On the Cohomology of Contextuality

Recent work by Abramsky and Brandenburger used sheaf theory to give a mathematical formulation of non-locality and contextuality. By adopting this viewpoint, it has been possible to define cohomological obstructions to the existence of global sections. In the present work, we illustrate new insights into different aspects of this theory. We shed light on the power of detection of the cohomological obstruction by showing that it is not a complete invariant for strong contextuality even under symmetry and connectedness restrictions on the measurement cover, disproving a previous conjecture. We generalise obstructions to higher cohomology groups and show that they give rise to a refinement of the notion of cohomological contextuality: different "levels" of contextuality are organised in a hierarchy of logical implications. Finally, we present an alternative description of the first cohomology group in terms of torsors, resulting in a new interpretation of the cohomological obstructions.Comment: In Proceedings QPL 2016, arXiv:1701.0024

Putting the Semantics into Semantic Versioning

The long-standing aspiration for software reuse has made astonishing strides in the past few years. Many modern software development ecosystems now come with rich sets of publicly-available components contributed by the community. Downstream developers can leverage these upstream components, boosting their productivity. However, components evolve at their own pace. This imposes obligations on and yields benefits for downstream developers, especially since changes can be breaking, requiring additional downstream work to adapt to. Upgrading too late leaves downstream vulnerable to security issues and missing out on useful improvements; upgrading too early results in excess work. Semantic versioning has been proposed as an elegant mechanism to communicate levels of compatibility, enabling downstream developers to automate dependency upgrades. While it is questionable whether a version number can adequately characterize version compatibility in general, we argue that developers would greatly benefit from tools such as semantic version calculators to help them upgrade safely. The time is now for the research community to develop such tools: large component ecosystems exist and are accessible, component interactions have become observable through automated builds, and recent advances in program analysis make the development of relevant tools feasible. In particular, contracts (both traditional and lightweight) are a promising input to semantic versioning calculators, which can suggest whether an upgrade is likely to be safe.Comment: to be published as Onward! Essays 202

Efficient Iterative Programs with Distributed Data Collections

Big data programming frameworks have become increasingly important for the development of applications for which performance and scalability are critical. In those complex frameworks, optimizing code by hand is hard and time-consuming, making automated optimization particularly necessary. In order to automate optimization, a prerequisite is to find suitable abstractions to represent programs; for instance, algebras based on monads or monoids to represent distributed data collections. Currently, however, such algebras do not represent recursive programs in a way which allows for analyzing or rewriting them. In this paper, we extend a monoid algebra with a fixpoint operator for representing recursion as a first class citizen and show how it enables new optimizations. Experiments with the Spark platform illustrate performance gains brought by these systematic optimizations.Comment: 36 page

On monogamy of non-locality and macroscopic averages: examples and preliminary results

We explore a connection between monogamy of non-locality and a weak macroscopic locality condition: the locality of the average behaviour. These are revealed by our analysis as being two sides of the same coin. Moreover, we exhibit a structural reason for both in the case of Bell-type multipartite scenarios, shedding light on but also generalising the results in the literature [Ramanathan et al., Phys. Rev. Lett. 107, 060405 (2001); Pawlowski & Brukner, Phys. Rev. Lett. 102, 030403 (2009)]. More specifically, we show that, provided the number of particles in each site is large enough compared to the number of allowed measurement settings, and whatever the microscopic state of the system, the macroscopic average behaviour is local realistic, or equivalently, general multipartite monogamy relations hold. This result relies on a classical mathematical theorem by Vorob'ev [Theory Probab. Appl. 7(2), 147-163 (1962)] about extending compatible families of probability distributions defined on the faces of a simplicial complex -- in the language of the sheaf-theoretic framework of Abramsky & Brandenburger [New J. Phys. 13, 113036 (2011)], such families correspond to no-signalling empirical models, and the existence of an extension corresponds to locality or non-contextuality. Since Vorob'ev's theorem depends solely on the structure of the simplicial complex, which encodes the compatibility of the measurements, and not on the specific probability distributions (i.e. the empirical models), our result about monogamy relations and locality of macroscopic averages holds not just for quantum theory, but for any empirical model satisfying the no-signalling condition. In this extended abstract, we illustrate our approach by working out a couple of examples, which convey the intuition behind our analysis while keeping the discussion at an elementary level.Comment: In Proceedings QPL 2014, arXiv:1412.810