Stabilizing isomorphisms from ℓp(ℓ2)\ell_p(\ell_2) into Lp[0,1]L_p[0,1]

Let $10$ and let $T:\ell_p(\ell_2)\overset{into}{\rightarrow}L_p[0,1]$ be an isomorphism. Then there is a subspace $Y\subset \ell_p(\ell_2)$ $(1+\epsilon)$-isomorphic to $\ell_p(\ell_2)$ such that: $T_{|Y}$ is an $(1+\epsilon)$-isomorphism and $T(Y)$ is $K_p$-complemented in $L_p[0,1]$, with $K_p$ depending only on $p$. Moreover, $K_p\le (1+\epsilon)\gamma_p$ if $p>2$ and $K_p\le (1+\epsilon)\gamma_{p/(p-1)}$ if $1<p<2$, where $\gamma_r$ is the $L_r$ norm of a standard Gaussian variable

Readable semi-automatic formal proofs of Depth-First Search in graphs using Why3

Quite often formal proofs are not published in conferences or journal articles, because formal proofs are usually too long. A typical article states the ability of having implemented a formal proof, but the proof itself is often sketched in terms of a natural language. At best, some formal lemmas and definitions are stated. Can we do better ? We try here to publish the details of a formal proof of the white-paths theorem about depth-first search in graphs. We use Why3 as the proving platform, because Why3 uses first-order logic augmented with inductive definitions of predicates and because Why3 makes possible to delegate bits of proofs to on-the-shelf automatic provers at same time as Why3 provides interfaces with interactive proof checkers such that Coq, PVS or Isabelle. Algorithms on graphs are also a good testbed since graphs are combinatorial structures whose algebraic properties are not fully obvious. Depth-first search may look over-simple, but it is the first step of the construction of a library of readable formal proofs for more complex algorithms on graphs with more realistic data structures

Quantum Kagome antiferromagnet ZnCu3(OH)6Cl2

The frustration of antiferromagnetic interactions on the loosely connected kagome lattice associated to the enhancement of quantum fluctuations for S=1/2 spins was acknowledged long ago as a keypoint to stabilize novel ground states of magnetic matter. Only very recently, the model compound Herbersmithite, ZnCu3(OH)6Cl2, a structurally perfect kagome antiferromagnet, could be synthesized and enables a close comparison to theories. We review and classify various experimental results obtained over the past years and underline some of the pending issues.Comment: 23 pages, 16 figures, invited paper in J. Phys. Soc. Jpn, special topics issue on "Novel States of Matter Induced by Frustration", to be published in Jan. 201

Anisotropic Impurity-States, Quasiparticle Scattering and Nematic Transport in Underdoped Ca(Fe1-xCox)2As2

Iron-based high temperature superconductivity develops when the `parent' antiferromagnetic/orthorhombic phase is suppressed, typically by introduction of dopant atoms. But their impact on atomic-scale electronic structure, while in theory quite complex, is unknown experimentally. What is known is that a strong transport anisotropy with its resistivity maximum along the crystal b-axis, develops with increasing concentration of dopant atoms; this `nematicity' vanishes when the `parent' phase disappears near the maximum superconducting Tc. The interplay between the electronic structure surrounding each dopant atom, quasiparticle scattering therefrom, and the transport nematicity has therefore become a pivotal focus of research into these materials. Here, by directly visualizing the atomic-scale electronic structure, we show that substituting Co for Fe atoms in underdoped Ca(Fe1-xCox)2As2 generates a dense population of identical anisotropic impurity states. Each is ~8 Fe-Fe unit cells in length, and all are distributed randomly but aligned with the antiferromagnetic a-axis. By imaging their surrounding interference patterns, we further demonstrate that these impurity states scatter quasiparticles in a highly anisotropic manner, with the maximum scattering rate concentrated along the b-axis. These data provide direct support for the recent proposals that it is primarily anisotropic scattering by dopant-induced impurity states that generates the transport nematicity; they also yield simple explanations for the enhancement of the nematicity proportional to the dopant density and for the occurrence of the highest resistivity along the b-axis

Multi-Review Fusion-in-Context

Grounded text generation, encompassing tasks such as long-form question-answering and summarization, necessitates both content selection and content consolidation. Current end-to-end methods are difficult to control and interpret due to their opaqueness. Accordingly, recent works have proposed a modular approach, with separate components for each step. Specifically, we focus on the second subtask, of generating coherent text given pre-selected content in a multi-document setting. Concretely, we formalize Fusion-in-Context (FiC) as a standalone task, whose input consists of source texts with highlighted spans of targeted content. A model then needs to generate a coherent passage that includes all and only the target information. Our work includes the development of a curated dataset of 1000 instances in the reviews domain, alongside a novel evaluation framework for assessing the faithfulness and coverage of highlights, which strongly correlate to human judgment. Several baseline models exhibit promising outcomes and provide insightful analyses. This study lays the groundwork for further exploration of modular text generation in the multi-document setting, offering potential improvements in the quality and reliability of generated content. Our benchmark, FuseReviews, including the dataset, evaluation framework, and designated leaderboard, can be found at https://fusereviews.github.io/.Comment: NAACL 2024, finding

Debating Technology for Dialogical Argument:Sensemaking, Engagement and Analytics

Debating technologies, a newly emerging strand of research into computational technologies to support human debating, offer a powerful way of providing naturalistic, dialogue-based interaction with complex information spaces. The full potential of debating technologies for dialogical argument can, however, only be realized once key technical and engineering challenges are overcome, namely data structure, data availability, and interoperability between components. Our aim in this article is to show that the Argument Web, a vision for integrated, reusable, semantically rich resources connecting views, opinions, arguments, and debates online, offers a solution to these challenges. Through the use of a running example taken from the domain of citizen dialogue, we demonstrate for the first time that different Argument Web components focusing on sensemaking, engagement, and analytics can work in concert as a suite of debating technologies for rich, complex, dialogical argument

Formal Proofs of Tarjan\u27s Strongly Connected Components Algorithm in Why3, Coq and Isabelle

Comparing provers on a formalization of the same problem is always a valuable exercise. In this paper, we present the formal proof of correctness of a non-trivial algorithm from graph theory that was carried out in three proof assistants: Why3, Coq, and Isabelle

Formal proofs of two algorithms for strongly connected components in graphs

We present formal proofs for the two classical Tarjan-1972 and Kosaraju-1978 algorithms for finding strongly connected components in directed graphs. We describe the two algorithms in a functional programming style with abstract values for vertices in graphs, with functions between vertices and their successors, and with data types such that lists (for representing immutable stacks) and sets. We use the Why3 system and the Why3-logic to express these proofs and fully check them by computer. The Why3-logic is a simple multi-sorted first-order logic augmented by inductively defined predicates. Furthermore it provides useful libraries for lists and sets. The Why3 system allows description of programs in a Why3-ML programming language (a first-order programming language with ML syntax) and provides interfaces to various state-of-the-art automatic provers and to manual interactive proof-checkers (we use mainly Coq). One important point of our article is that our proofs are intuitive and human readable