Andrea Asperti

A web interface for matita

This article describes a prototype implementation of a web interface for the Matita proof assistant [2]. The motivations behind our work are similar to those of several recent, related efforts [7,9,1,8] (see also [6]).

A machine-checked proof of the odd order theorem

This paper reports on a six-year collaborative effort that culminated in a complete formalization of a proof of the Feit-Thompson Odd Order Theorem in the Coq proof assistant. The formalized proof is constructive, and relies on nothing but the axioms and rules of the foundational framework implemented by Coq. To support the formalization, we developed a comprehensive set of reusable libraries of formalized mathematics, including results in finite group theory, linear algebra, Galois theory, and the theories of the real and complex algebraic numbers.

Intuitionistic Light Affine Logic

This article is a structured introduction to Intuitionistic Light Affine Logic (ILAL). ILAL has a polynomially costing normalization, and it is expressive enough to encode, and simulate, all PolyTime Turing machines. The bound on the normalization cost is proved by introducing the proof-nets for ILAL. The bound follows from a suitable normalization strategy that exploits structural properties of the proof-nets. This allows us to have a good understanding of the meaning of the § modality, which is a peculiarity of light logics. The expressive power of ILAL is demonstrated in full detail. Such a proof gives a hint of the nontrivial task of programming with resource limitations, using ILAL derivations as programs.

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.

Mobile petri nets

We add mobility to Place-Transition Petri nets: tokens are names for places, and an input token of a transition can be used in its postset to specify a destination. Mobile Petri nets are then further extended to dynamic nets by adding the possibility of creating new nets during the firing of a transition. In this way, starting from Petri nets, we define a simple hierarchy of nets with increasing degrees of dynamicity. For each class in this hierarchy, we provide its encoding in the former class. Our work was largely inspired by the join-calculus of Fournet and Gonthier, which turns out to be a (well-motivated) particular case of dynamic Petri nets. The main difference is that, in the preset of a transition, we allow both non-linear patterns (name unification) and (locally) free names for input places (that is, we remove the locality constraint, and preserve reflexion).

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.

Paths, computations and labels in the l-calculus

We provide a new characterization of Levys redex-families in the λ-calculus [11] as suitable paths in the initial term of the derivation. The idea is that redexes in a same family are created by “contraction” (via β-reduction) of a unique common path in the initial term. This fact gives new evidence about the “common nature” of redexes in a same family, and about the possibility of sharing their reduction. From this point of view, our characterization underlies all recent works on optimal graph reduction techniques for the λ-calculus [9,6,7,1], providing an original and intuitive understanding of optimal implementations.

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.