Searching and retrieving in content-based repositories of formal mathematical knowledge

In this thesis, the author presents a query language for an RDF (Resource Description Framework) database and discusses its applications in the context of the HELM project (the Hypertextual Electronic Library of Mathematics). This language aims at meeting the main requirements coming from the RDF community. in particular it includes: a human readable textual syntax and a machine-processable XML (Extensible Markup Language) syntax both for queries and for query results, a rigorously exposed formal semantics, a graph-oriented RDF data access model capable of exploring an entire RDF graph (including both RDF Models and RDF Schemata), a full set of Boolean operators to compose the query constraints, fully customizable and highly structured query results having a 4-dimensional geometry, some constructions taken from ordinary programming languages that simplify the formulation of complex queries. The HELM project aims at integrating the modern tools for the automation of formal reasoning with the most recent electronic publishing technologies, in order create and maintain a hypertextual, distributed virtual library of formal mathematical knowledge. In the spirit of the Semantic Web, the documents of this library include RDF metadata describing their structure and content in a machine-understandable form. Using the author's query engine, HELM exploits this information to implement some functionalities allowing the interactive and automatic retrieval of documents on the basis of content-aware requests that take into account the mathematical nature of these documents.

User interaction widgets for interactive theorem proving

Matita (that means pencil in Italian) is a new interactive theorem prover under development at the University of Bologna. When compared with state-of-the-art proof assistants, Matita presents both traditional and innovative aspects. The underlying calculus of the system, namely the Calculus of (Co)Inductive Constructions (CIC for short), is well-known and is used as the basis of another mainstream proof assistant—Coq—with which Matita is to some extent compatible. In the same spirit of several other systems, proof authoring is conducted by the user as a goal directed proof search, using a script for storing textual commands for the system. In the tradition of LCF, the proof language of Matita is procedural and relies on tactic and tacticals to proceed toward proof completion. The interaction paradigm offered to the user is based on the script management technique at the basis of the popularity of the Proof General generic interface for interactive theorem provers: while editing a script the user can move forth the execution point to deliver commands to the system, or back to retract (or “undo”) past commands. Matita has been developed from scratch in the past 8 years by several members of the Helm research group, this thesis author is one of such members. Matita is now a full-fledged proof assistant with a library of about 1.000 concepts. Several innovative solutions spun-off from this development effort. This thesis is about the design and implementation of some of those solutions, in particular those relevant for the topic of user interaction with theorem provers, and of which this thesis author was a major contributor. Joint work with other members of the research group is pointed out where needed. The main topics discussed in this thesis are briefly summarized below. Disambiguation. Most activities connected with interactive proving require the user to input mathematical formulae. Being mathematical notation ambiguous, parsing formulae typeset as mathematicians like to write down on paper is a challenging task; a challenge neglected by several theorem provers which usually prefer to fix an unambiguous input syntax. Exploiting features of the underlying calculus, Matita offers an efficient disambiguation engine which permit to type formulae in the familiar mathematical notation. Step-by-step tacticals. Tacticals are higher-order constructs used in proof scripts to combine tactics together. With tacticals scripts can be made shorter, readable, and more resilient to changes. Unfortunately they are de facto incompatible with state-of-the-art user interfaces based on script management. Such interfaces indeed do not permit to position the execution point inside complex tacticals, thus introducing a trade-off between the usefulness of structuring scripts and a tedious big step execution behavior during script replaying. In Matita we break this trade-off with tinycals: an alternative to a subset of LCF tacticals which can be evaluated in a more fine-grained manner. Extensible yet meaningful notation. Proof assistant users often face the need of creating new mathematical notation in order to ease the use of new concepts. The framework used in Matita for dealing with extensible notation both accounts for high quality bidimensional rendering of formulae (with the expressivity of MathMLPresentation) and provides meaningful notation, where presentational fragments are kept synchronized with semantic representation of terms. Using our approach interoperability with other systems can be achieved at the content level, and direct manipulation of formulae acting on their rendered forms is possible too. Publish/subscribe hints. Automation plays an important role in interactive proving as users like to delegate tedious proving sub-tasks to decision procedures or external reasoners. Exploiting the Web-friendliness of Matita we experimented with a broker and a network of web services (called tutors) which can try independently to complete open sub-goals of a proof, currently being authored in Matita. The user receives hints from the tutors on how to complete sub-goals and can interactively or automatically apply them to the current proof. Another innovative aspect of Matita, only marginally touched by this thesis, is the embedded content-based search engine Whelp which is exploited to various ends, from automatic theorem proving to avoiding duplicate work for the user. We also discuss the (potential) reusability in other systems of the widgets presented in this thesis and how we envisage the evolution of user interfaces for interactive theorem provers in the Web 2.0 era.

Humour, Hate Crimes and British Radical Right Users on Twitter: A longitudinal Post-Brexit study

This dissertation explores the link between hate crimes that occurred in the United Kingdom in June 2017, June 2018 and June 2019 through the posts of a robust sample of Conservative and radical right users on Twitter. In order to avoid the traditional challenges of this kind of research, I adopted a four staged research protocol that enabled me to merge content produced by a group of randomly selected users to observe the phenomenon from different angles. I collected tweets from thirty Conservative/right wing accounts for each month of June over the three years with the help of programming languages such as Python and CygWin tools. I then examined the language of my data focussing on humorous content in order to reveal whether, and if so how, radical users online often use humour as a tool to spread their views in conditions of heightened disgust and wide-spread political instability. A reflection on humour as a moral occurrence, expanding on the works of Christie Davies as well as applying recent findings on the behavioural immune system on online data, offers new insights on the overlooked humorous nature of radical political discourse. An unorthodox take on the moral foundations pioneered by Jonathan Haidt enriched my understanding of the analysed material through the addition of a moral-based layer of enquiry to my more traditional content-based one. This convergence of theoretical, data driven and real life events constitutes a viable “collection of strategies” for academia, data scientists; NGO’s fighting hate crimes and the wider public alike. Bringing together the ideas of Davies, Haidt and others to my data, helps us to perceive humorous online content in terms of complex radical narratives that are all too often compressed into a single tweet.

Retrieval of atmospheric trace gases from DOAS/MAX-DOAS ground-based spectra

The study of the atmospheric chemical composition is crucial to understand the climate changes that we are experiencing in the last decades and to monitor the air quality over industrialized areas. The Multi-AXis Differential Optical Absorption Spectroscopy (MAX-DOAS) ground-based instruments are particularly suitable to derive the concentration of some trace gases that absorb the Visible (VIS) and Ultra-Violet (UV) solar radiation. The zenith-sky spectra acquired by the Gas Analyzer Spectrometer Correlating Optical Differences / New Generation 4 (GASCOD/NG4) instrument are exploited to retrieve the NO2 and O3 total Vertical Column Densities (VCDs) over Lecce. The results show that the NO2 total VCDs are significantly affected by the tropospheric content, consequence of the anthropogenic activity. Indeed, they present systematically lower values during Sunday, when less traffic is generally present around the measurement site, and during windy days, especially when the wind direction measured at 2 m height is not from the city of Lecce. Another MAX-DOAS instrument (SkySpec-2D) is exploited to create the first Italian MAX-DOAS site compliant to the Fiducial Reference Measurements for DOAS (FRM4DOAS) standards, in San Pietro Capofiume (SPC), located in the middle of the Po Valley. After the assessment of the SkySpec-2D’s performances through two measurement campaigns taken place in Bologna and in Rome, SkySpec-2D is installed in SPC on the 1st October 2021. Its MAX-DOAS spectra are used to retrieve the NO2 and O3 total VCDs, and aerosol extinction and NO2 tropospheric vertical profiles over the Po Valley exploiting the Bremen Optimal estimation REtrieval for Aerosol and trace gaseS (BOREAS) algorithm. Promising results are found, with high correlations against both in-situ and satellite data. In the future, these data will play an important role for air quality studies over the Po Valley and for satellite validation purposes.