The algebra of entanglement and the geometry of composition

String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another. We individuate, at the root of this fact, the dual nature of polygraphs as presentations of higher algebraic theories, and as combinatorial descriptions of "directed spaces". Operations of polygraphs modelled on operations of topological spaces are used as the foundation of a compositional universal algebra, where sliding moves arise from tensor products of polygraphs. We reconstruct several higher algebraic theories in this framework. In this regard, the standard formalism of polygraphs has some technical problems. We propose a notion of regular polygraph, barring cell boundaries that are not homeomorphic to a disk of the appropriate dimension. We define a category of non-degenerate shapes, and show how to calculate their tensor products. Then, we introduce a notion of weak unit to recover weakly degenerate boundaries in low dimensions, and prove that the existence of weak units is equivalent to a representability property. We then turn to applications of diagrammatic algebra to quantum theory. We re-evaluate the category of Hilbert spaces from the perspective of categorical universal algebra, which leads to a bicategorical refinement. Then, we focus on the axiomatics of fragments of quantum theory, and present the ZW calculus, the first complete diagrammatic axiomatisation of the theory of qubits. The ZW calculus has several advantages over ZX calculi, including a computationally meaningful normal form, and a fragment whose diagrams can be read as setups of fermionic oscillators. Moreover, its generators reflect an operational classification of entangled states of 3 qubits. We conclude with generalisations of the ZW calculus to higher-dimensional systems, including the definition of a universal set of generators in each dimension.Comment: v2: changes to end of Chapter 3. v1: 214 pages, many figures; University of Oxford doctoral thesi

Detecting referral and selection bias by the anonymous linkage of practice, hospital and clinic data using Secure and Private Record Linkage (SAPREL): case study from the evaluation of the Improved Access to Psychological Therapy (IAPT) service

Background: The evaluation of demonstration sites set up to provide improved access to psychological therapies (IAPT) comprised the study of all people identified as having common mental health problems (CMHP), those referred to the IAPT service, and a sample of attenders studied in-depth. Information technology makes it feasible to link practice, hospital and IAPT clinic data to evaluate the representativeness of these samples. However, researchers do not have permission to browse and link these data without the patients’ consent. Objective: To demonstrate the use of a mixed deterministic-probabilistic method of secure and private record linkage (SAPREL) - to describe selection bias in subjects chosen for in-depth evaluation. Method: We extracted, pseudonymised and used fuzzy logic to link multiple health records without the researcher knowing the patient’s identity. The method can be characterised as a three party protocol mainly using deterministic algorithms with dynamic linking strategies; though incorporating some elements of probabilistic linkage. Within the data providers’ safe haven we extracted: Demographic data, hospital utilisation and IAPT clinic data; converted post code to index of multiple deprivation (IMD); and identified people with CMHP. We contrasted the age, gender, ethnicity and IMD for the in-depth evaluation sample with people referred to IAPT, use hospital services, and the population as a whole. Results: The in IAPT-in-depth group had a mean age of 43.1 years; CI: 41.0 - 45.2 (n = 166); the IAPT-referred 40.2 years; CI: 39.4 - 40.9 (n = 1118); and those with CMHP 43.6 years SEM 0.15. (n = 12210). Whilst around 67% of those with a CMHP were women, compared to 70% of those referred to IAPT, and 75% of those subject to indepth evaluation (Chi square p< 0.001). The mean IMD score for the in-depth evaluation group was 36.6; CI: 34.2 - 38.9; (n = 166); of those referred to IAPT 38.7; CI: 37.9 - 39.6; (n = 1117); and of people with CMHP 37.6; CI 37.3- 37.9; (n = 12143). Conclusions: The sample studied in-depth were older, more likely female, and less deprived than people with CMHP, and fewer had recorded ethnic minority status. Anonymous linkage using SAPREL provides insight into the representativeness of a study population and possible adjustment for selection bias

Algorithmic and Combinatorial Results in Selection and Computational Geometry

This dissertation investigates two sets of algorithmic and combinatorial problems. Thefirst part focuses on the selection problem under the pairwise comparison model. For the classic “median of medians” scheme, contrary to the popular belief that smaller group sizes cause superlinear behavior, several new linear time algorithms that utilize small groups are introduced. Then the exact number of comparisons needed for an optimal selection algorithm is studied. In particular, the implications of a long standing conjecture known as Yao’s hypothesis are explored. For the multiparty model, we designed low communication complexity protocols for selecting an exact or an approximate median of data that is distributed among multiple players. In the second part, three computational geometry problems are studied. For the longestspanning tree with neighborhoods, approximation algorithms are provided. For the stretch factor of polygonal chains, upper bounds are proved and almost matching lower bound constructions in \mathbb{R}^2 and higher dimensions are developed. For the piercing number τ and independence number ν of a family of axis-parallel rectangles in the plane, a lower bound construction for ν = 4 that matches Wegner’s conjecture is analyzed. The previous matching construction for ν = 3, due to Wegner himself, dates back to 1968

Principles of Security and Trust

This open access book constitutes the proceedings of the 8th International Conference on Principles of Security and Trust, POST 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 10 papers presented in this volume were carefully reviewed and selected from 27 submissions. They deal with theoretical and foundational aspects of security and trust, including on new theoretical results, practical applications of existing foundational ideas, and innovative approaches stimulated by pressing practical problems

Principles of Security and Trust

This open access book constitutes the proceedings of the 8th International Conference on Principles of Security and Trust, POST 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 10 papers presented in this volume were carefully reviewed and selected from 27 submissions. They deal with theoretical and foundational aspects of security and trust, including on new theoretical results, practical applications of existing foundational ideas, and innovative approaches stimulated by pressing practical problems

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems