A Dependent Type Theory with Abstractable Names

This paper describes a version of Martin-Löf's dependent type theory extended with names and constructs for freshness and name-abstraction derived from the theory of nominal sets. We aim for a type theory for computing and proving (via a Curry-Howard correspondence) with syntactic structures which captures familiar, but informal, ‘nameful’ practices when dealing with binders.Partially supported by the UK EPSRC program grant EP/K008528/1, Rigorous Engineering for Mainstream Systems (REMS). Supported by the UK EPSRC leadership fellowship (Peter Sewell) grant EP/H005633/1, Semantic Foundations for Real-World Systems.This is the final published version of the article. It was originally published in Electronic Notes in Theoretical Computer Science (Pitts AM, Matthiesen J, Derikx J, Electronic Notes in Theoretical Computer Science 2015, 312, 19–50, doi:10.1016/j.entcs.2015.04.003) http://dx.doi.org/10.1016/j.entcs.2015.04.00

Non-producibility of arbitrary non-Gaussian states using zero-mean Gaussian states and partial photon number resolving detection

Gaussian states and measurements collectively are not powerful-enough resources for quantum computing, as any Gaussian dynamics can be simulated efficiently, classically. However, it is known that any one non-Gaussian resource -- either a state, a unitary operation, or a measurement -- together with Gaussian unitaries, makes for universal quantum resources. Photon number resolving (PNR) detection, a readily-realizable non-Gaussian measurement, has been a popular tool to try and engineer non-Gaussian states for universal quantum processing. In this paper, we consider PNR detection of a subset of the modes of a zero-mean pure multi-mode Gaussian state as a means to herald a target non-Gaussian state on the undetected modes. This is motivated from the ease of scalable preparation of Gaussian states that have zero mean, using squeezed vacuum and passive linear optics. We calculate upper bounds on the fidelity between the actual heralded state and the target state. We find that this fidelity upper bound is $1/2$ when the target state is a multi-mode coherent cat-basis cluster state, a resource sufficient for universal quantum computing. This proves that there exist non-Gaussian states that are not producible by this method. Our fidelity upper bound is a simple expression that depends only on the target state represented in the photon-number basis, which could be applied to other non-Gaussian states of interest.Comment: Revised version which now considers state engineering based on partial PNR detection, which subsumes subtraction and addition of photons. Said generalization allowed for cleaner and easier mathematical derivations. Appendix was taken from arXiv:2108.08290, co-authored by present authors and collaborators. Comments welcome and appreciate

Universality of slow decorrelation in KPZ growth

There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar-Parisi-Zhang (KPZ) universality class. A proper rescaling of time should introduce a non-trivial temporal dimension to these limiting fluctuations. In one-dimension, the KPZ class has the dynamical scaling exponent $z=3/2$, that means one should find a universal space-time limiting process under the scaling of time as $t\,T$, space like $t^{2/3} X$ and fluctuations like $t^{1/3}$ as $t\to\infty$. In this paper we provide evidence for this belief. We prove that under certain hypotheses, growth models display temporal slow decorrelation. That is to say that in the scalings above, the limiting spatial process for times $t\, T$ and $t\, T+t^{\nu}$ are identical, for any $\nu<1$. The hypotheses are known to be satisfied for certain last passage percolation models, the polynuclear growth model, and the totally / partially asymmetric simple exclusion process. Using slow decorrelation we may extend known fluctuation limit results to space-time regions where correlation functions are unknown. The approach we develop requires the minimal expected hypotheses for slow decorrelation to hold and provides a simple and intuitive proof which applied to a wide variety of models.Comment: Exposition improved, typos correcte

Formally Verified Approximations of Definite Integrals

International audienceFinding an elementary form for an antiderivative is often a difficult task, so numerical integration has become a common tool when it comes to making sense of a definite integral. Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. Yet numerical integration is still missing from the toolbox when performing formal proofs in analysis. This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system. Our approach is not based on traditional quadrature methods such as Newton-Cotes formulas. Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting. This work has been integrated to the CoqInterval library

Formal certification of arithmetic filters for geometric predicates

International audienceFloating-point arithmetic provides a fast but inexact way of computing geometric predicates. In order for these predicates to be exact, it is important to rule out all the numerical situations where floating-point computations could lead to wrong results. Taking into account all the potential problems is a tedious work to do by hand. We study in this paper a floating-point implementation of a filter for the orientation-2 predicate, and how a formal and partially automatized verification of this algorithm avoided many pitfalls. The presented method is not limited to this particular predicate, it can easily be used to produce correct semi-static floating-point filters for other geometric predicates

Formally Certified Floating-Point Filters For Homogeneous Geometric Predicates

International audienceFloating-point arithmetic provides a fast but inexact way of computing geometric predicates. In order for these predicates to be exact, it is important to rule out all the numerical situations where floating-point computations could lead to wrong results. Taking into account all the potential problems is a tedious work to do by hand. We study in this paper a floating-point implementation of a filter for the orientation-2 predicate, and how a formal and partially automatized verification of this algorithm avoided many pitfalls. The presented method is not limited to this particular predicate, it can easily be used to produce correct semi-static floating-point filters for other geometric predicates

eMath 3.0: building blocks for a social and semantic Web for online mathematics & elearning

In this paper we present recent developments in content markup for mathematics, and a corresponding software stack that functions as an enabling technology for a social and semantic web for the STEM disciplines. We show the potential of this technology in two eMath 3.0 applications: PlanetMathRedux, a re-implementation of the mathematical encyclopedia PlanetMath.org, and PantaRheiRedux, a community reader for course materials. These applications indicate both present and potential uses for this software as a basis for eLearning applications in Science, Technology, Engineering and Mathematics through the addition of suitable pedagogies

Colorful linear programming, Nash equilibrium , and pivots

The colorful Carathéodory theorem, proved by Barany in 1982, states that given d+1 sets of points S_1,...,S_{d+1} in R^d, such that each S_i contains 0 in its convex hull, there exists a set subset T in the union of the S_i containing 0 in its convex hull and such that T intersects each S_i at most once. An intriguing question - still open - is whether such a set T, whose existence is ensured, can be found in polynomial time. In 1997, Barany and Onn defined colorful linear programming as algorithmic questions related to the colorful Carathéodory theorem. The question we just mentioned comes under colorful linear programming, and there are also other problems. We present new complexity results for colorful linear programming problems and propose a variant of the "Barany-Onn" algorithm, which is an algorithm computing a set T whose existence is ensured by the colorful Carathéodory theorem. Our algorithm makes a clear connection with the simplex algorithm. Some combinatorial applications of the colorful Carathéodory theorem are also discussed from an algorithmic point of view. Finally, we show that computing a Nash equilibrium in a bimatrix game is polynomially reducible to a colorful linear programming problem. On our track, we found a new way to prove that a complementarity problem belongs to the PPAD class with the help of Sperner's lemma