Claudio Sacerdoti Coen

User Interaction with the Matita Proof Assistant

Matita is a new, document-centric, tactic-based interactive theorem prover. This paper focuses on some of the distinctive features of the user interaction with Matita, characterized mostly by the organization of the library as a searchable knowledge base, the emphasis on a high-quality notational rendering, and the complex interplay between syntax, presentation, and semantics.

A content based mathematical search engine: whelp

The prototype of a content based search engine for mathematical knowledge supporting a small set of queries requiring matching and/or typing operations is described. The prototype — called Whelp — exploits a metadata approach for indexing the information that looks far more flexible than traditional indexing techniques for structured expressions like substitution, discrimination, or context trees. The prototype has been instantiated to the standard library of the Coq proof assistant extended with many user contributions.

The Matita interactive theorem prover

Matita is an interactive theorem prover being developed by the Helm team at the University of Bologna. Its stable version 0.5.x may be downloaded at http://matita.cs.unibo.it. The tool originated in the European project MoWGLI as a set of XML-based tools aimed to provide a mathematician-friendly web-interface to repositories of formal mathematical knoweldge, supporting advanced content-based functionalities for querying, searching and browsing the library. It has since then evolved into a fully fledged ITP, specifically designed as a light-weight, but competitive system, particularly suited for the assessment of innovative ideas, both at foundational and logical level. In this paper, we give an account of the whole system, its peculiarities and its main applications.

Schemapath, a minimal extension to xml schema for conditional constraints

In the past few years, a number of constraint languages for XML documents has been proposed. They are cumulatively called schema languages or validation languages and they comprise, among others, DTD, XML Schema, RELAX NG, Schematron, DSD, xlinkit. One major point of discrimination among schema languages is the support of co-constraints, or co-occurrence constraints, e.g., requiring that attribute A is present if and only if attribute B is (or is not) presentin the same element. Although there is no way in XML Schema to express these requirements, they are in fact frequently used in many XML document types, usually only expressed in plain human-readable text, and validated by means of special code modules by the relevant applications. In this paper we propose SchemaPath, a light extension of XML Schema to handle conditional constraints on XML documents. Two new constructs have been added to XML Schema: conditions -- based on XPath patterns -- on type assignments for elements and attributes; and a new simple type, xsd:error, for the direct expression of negative constraints (e.g. it is prohibited for attribute A to be present if attribute B is also present). A proof-of-concept implementation is provided. A Web interface is publicly accessible for experiments and assessments of the real expressiveness of the proposed extension.

Hints in Unification

Several mechanisms such as Canonical Structures [14], Type Classes [13,16], or Pullbacks [10] have been recently introduced with the aim to improve the power and flexibility of the type inference algorithm for interactive theorem provers. We claim that all these mechanisms are particular instances of a simpler and more general technique, just consisting in providing suitable hints to the unification procedure underlying type inference. This allows a simple, modular and not intrusive implementation of all the above mentioned techniques, opening at the same time innovative and unexpected perspectives on its possible applications.

HELM and the Semantic Math-Web

The eXtensible Markup Language (XML) opens the possibility to start anew, on a solid technological ground, the ambitious goal of developing a suitable technologyf or the creation and maintenance of a virtual, distributed, hypertextual library of formal mathematical knowledge. In particular, XML provides a central technology for storing, retrieving and processing mathematical documents, comprising sophisticated web-publishing mechanisms (stylesheets) covering notational and stylistic issues. By the application of XML technology to the large repositories of structured, content oriented information offered by Logical Frameworks we meet the ultimate goal of the Semantic Web, that is to allow machines the sharing and exploitation of knowledge in the Web way, i.e. without central authority, with few basic rules, in a scalable, adaptable, extensible manner.

Tinycals: Step by Step Tacticals

Most of the state-of-the-art proof assistants are based on procedural proof languages, scripts, and rely on LCF tacticals as the primary tool for tactics composition. In this paper we discuss how these ingredients do not interact well with user interfaces based on the same interaction paradigm of Proof General (the de facto standard in this field), identifying in the coarse-grainedness of tactical evaluation the key problem. We propose Tinycals as an alternative to a subset of LCF tacticals, showing that the user does not experience the same problem if tacticals are evaluated in a more fine-grained manner. We present the formal operational semantics of tinycals as well as their implementation in the Matita proof assistant.

Crafting a proof assistant

Proof assistants are complex applications whose development has never been properly systematized or documented. This work is a contribution in this direction, based on our experience with the development of Matita: a new interactive theorem prover based--as Coq--on the Calculus of Inductive Constructions (CIC). In particular, we analyze its architecture focusing on the dependencies of its components, how they implement the main functionalities, and their degree of reusability. The work is a first attempt to provide a ground for a more direct comparison between different systems and to highlight the common functionalities, not only in view of reusability but also to encourage a more systematic comparison of different softwares and architectural solutions.

Efficient Ambiguous Parsing of Mathematical Formulae

Mathematical notation has the characteristic of being ambiguous: operators can be overloaded and information that can be deduced is often omitted. Mathematicians are used to this ambiguity and can easily disambiguate a formula making use of the context and of their ability to find the right interpretation.

On the Value of Variables

Call-by-value and call-by-need l-calculi are defined using the distinguished syntactic category of values. In theoretical studies, values are variables and abstractions. In more practical works, values are usually defined simply as abstractions. This paper shows that practical values lead to a more efficient process of substitution--for both call-by-value and call-by-need--once the usual hypothesis for implementations hold terms are closed, reduction does not go under abstraction, and substitution is done in micro steps, replacing one variable occurrence at the time. Namely, the number of substitution steps becomes linear in the number of β-redexes, while theoretical values only provide a quadratic bound.