Sheaf semantics of termination-insensitive noninterference

We propose a new sheaf semantics for secure information flow over a space of abstract behaviors, based on synthetic domain theory: security classes are open/closed partitions, types are sheaves, and redaction of sensitive information corresponds to restricting a sheaf to a closed subspace. Our security-aware computational model satisfies termination-insensitive noninterference automatically, and therefore constitutes an intrinsic alternative to state of the art extrinsic/relational models of noninterference. Our semantics is the latest application of Sterling and Harper's recent re-interpretation of phase distinctions and noninterference in programming languages in terms of Artin gluing and topos-theoretic open/closed modalities. Prior applications include parametricity for ML modules, the proof of normalization for cubical type theory by Sterling and Angiuli, and the cost-aware logical framework of Niu et al. In this paper we employ the phase distinction perspective twice: first to reconstruct the syntax and semantics of secure information flow as a lattice of phase distinctions between "higher" and "lower" security, and second to verify the computational adequacy of our sheaf semantics vis-\`a-vis an extension of Abadi et al.'s dependency core calculus with a construct for declassifying termination channels.Comment: Extended version of FSCD '22 paper with full technical appendice

Pseudo-commutative Monads

AbstractWe introduce the notion of pseudo-commutative monad together with that of pseudo-closed 2-category, the leading example being given by the 2-monad on Cat whose 2-category of algebras is the 2-category of small symmetric monoidal categories. We prove that for any pseudo-commutative 2-monad on Cat, its 2-category of algebras is pseudo-closed. We also introduce supplementary definitions and results, and we illustrate this analysis with further examples such as those of small categories with finite products, and examples arising from wiring, interaction, contexts, and the logic of Bunched Implication

Recursion and Sequentiality in Categories of Sheaves

We present a fully abstract model of a call-by-value language with higher-order functions, recursion and natural numbers, as an exponential ideal in a topos. Our model is inspired by the fully abstract models of O'Hearn, Riecke and Sandholm, and Marz and Streicher. In contrast with semantics based on cpo's, we treat recursion as just one feature in a model built by combining a choice of modular components

An equational notion of lifting monad

We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the equational properties of partial maps as induced by partial map classifiers. The representation theorem also provides a tool for transferring non-equational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of Kleisli categories of equational lifting monads. This axiomatization is of interest in its own right.

Categories and Types for Axiomatic Domain Theory

Submitted for the degree of Doctor of Philosophy, University of londo

Optimising Spatial and Tonal Data for PDE-based Inpainting

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods

Ways of seeing geometrical meaning in different situations.

This thesis set out to challenge the traditional approach to the study of\ud geometrical understanding which has assumed that conceiving and interpreting\ud shapes or forms is the result of logical and mental interaction between an\ud individual and geometrical objects and that the production of geometrical meaning\ud is motivated by the stimulus of the external structure of a visual text. By way of\ud contrast, this study makes the case that geometrical meaning is socially and\ud contextually produced.\ud The research has two interconnected strands. The first strand is\ud theoretical aiming to develop a framework for the study of geometrical\ud understanding drawing on concepts from Mikhail Bakhtin, Umberto Eco and\ud Gunther Kress. The second is empirical aiming to collect data whose analysis will\ud inform and be informed by this theoretical framework. For this study, three\ud groups of people who differed radically in terms of their geometrical\ud experiences, socio-economic and educational backgrounds were interviewed in\ud order to examine their interpretations of geometrical elements exhibited in\ud different settings.\ud The theoretical work of this thesis led to a framework for understanding\ud geometry comprising 'sign', 'sign-functions', 'visual text', and 'heteroglossia'.\ud Analysis of the data from empirical study in terms of this framework revealed\ud the importance of the dynamics for visual experience as a process for\ud communicating and of signifying, and how this relationship was itself dependent\ud on the material conditions and contextual dynamics in which the meanings were\ud constructed. The thesis concludes with an assessment of its potential contribution\ud to redress the balance between learning about geometry and learning through\ud geometry

Process Models for Laser Engineered Net Shaping

The goal of this dissertation is to develop a model relating LENSâ„¢ process parameters to deposited thickness, incorporating the effect of substrate heating. A design review was carried out, adapting the technique of functional decomposition borrowed from axiomatic design. The review revealed that coupling between the laser path and laser power caused substrate heating. The material delivery mechanism was modeled and verified using experimental data. It was used in the derivation of the average deposition model which predicted deposition based on build parameters, but did not incorporate substrate heating. The average deposition model appeared capable of predicting deposited thickness for single line, 1- layer and 2-layer builds, performing best for the 1- layer builds which were built under essentially isothermal conditions. This model was extended to incorporate the effect of substrate heating, estimated using an energy partition approach. The energy used for substrate heating was modeled as a series of timed heating events from an instantaneous point heat source along the path of the laser. The result was called the spatial deposition model, and was verified using the same set of experimental data. The model appeared capable of predicting deposited thickness for single line, 1- layer and 2- layer builds and was able to predict the characteristic temperature rise near the borders as the laser reversed direction

Computational Adequacy for Recursive Types in Models of Intuitionistic Set Theory

This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic Zermelo-Fraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambda-calculus with callby-value operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal ” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external ” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1-consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domain-theoretic models