ\in[0,1]$, revealing a crucial trade-off: lower domain regularity can be compensated by enlarging the weight exponent. As a primary consequence, we establish maximal regularity f…

Closure Conversion, Flat Environments, and the Complexity of Abstract Machines
Beniamino Accattoli (Inria & LIX, \'Ecole Polytechnique), Dan Ghica (Huawei Central Software Institute, University of Birmingham), Giulio Guerrieri (University of Sussex), Cl\'audio Belo Louren\c{c}o (Huawei Central Software Institute), Claudio Sacerdoti Coen (Universit\`a di Bologna)

Closure Conversion, Flat Environments, and the Complexity of Abstract Machines
Closure conversion is a program transformation at work in compilers for functional languages to turn inner functions into global ones, by building closures pairing the transformed functions with the environment of their free variables. Abstract machines rely on similar and yet different concepts of closures and environments. In this paper, we study the relationship between the two approaches. We adopt a very simple λ-calculus with tuples as source language and study abstract machines for bot…

Term Assignment and Categorical Models for Intuitionistic Linear Logic with Subexponentials
Daniel Rogozin
https://arxiv.org/abs/2507.12360 https://arxiv.o…

Term Assignment and Categorical Models for Intuitionistic Linear Logic with Subexponentials
In this paper, we present a typed lambda calculus ${\bf SILL}(λ)_Σ$, a type-theoretic version of intuitionistic linear logic with subexponentials, that is, we have many resource comonadic modalities with some interconnections between them given by a subexponential signature. We also give proof normalisation rules and prove the strong normalisation and Church-Rosser properties for $β$-reduction by adapting the Tait-Girard method to subexponential modalities. Further, we analyse subexponential…

A simple formalization of alpha-equivalence
Kalmer Apinis, Danel Ahman
https://arxiv.org/abs/2507.10181 https://arxiv.org/pdf/2507.10…

A simple formalization of alpha-equivalence
While teaching untyped $λ$-calculus to undergraduate students, we were wondering why $α$-equivalence is not directly inductively defined. In this paper, we demonstrate that this is indeed feasible. Specifically, we provide a grounded, inductive definition for $α$-equivalence and show that it conforms to the specification provided in the literature. The work presented in this paper is fully formalized in the Rocq Prover.

From Partial to Monadic: Combinatory Algebra with Effects
Liron Cohen (BGU), Ariel Grunfeld (BGU), Dominik Kirst (PICUBE), \'Etienne Miquey (I2M)
https://arxiv.org/abs/2506.09453

From Partial to Monadic: Combinatory Algebra with Effects
Partial Combinatory Algebras (PCAs) provide a foundational model of the untyped $λ$-calculus and serve as the basis for many notions of computability, such as realizability theory. However, PCAs support a very limited notion of computation by only incorporating non-termination as a computational effect. To provide a framework that better internalizes a wide range of computational effects, this paper puts forward the notion of Monadic Combinatory Algebras (MCAs). MCAs generalize the notion of P…