46 research outputs found
Computing Persistent Homology within Coq/SSReflect
Persistent homology is one of the most active branches of Computational
Algebraic Topology with applications in several contexts such as optical
character recognition or analysis of point cloud data. In this paper, we report
on the formal development of certified programs to compute persistent Betti
numbers, an instrumental tool of persistent homology, using the Coq proof
assistant together with the SSReflect extension. To this aim it has been
necessary to formalize the underlying mathematical theory of these algorithms.
This is another example showing that interactive theorem provers have reached a
point where they are mature enough to tackle the formalization of nontrivial
mathematical theories
Pure Type System conversion is always typable
International audiencePure Type Systems are usually described in two different ways, one that uses an external notion of computation like beta-reduction, and one that relies on a typed judgment of equality, directly in the typing system. For a long time, the question was open to know whether both presentations described the same theory. A first step toward this equivalence has been made by Adams for a particular class of \emph{Pure Type Systems} (PTS) called functional. Then, his result has been relaxed to all semi-full PTS in previous work. In this paper, we finally give a positive answer to the general issue, and prove that equivalence holds for any Pure Type System.Les Systèmes de Types Purs (PTS) sont habituellement présentés de deux manières différentes, une qui utilise une notion de calcul indépendante du typage, comme la béta-reduction, et une qui défini un jugement d'égalité typée au sein du système de types. La question de savoir si ces deux présentations représentaient la même théorie est restée ouverte pendant de nombreuses années. Une première réponse partielle à cette question a été apportée par Adams pour une classe particulière de PTS dit "fonctionnels". Nous avons récement étendu ce résultat à tous les PTS "semi-complets" . Dans cet article, nous pouvons finalement donner une réponse positive à la question dans toute sa généralité: l'équivalence entre les deux présentations est prouvée correcte pour n'importe quel Système de Types Purs
Formalized linear algebra over Elementary Divisor Rings in Coq
This paper presents a Coq formalization of linear algebra over elementary
divisor rings, that is, rings where every matrix is equivalent to a matrix in
Smith normal form. The main results are the formalization that these rings
support essential operations of linear algebra, the classification theorem of
finitely presented modules over such rings and the uniqueness of the Smith
normal form up to multiplication by units. We present formally verified
algorithms computing this normal form on a variety of coefficient structures
including Euclidean domains and constructive principal ideal domains. We also
study different ways to extend B\'ezout domains in order to be able to compute
the Smith normal form of matrices. The extensions we consider are: adequacy
(i.e. the existence of a gdco operation), Krull dimension and
well-founded strict divisibility
A refinement-based approach to computational algebra in COQ
International audienceWe describe a step-by-step approach to the implementation and formal verification of efficient algebraic algorithms. Formal specifications are expressed on rich data types which are suitable for deriving essential theoretical properties. These specifications are then refined to concrete implementations on more efficient data structures and linked to their abstract counterparts. We illustrate this methodology on key applications: matrix rank computation, Winograd's fast matrix product, Karatsuba's polynomial multiplication, and the gcd of multivariate polynomials
Towards a certified computation of homology groups for digital images
International audienceIn this paper we report on a project to obtain a verified computation of homology groups of digital images. The methodology is based on program- ming and executing inside the COQ proof assistant. Though more research is needed to integrate and make efficient more processing tools, we present some examples partially computed in COQ from real biomedical images
Search for Eccentric Black Hole Coalescences during the Third Observing Run of LIGO and Virgo
Despite the growing number of confident binary black hole coalescences
observed through gravitational waves so far, the astrophysical origin of these
binaries remains uncertain. Orbital eccentricity is one of the clearest tracers
of binary formation channels. Identifying binary eccentricity, however, remains
challenging due to the limited availability of gravitational waveforms that
include effects of eccentricity. Here, we present observational results for a
waveform-independent search sensitive to eccentric black hole coalescences,
covering the third observing run (O3) of the LIGO and Virgo detectors. We
identified no new high-significance candidates beyond those that were already
identified with searches focusing on quasi-circular binaries. We determine the
sensitivity of our search to high-mass (total mass ) binaries
covering eccentricities up to 0.3 at 15 Hz orbital frequency, and use this to
compare model predictions to search results. Assuming all detections are indeed
quasi-circular, for our fiducial population model, we place an upper limit for
the merger rate density of high-mass binaries with eccentricities at Gpc yr at 90\% confidence level.Comment: 24 pages, 5 figure
Open data from the third observing run of LIGO, Virgo, KAGRA and GEO
The global network of gravitational-wave observatories now includes five
detectors, namely LIGO Hanford, LIGO Livingston, Virgo, KAGRA, and GEO 600.
These detectors collected data during their third observing run, O3, composed
of three phases: O3a starting in April of 2019 and lasting six months, O3b
starting in November of 2019 and lasting five months, and O3GK starting in
April of 2020 and lasting 2 weeks. In this paper we describe these data and
various other science products that can be freely accessed through the
Gravitational Wave Open Science Center at https://gwosc.org. The main dataset,
consisting of the gravitational-wave strain time series that contains the
astrophysical signals, is released together with supporting data useful for
their analysis and documentation, tutorials, as well as analysis software
packages.Comment: 27 pages, 3 figure
Open data from the third observing run of LIGO, Virgo, KAGRA, and GEO
The global network of gravitational-wave observatories now includes five detectors, namely LIGO Hanford, LIGO Livingston, Virgo, KAGRA, and GEO 600. These detectors collected data during their third observing run, O3, composed of three phases: O3a starting in 2019 April and lasting six months, O3b starting in 2019 November and lasting five months, and O3GK starting in 2020 April and lasting two weeks. In this paper we describe these data and various other science products that can be freely accessed through the Gravitational Wave Open Science Center at https://gwosc.org. The main data set, consisting of the gravitational-wave strain time series that contains the astrophysical signals, is released together with supporting data useful for their analysis and documentation, tutorials, as well as analysis software packages
Etude sur le typage de l'égalité dans les systèmes de types
This dissertation is about the investigation of the concept of conversion that lies within any kind of dependently typed systems. Several presentations of those systems have been studied for various purposes: typing, proof search, logical coherence... Each presentation embeds its own notion of equality, tuned for its own goal. However, the question to know whereas all those representations in fact reflecting the same theory is not yet proved. We will focus here on a quite large family of systems called Pure Type Systems, and we will prove that all of their usual presentations are equivalent, by exhibiting a constructive translation from a representation to the other. All of those translations are based on the behaviour of the equality and the ways to translate them. Therefore, this work will focus on the properties of the equality, and we will prove that it is possible to type any syntactic equality into a semantic one, which is enough to translate any system into another one. Moreover, all of this thesis has been verified and proved correct with the Coq proof assistant, which has also been used during the development of the proofs.Le travail présenté dans cette thèse concerne l'étude de la notion de conversion inhérente à tous système de types dépendants. Plusieurs présentations de ces systèmes ont été étudiées pour des usages variés: typage, recherche de preuve, cohérence de logique... Chacune de ces représentation est accompagnée d'une notion d'égalité différente, suivant les besoins du moment. Mais il n'est pas certains que toutes ces représentations parlent en fin de compte d'une seule et même logique. Nous nous intéressons ici à une famille assez conséquente de systèmes de types, appelés Systèmes de Types Purs, et nous allons prouver que pour ces systèmes, toutes les représentations habituellement utilisées sont en fait équivalentes, c'est à dire qu'il existe des traductions constructives entre chacune de ces présentations. Ces traductions reposent toutes sur la manière de porter une égalité d'un système à l'autre. Ce travail se concentre donc sur les mécanismes de ces égalités, et prouve qu'il est possible de typer n'importe quelle égalité syntaxique en égalité sémantique, et ainsi qu'il est donc possible de passer d'un système à l'autre. L'intégralité de cette thèse a en outre été vérifiée et certifiée correcte à l'aide de l'assistant à la preuve Coq, qui a activement été utilisé tout au long de l'élaboration des preuves.