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

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 $\leq 1$ and well-founded strict divisibility

Cubical Synthetic Homotopy Theory

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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

Refinements for Free!

International audienceFormal verification of algorithms often requires a choice be-tween definitions that are easy to reason about and definitions that are computationally efficient. One way to reconcile both consists in adopt-ing a high-level view when proving correctness and then refining stepwise down to an efficient low-level implementation. Some refinement steps are interesting, in the sense that they improve the algorithms involved, while others only express a switch from data representations geared towards proofs to more efficient ones geared towards computations. We relieve the user of these tedious refinements by introducing a framework where correctness is established in a proof-oriented context and automatically transported to computation-oriented data structures. Our design is gen-eral enough to encompass a variety of mathematical objects, such as rational numbers, polynomials and matrices over refinable structures. Moreover, the rich formalism of the Coq proof assistant enables us to develop this within Coq, without having to maintain an external tool

Cubical Type Theory: a constructive interpretation of the univalence axiom

International audienceThis paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory

A Spatially Explicit Decision Support System for Assessment of Tree Stump Harvest Using Biodiversity and Economic Criteria

Stump harvesting is predicted to increase with future increasing demands for renewable energy. This may affect deadwood affiliate biodiversity negatively, given that stumps constitute a large proportion of the coarse deadwood in young managed forests. Spatial decision support for evaluating the integrated effects on biodiversity and production of stump harvesting is needed. We developed a spatially explicit decision support system (called MapStump-DSS), for assessment of tree stump harvesting using biodiversity and economic criteria together with different scenarios for biodiversity conservation and bioenergy market prices. Two novel key aspects of the MAPStump-DSS is that it (1) merges and utilizes georeferenced stump-level data (e.g., tree species and diameter) directly from the harvester with stand data that are increasingly available to forest managers and (2) is flexible toward incorporating both quantitative and qualitative criteria based on emerging knowledge (here biodiversity criteria) or underlying societal drivers and end-user preferences. We tested the MAPStump-DSS on a 45 ha study forest, utilizing harvester data on characteristics and geographical positions for >26,000 stumps. The MAPStump-DSS produced relevant spatially explicit information on the biodiversity and economic values of individual stumps, where amounts of "conflict stumps" (with both high biodiversity and economical value) increased with bioenergy price levels and strengthened biodiversity conservation measures. The MAPStump-DSS can be applied in practice for any forest site, allowing the user to examine the spatial distribution of stumps and to obtain summaries for whole forest stands. Information depicted by the MAPStump-DSS includes amounts, characteristics, biodiversity values and costs of stumps in relation to different scenarios, which also allow the user to explore and optimize biodiversity and economy trade-offs prior to stump harvest

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