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Computer-Assisted Reasoning Based on Type Theory

Objective

The subject of this project is Computer-Assisted Formal Reasoning, an area of vital concern for industry. The aim of our research activities is to develop the technology of formal reasoning based on Type Theory by improving the languages and tools of reasoning and by applying the technology in several domains such as programming languages, certified software, and formalisation of mathematics. The proposal is based on strong collaboration and achievements in three successful European projects (ESPRIT BRA and Working Group), in which we have built several computer systems for proof development and used them in applications. Our consortium defines the state of the art in type theory and its applications. The proposed Working Group is essential to keep our community together so that the sites doing related and complementary work can continue to collaborate fruitfully. The subject of this project is Computer-Assisted Formal Reasoning, an area of vital concern for industry. The aim of our research activities is to develop the technology of formal reasoning based on Type Theory by improving the languages and tools of reasoning and by applying the technology in several domains such as programming languages, certified software, and formalisation of mathematics. The proposal is based on strong collaboration and achievements in three successful European projects (ESPRIT BRA and Working Group), in which we have built several computer systems for proof development and used them in applications. Our consortium defines the state of the art in type theory and its applications. The proposed Working Group is essential to keep our community together so that the sites doing related and complementary work can continue to collaborate fruitfully.

OBJECTIVES
The subject of this project is Computer-Assisted Formal Reasoning, an area of vital concern for industry. Our research aims to develop the technology of formal reasoning based on Type Theory. In past EU projects (ESPRIT BRA and Working Group) we have made significant progress in developing type theory and tools for formal reasoning. These have been successfully used in applications, but have been limited by lack of adequate support for some aspects of formal reasoning. We will now focus on: raising tee level of formal discourse, improving our formal vernacular and implementations to capture more of the richness of conventional mathematical language, and applications which test the utility of the formal vernacular in areas that have defied adequate formal treatment to date.

DESCRIPTION OF WORK
Our research themes may be classified into two broad areas: raising the level of formal discourse, and applications of the improved formal vernacular to domains of interest. These areas are mutually supporting: existing applications of computer aided proof are limited by expressiveness of the formal vernacular, and utility of a formal vernacular can only be determined in practice, through applications.

Raising the level of formal discourse will be addressed in four ways:
- Proof in the large: Structuring mathematical knowledge and the process of formal development to support building, maintenance and use of an ever expanding base of formal knowledge;
-Automation: Large parts of most proofs can be mechanically filled in once a user has given the main steps;
-User interfaces: Current user interfaces lack the freedom of pencil and paper, but improved interfaces could provide much assistance that pencil and paper cannot, for both novice and skilled users;
-Mathematical vernacular: Other spects of mathematical discourse, especially representation of informal ideas in the formal language.

Application areas of special interest include:
-Formal mathematics: Building libraries of abstract mathematics to support all kinds of formal reasoning. Interactive mathematical "textbooks" based on such libraries is a challenging application;
-Programming languages and certified software: Work by our community has contributed to formal understanding of widely used languages (e.g. Java) and commercially important software (e.g. security protocols). Programme extraction from proofs, and programming directly with dependent types are other approaches to certified software, ideally suited for type theory tools.

Funding Scheme

THN - Thematic network contracts
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Coordinator

UNIVERSITY OF DURHAM
Address
University Office
DH1 3HP Durham
United Kingdom

Participants (16)

CHALMERS TEKNISKA HOEGSKOLA AKTIEBOLAG
Sweden
Address

412 96 Goeteborg
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
France
Address
Domaine De Voluceau
78153 Le Chesnay
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
France
Address
Domaine De Voluceau
78153 Le Chesnay
LUDWIG-MAXIMILIANS-UNIVERSITAET MUENCHEN
Germany
Address
Geschwister-scholl-platz 1
80539 Muenchen
QUEEN MARY AND WESTFIELD COLLEGE, UNIVERSITY OF LONDON
United Kingdom
Address
Mile End Road
E1 4NS London
STICHTING KATHOLIEKE UNIVERSITEIT
Netherlands
Address
Geert Grooteplein-noord 9
6525 EZ Nijmegen
TECHNISCHE UNIVERSITAET MUENCHEN
Germany
Address
Arcisstrasse 21
80333 Muenchen
THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE
United Kingdom
Address
The Old Schools, Trinity Lane
CB2 1TS Cambridge
THE VICTORIA UNIVERSITY OF MANCHESTER
United Kingdom
Address
Oxford Road
M13 9PL Manchester
The University of Edinburgh
United Kingdom
Address
Old College, South Bridge
EH8 9YL Edinburgh
UNIVERSITA DEGLI STUDI DI TORINO
Italy
Address
Via Giuseppe Verdi 8
10124 Torino
UNIVERSITA DEGLI STUDI DI UDINE
Italy
Address
Via Palladio 8
33100 Udine
UNIVERSITE DE PARIS XI PARIS-SUD
France
Address
15, Rue Georges Clemenceau
91405 Orsay Cedex
UNIVERSITE PARIS 7 DENIS DIDEROT
France
Address
Place Jussieu 2
75251 Paris Cedex 05
UNIWERSYTET W BIALYMSTOKU
Poland
Address
Marii Sklodowskiej-curie 14
15097 Bialystok
UNIWERSYTET WARSZAWSKI
Poland
Address
Krakowskie Przedmiescie 26/28
00-927 Warszawa