The combinatorics of minimal unsatisfiability: connecting to graph theory

Minimally Unsatisfiable CNFs (MUs) are unsatisfiable CNFs where removing any clause destroys unsatisfiability. MUs are the building blocks of unsatisfia-bility, and our understanding of them can be very helpful in answering various algorithmic and structural questions relating to unsatisfiability. In this thesis we study MUs from a combinatorial point of view, with the aim of extending the understanding of the structure of MUs. We show that some important classes of MUs are very closely related to known classes of digraphs, and using arguments from logic and graph theory we characterise these MUs.Two main concepts in this thesis are isomorphism of CNFs and the implica-tion digraph of 2-CNFs (at most two literals per disjunction). Isomorphism of CNFs involves renaming the variables, and flipping the literals. The implication digraph of a 2-CNF F has both arcs (¬a → b) and (¬b → a) for every binary clause (a ∨ b) in F .In the first part we introduce a novel connection between MUs and Minimal Strong Digraphs (MSDs), strongly connected digraphs, where removing any arc destroys the strong connectedness. We introduce the new class DFM of special MUs, which are in close correspondence to MSDs. The known relation between 2-CNFs and implication digraphs is used, but in a simpler and more direct way, namely that we have a canonical choice of one of the two arcs. As an application of this new framework we provide short and intuitive new proofs for two im-portant but isolated characterisations for nonsingular MUs (every literal occurs at least twice), both with ingenious but complicated proofs: Characterising 2-MUs (minimally unsatisfiable 2-CNFs), and characterising MUs with deficiency 2 (two more clauses than variables).In the second part, we provide a fundamental addition to the study of 2-CNFs which have efficient algorithms for many interesting problems, namely that we provide a full classification of 2-MUs and a polytime isomorphism de-cision of this class. We show that implication digraphs of 2-MUs are “Weak Double Cycles” (WDCs), big cycles of small cycles (with possible overlaps). Combining logical and graph-theoretical methods, we prove that WDCs have at most one skew-symmetry (a self-inverse fixed-point free anti-symmetry, re-versing the direction of arcs). It follows that the isomorphisms between 2-MUs are exactly the isomorphisms between their implication digraphs (since digraphs with given skew-symmetry are the same as 2-CNFs). This reduces the classifi-cation of 2-MUs to the classification of a nice class of digraphs.Finally in the outlook we discuss further applications, including an alter-native framework for enumerating some special Minimally Unsatisfiable Sub-clause-sets (MUSs)

Cohort study of cardiovascular safety of different COVID-19 vaccination doses among 46 million adults in England

The first dose of COVID-19 vaccines led to an overall reduction in cardiovascular events, and in rare cases, cardiovascular complications. There is less information about the effect of second and booster doses on cardiovascular diseases. Using longitudinal health records from 45.7 million adults in England between December 2020 and January 2022, our study compared the incidence of thrombotic and cardiovascular complications up to 26 weeks after first, second and booster doses of brands and combinations of COVID-19 vaccines used during the UK vaccination program with the incidence before or without the corresponding vaccination. The incidence of common arterial thrombotic events (mainly acute myocardial infarction and ischaemic stroke) was generally lower after each vaccine dose, brand and combination. Similarly, the incidence of common venous thrombotic events, (mainly pulmonary embolism and lower limb deep venous thrombosis) was lower after vaccination. There was a higher incidence of previously reported rare harms after vaccination: vaccine-induced thrombotic thrombocytopenia after first ChAdOx1 vaccination, and myocarditis and pericarditis after first, second and transiently after booster mRNA vaccination (BNT-162b2 and mRNA-1273). These findings support the wide uptake of future COVID-19 vaccination programs

Clinical coding of long Covid in Wales: A cohort study of 3.5 million people using linked health and demographic data

Objectives ‘Long COVID’ (LC) is broadly defined as signs and symptoms that continue or develop after the acute phase of COVID-19, and can affect cardiovascular, respiratory and other organ systems. Using electronic health records, we investigated clinical coding of LC in primary and secondary care for the population of Wales. Methods We conducted a cohort study for the population of Wales, using anonymised individual-level linked data in the Secure Anonymised Information Linkage (SAIL) Databank. We used the Welsh COVID-19 e-cohort (doi:10.1136/bmjopen-2020-043010), which consists of all people (adults and children) alive and resident in Wales from 1st January 2020. To this e-cohort we linked primary and secondary care, COVID-19 testing, and ethnic group data. We then calculated the proportion of people with a LC diagnosis code (in primary and secondary care data) overall and stratified by demographic variables. Results Of 3.5m residents, 7,696 (0.2%) had a LC clinical diagnosis. Compared with the general population, a higher proportion of people with LC were female, middle age, white, and hospitalised within 28 days of a confirmed COVID-19 infection. LC affected all socioeconomic groups, as assessed using the Welsh Index of Multiple Deprivation. When looking at LC diagnosis codes in primary care, 30.9% of practices in SAIL have not used these codes at all. And the number of recorded events was low until the end of January 2021, after which there was an increase in coding. These findings are likely a substantial underestimate of LC prevalence in Wales. Earlier estimates from self-reported surveys, such as the Office for National Statistics, are much higher, ranging anywhere between 3-5%. Conclusion Low recording rates of LC and variation between practices could be due to a delay in introducing clinical coding and lack of presentation/recording. Understanding prevalence of LC is vital for addressing the scale of the problem. Therefore developing additional data-driven approaches is necessary to obtain an accurate prevalence estimate

Harmonising electronic health records for reproducible research: challenges, solutions and recommendations from a UK-wide COVID-19 research collaboration

BackgroundThe CVD-COVID-UK consortium was formed to understand the relationship between COVID-19 and cardiovascular diseases through analyses of harmonised electronic health records (EHRs) across the four UK nations. Beyond COVID-19, data harmonisation and common approaches enable analysis within and across independent Trusted Research Environments. Here we describe the reproducible harmonisation method developed using large-scale EHRs in Wales to accommodate the fast and efficient implementation of cross-nation analysis in England and Wales as part of the CVD-COVID-UK programme. We characterise current challenges and share lessons learnt.MethodsServing the scope and scalability of multiple study protocols, we used linked, anonymised individual-level EHR, demographic and administrative data held within the SAIL Databank for the population of Wales. The harmonisation method was implemented as a four-layer reproducible process, starting from raw data in the first layer. Then each of the layers two to four is framed by, but not limited to, the characterised challenges and lessons learnt. We achieved curated data as part of our second layer, followed by extracting phenotyped data in the third layer. We captured any project-specific requirements in the fourth layer.ResultsUsing the implemented four-layer harmonisation method, we retrieved approximately 100 health-related variables for the 3.2 million individuals in Wales, which are harmonised with corresponding variables for > 56 million individuals in England. We processed 13 data sources into the first layer of our harmonisation method: five of these are updated daily or weekly, and the rest at various frequencies providing sufficient data flow updates for frequent capturing of up-to-date demographic, administrative and clinical information.ConclusionsWe implemented an efficient, transparent, scalable, and reproducible harmonisation method that enables multi-nation collaborative research. With a current focus on COVID-19 and its relationship with cardiovascular outcomes, the harmonised data has supported a wide range of research activities across the UK

Exploring ethnicity dynamics in Wales: a longitudinal population-scale linked data study and development of a harmonised ethnicity spine

Objective This study aims to create a national ethnicity spine based on all available ethnicity records in linkable anonymised electronic health record and administrative data sources.Design A longitudinal study using anonymised individual-level population-scale ethnicity data from 26 data sources available within the Secure Anonymised Information Linkage Databank.Setting The national ethnicity spine is created based on longitudinal national data for the population of Wales-UK over 22 years (between 2000 and 2021).Procedure and participants A total of 46 million ethnicity records for 4 297 694 individuals have been extracted, harmonised, deduplicated and made available within a longitudinal research ready data asset.Outcome measures (1) Comparing the distribution of ethnicity records over time for four different selection approaches (latest, mode, weighted mode and composite) across age bands, sex, deprivation quintiles, health board and residential location and (2) distribution and completeness of records against the ONS census 2011.Results The distribution of the dominant group (white) is minimally affected based on the four different selection approaches. Across all other ethnic group categorisations, the mixed group was most susceptible to variation in distribution depending on the selection approach used and varied from a 0.6% prevalence across the latest and mode approach to a 1.1% prevalence for the weighted mode, compared with the 3.1% prevalence for the composite approach. Substantial alignment was observed with ONS 2011 census with the Latest group method (kappa=0.68, 95% CI (0.67 to 0.71)) across all subgroups. The record completeness rate was over 95% in 2021.Conclusion In conclusion, our development of the population-scale ethnicity spine provides robust ethnicity measures for healthcare research in Wales and a template which can easily be deployed in other trusted research environments in the UK and beyond

Staff-Pupil SARS-CoV-2 Infection Pathways in Schools: A Population Level Linked Data Approach

Background Better understanding of the role that children and school staff play in the transmission of SARS-CoV-2 is essential to guide policy development on controlling infection while minimising disruption to children’s education and well-being.Methods Our national e-cohort (n=464531) study used anonymised linked data for pupils, staff and associated households linked via educational settings in Wales. We estimated the odds of testing positive for SARS-CoV-2 infection for staff and pupils over the period August– December 2020, dependent on measures of recent exposure to known cases linked to their educational settings.Results The total number of cases in a school was not associated with a subsequent increase in the odds of testing positive (staff OR per case: 0.92, 95% CI 0.85 to 1.00; pupil OR per case: 0.98, 95% CI 0.93 to 1.02). Among pupils, the number of recent cases within the same year group was significantly associated with subsequent increased odds of testing positive (OR per case: 1.12, 95% CI 1.08 to 1.15). These effects were adjusted for a range of demographic covariates, and in particular any known cases within the same household, which had the strongest association with testing positive (staff OR: 39.86, 95% CI 35.01 to 45.38; pupil OR: 9.39, 95% CI 8.94 to 9.88).Conclusions In a national school cohort, the odds of staff testing positive for SARS-CoV-2 infection were not significantly increased in the 14-day period after case detection in the school. However, pupils were found to be at increased odds, following cases appearing within their own year group, where most of their contacts occur. Strong mitigation measures over the whole of the study period may have reduced wider spread within the school environment

Risk of thrombocytopenic, haemorrhagic and thromboembolic disorders following COVID-19 vaccination and positive test: a self-controlled case series analysis in Wales

There is a need for better understanding of the risk of thrombocytopenic, haemorrhagic, thromboembolic disorders following first, second and booster vaccination doses and testing positive for SARS-CoV-2. Self-controlled cases series analysis of 2.1 million linked patient records in Wales between 7th December 2020 and 31st December 2021. Outcomes were the first diagnosis of thrombocytopenic, haemorrhagic and thromboembolic events in primary or secondary care datasets, exposure was defined as 0–28 days post-vaccination or a positive reverse transcription polymerase chain reaction test for SARS-CoV-2. 36,136 individuals experienced either a thrombocytopenic, haemorrhagic or thromboembolic event during the study period. Relative to baseline, our observations show greater risk of outcomes in the periods post-first dose of BNT162b2 for haemorrhagic (IRR 1.47, 95%CI: 1.04–2.08) and idiopathic thrombocytopenic purpura (IRR 2.80, 95%CI: 1.21–6.49) events; post-second dose of ChAdOx1 for arterial thrombosis (IRR 1.14, 95%CI: 1.01–1.29); post-booster greater risk of venous thromboembolic (VTE) (IRR-Moderna 3.62, 95%CI: 0.99–13.17) (IRR-BNT162b2 1.39, 95%CI: 1.04–1.87) and arterial thrombosis (IRR-Moderna 3.14, 95%CI: 1.14–8.64) (IRR-BNT162b2 1.34, 95%CI: 1.15–1.58). Similarly, post SARS-CoV-2 infection the risk was increased for haemorrhagic (IRR 1.49, 95%CI: 1.15–1.92), VTE (IRR 5.63, 95%CI: 4.91, 6.4), arterial thrombosis (IRR 2.46, 95%CI: 2.22–2.71). We found that there was a measurable risk of thrombocytopenic, haemorrhagic, thromboembolic events after COVID-19 vaccination and infection