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.

Come città dell'infanzia. Architettura, educazione e immagine in due colonie per i figli degli Italiani all'estero a Cattolica e Tirrenia

The present doctoral dissertation deals with two significant case studies of Italian holiday camps which appear as interesting architectural experiences that reflect both the pedagogic and educational programmes of the fascist regime, and the discussion on the constructive and expressive principles that characterize the Italian architecture during the Thirties. The research explains the colony "XXVIII October for the sons of the Italian workers living abroad", today known as "Le Navi" ("The Ships"), built in Cattolica in 1934 and projected by the Roman architect Clemente Busiri Vici, and the feminine colony "for the sons of the Italian workers living abroad" built in 1934 in Tirrenia and projected by the architects Mario Paniconi and Giulio Pediconi. These holiday camps are the sole buildings commissioned directly by the Department in the Italian Foreign Office with the aim of offering a seaside stay to the sons of the Italians living in the colonies who, probably, could visit Italy only one time in their life. Firstly, the work illustrates the most relevant themes concerning these holiday camps, such as the representative intents that the buildings evoked to the children attending the places. Sun-bathing and group gymnastics were some of the rituals in the communal life, where order and discipline gave a precise internal organization to the spaces. Over the correspondence to practical functions, the figures and the forms of the different spaces of the buildings involve the children in an educational dimension. Subsequently, the function of the Department in the Italian Foreign Office and the planning and constructive ideas of the two colonies will be introduced. These colonies were conceived by a precise social project with educational, welfare and therapeutic aims. The elements, the spaces and the volumes create a fixed and theatrical scene of the life, full of ideological, political and celebratory overtones. Finally, the research shows that the relation between the architectural shape of the buildings and the rituals performed by fascist tutors produces an ideal space, extraneous to the external world that could influence the behavior of the children. The plan is to transmit to the children an image of Italy that will remain engraved in their minds once they have returned to their countries. In these projects there is the intent to transmit the image of "italianity" abroad. The way to do this was to plan for them a scenery which contains all the architectural elements of Italian cities. The holiday camps are proposed a sort of microcosm that appears as an "evocation" of the places and the spaces of Italian cities. The buildings appear as veritable "cities of childhood".