Formalizing Size-Optimal Sorting Networks: Extracting a Certified Proof Checker

Since the proof of the four color theorem in 1976, computer-generated proofs have become a reality in mathematics and computer science. During the last decade, we have seen formal proofs using verified proof assistants being used to verify the validity of such proofs. In this paper, we describe a formalized theory of size-optimal sorting networks. From this formalization we extract a certified checker that successfully verifies computer-generated proofs of optimality on up to 8 inputs. The checker relies on an untrusted oracle to shortcut the search for witnesses on more than 1.6 million NP-complete subproblems.Comment: IMADA-preprint-c

Optimizing a Certified Proof Checker for a Large-Scale Computer-Generated Proof

In recent work, we formalized the theory of optimal-size sorting networks with the goal of extracting a verified checker for the large-scale computer-generated proof that 25 comparisons are optimal when sorting 9 inputs, which required more than a decade of CPU time and produced 27 GB of proof witnesses. The checker uses an untrusted oracle based on these witnesses and is able to verify the smaller case of 8 inputs within a couple of days, but it did not scale to the full proof for 9 inputs. In this paper, we describe several non-trivial optimizations of the algorithm in the checker, obtained by appropriately changing the formalization and capitalizing on the symbiosis with an adequate implementation of the oracle. We provide experimental evidence of orders of magnitude improvements to both runtime and memory footprint for 8 inputs, and actually manage to check the full proof for 9 inputs.Comment: IMADA-preprint-c

A certifying frontend for (sub)polyhedral abstract domains

Convex polyhedra provide a relational abstraction of numerical properties for static analysis of programs by abstract interpretation. We describe a lightweight certification of polyhedral abstract domains using the Coq proof assistant. Our approach consists in delegating most computations to an untrusted backend and in checking its outputs with a certified frontend. The backend is free to implement relaxations of domain operators in order to trade some precision for more efficiency, but must produce hints about the soundness of its results. Experiments with a full-precision backend show that the certification overhead is small and that the certified abstract domain has comparable performance to non-certifying state-of-the-art implementations

Vibration Modes of the Cello Tailpiece

The application of modern scientific methods and measuring techniques can ex- tend the empirical knowledge used for centuries by violinmakers for making and adjusting the sound of violins, violas, and cellos. Accessories such as strings and tailpieces have been studied recently with respect to style and historical coherence, after having been somehow neglected by researchers in the past. These fittings have played an important part in the history of these instruments, but have largely disappeared as they have been modernised. However, the mechanics of these accessories contribute significantly to sound production in ways that have changed over time with different musical aesthetics and in different technical contexts. There is a need to further elucidate the function and musical contribution of strings and tailpieces. With this research we are trying to understand the modifications of the cello’s sound as a consequence of tailpiece characteristics (shape of the tailpiece and types of attachments). Modal analysis was used to first investigate the vibration modes of the tailpiece when mounted on a non-reactive rig and then when mounted on a real cello where it can interact with the modes of the instrument’s corpus. A preliminary study of the effect of the tailpiece cord length will be presented

Scalable Minimizing-Operators on Polyhedra via Parametric Linear Programming

International audienceConvex polyhedra capture linear relations between variables. They are used in static analysis and optimizing compilation. Their high expressiveness is however barely used in verification because of their cost, often prohibitive as the number of variables involved increases. Our goal in this article is to lower this cost. Whatever the chosen representation of polyhedra – as constraints, as generators or as both – expensive operations are unavoidable. That cost is mostly due to four operations: conversion between representations, based on Chernikova’s algorithm, for libraries in double description; convex hull, projection and minimization, in the constraints-only representation of polyhedra. Libraries operating over generators incur exponential costs on cases common in program analysis. In the Verimag Polyhedra Library this cost was avoided by a constraints-only representation and reducingall operations to variable projection, classically done by Fourier-Motzkin elimination. Since Fourier-Motzkin generates many redundant constraints, minimization was however very expensive. In this article, we avoid this pitfall by expressing projection as a parametric linear programming problem. This dramatically improves efficiency, mainly because it avoids the post-processing minimization. We show how our new approach can be up to orders of magnitude faster than the previous approach implemented in the Verimag Polyhedra Library that uses only constraints and Fourier-Motzkin elimination, and on par with the conventional double description approach, as implemented in well-known libraries

A Formalization of Convex Polyhedra Based on the Simplex Method

International audienc

Formalizing the Face Lattice of Polyhedra

23 pages, 4 figures, minor revision. Extended version of hal-0315165.Faces play a central role in the combinatorial and computational aspects of polyhedra. In this paper, we present the first formalization of faces of polyhedra in the proof assistant Coq. This builds on the formalization of a library providing the basic constructions and operations over polyhedra, including projections, convex hulls and images under linear maps. Moreover, we design a special mechanism which automatically introduces an appropriate representation of a polyhedron or a face, depending on the context of the proof. We demonstrate the usability of this approach by establishing some of the most important combinatorial properties of faces, namely that they constitute a family of graded atomistic and coatomistic lattices closed under interval sublattices. We also prove a theorem due to Balinski on the d-connectedness of the adjacency graph of polytopes of dimension d