

The 9th International Conference on Mathematical Knowledge Management
MKM 2010
Paris, France 8th9th July 2010



Conference Programme

Thursday, July 8th 2010 
Welcome (13:5014:00)

Invited Talk (14:0015:00, Chair: Patrick Ion)
14:0015:00 
Can we make Mathematics universal as well as fully reliable?
Pierre Cartier
Abstract: These are times
of change for the mathematical profession
and for the practice of mathematics, with,
for instance, the advent of powerful
computing machinery and the extraordinary
immediacy of communication all over the
world. If the concept of globalization has
any meaning, this is now the time of
globalization in mathematics. The pretense
of mathematics of being universal can now
be made real, and mathematical knowledge
be shared by people from many different
backgrounds. In addition, there is now
mechanical help in reasoning, digitization
of large parts of the traditional
mathematical literature, and amazing
amounts of scientific data and problems
begging analysis which all help to offer a
change in the role of mathematics.
Checking proofs by computers is just one
of the many new possibilities. A new
paradigm is needed to understand a world
of TERABYTES, where the frontier between
"proof" and "guess" is quite fuzzy.
Between "finite" and "infinite", we have
to introduce in our axiomatics the world
of "very very large".


Break (15:0015:30)

Session 1 (15:3016:30, Chair: Bruce Miller)
15:3016:00 
Towards MKM in the Large: Modular Representation and Scalable Software Architecture
Michael Kohlhase, Florian Rabe, Vyacheslav Zholudev
Abstract: MKM has been
defined as the quest for technologies to
manage mathematical knowledge. MKM “in
the small” is wellstudied, so the
real problem is to scale up to large, highly
interconnected corpora: “MKM in the
large”. We contend that advances in
two areas are needed to reach this goal. We
need representation languages that support
incremental processing of all primitive MKM
operations, and we need software
architectures and implementations that
implement these operations scalably on large
knowledge bases.
We present instances of both in this paper:
the MMT framework for modular theorygraphs
that integrates metalogical foundations,
which forms the base of the next OMDOC
version; and TNTBase, a versioned storage
system for XMLbased document formats.
TNTBase becomes an MMT database by
instantiating it with special MKM operations
for MMT.
[Preprint] [Springerlink]

16:0016:30 
Electronic Geometry Textbook: A Geometric Textbook Knowledge Management System
Xiaoyu Chen
Abstract: Electronic
Geometry Textbook is a knowledge management
system that manages geometric textbook
knowledge to enable users to construct and
share dynamic geometry textbooks
interactively and efficiently. Based on a
knowledge base organizing and storing the
knowledge represented in specific languages,
the system implements interfaces for
maintaining the data representing that
knowledge as well as relations among those
data, for automatically generating readable
documents for viewing or printing, and for
automatically discovering the relations
among knowledge data. An interface has been
developed for users to create geometry
textbooks with automatic checking, in real
time, of the consistency of the structure of
each resulting textbook. By integrating an
external geometric theorem prover and an
external dynamic geometry software package,
the system offers the facilities for
automatically proving theorems and
generating dynamic figures in the created
textbooks. This paper provides a
comprehensive account of the current version
of Electronic Geometry Textbook.
[Preprint] [Springerlink]


Coffee Break (16:3017:00)

Session 2 (17:0018:30, Chair: Stephen Watt)
17:0017:30 
The Formulator MathML Editor Project: UserFriendly Authoring of Content Markup Documents
Andriy Kovalchuk, Vyacheslav Levitsky, Igor Samolyuk, Valentyn Yanchuk
Abstract: Implementation of
an editing process for Content MathML
formulas in common visual style is a real
challenge for a software developer who does
not really want the user to have to
understand the structure of Content MathML
in order to edit an expression, since it is
expected that users are often not that
technically minded. In this paper, we
demonstrate how this aim is achieved in the
context of the Formulator project and
discuss features of this MathML editor,
which provides a user with a WYSIWYG editing
style while authoring MathML documents with
Content or mixed markup. We also present the
approach taken to enhance availability of
the MathML editor to endusers,
demonstrating an online version of the
editor that runs inside a Web browser.
[Preprint] [Springerlink]

17:3018:00 
A Wiki for Mizar: Motivation, Considerations, and Initial Prototype
Josef Urban, Jesse Alama, Piotr Rudnicki, Herman Geuvers
Abstract: Formal
mathematics has so far not taken full
advantage of ideas from collaborative tools
such as wikis and distributed version
control systems (DVCS). We argue that the
field could profit from such tools, serving
both newcomers and experts alike. We
describe a preliminary system for such
collaborative development based on the Git
DVCS. We focus, initially, on the Mizar
system and its library of formalized
mathematics.

18:0018:30 
Proofs, proofs, proofs, and proofs
Manfred Kerber
Abstract: In logic there is
a clear concept of what constitutes a proof
and what not. A proof is essentially defined
as a finite sequence of formulae which are
either axioms or derived by proof rules from
formulae earlier in the sequence.
Sociologically, however, it is more
difficult to say what should constitute a
proof and what not. In this paper we will
look at different forms of proofs and try to
clarify the concept of proof in the wider
meaning of the term. This has implications
on how proofs should be represented
formally.
[Preprint] [Springerlink]


Friday, July 9th 2010 
Session 3 (09:0010:30, Chair: Michael Kohlhase)
09:0009:30 
Evidence Algorithm and System for Automated Deduction: A Retrospective View
Alexander Lyaletski, Konstantin Verchinine
Abstract: A research
project aimed at the development of an
automated theorem proving system was started
in Kiev (Ukraine) in early 1960s. The
mastermind of the project, Academician
V.Glushkov, baptized it “Evidence
Algorithm“, EA. The work on the
project lasted, off and on, more than 40
years. In the framework of the project, the
Russian and English versions of the System
for Automated Deduction, SAD, were
constructed. They may be already seen as
powerful theorem proving assistants. The
paper gives a retrospective view to the
whole history of the development of the EA
and SAD. Theoretical and practical results
obtained on the long way are systematized.
No comparison with similar projects is made.
[Preprint] [Springerlink]

09:3010:00 
On Building a Knowledge Base for Stability Theory
Agnieszka RowinskaSchwarzweller, Christoph Schwarzweller
Abstract: A lot of
mathematical knowledge has been formalized
and stored in repositories by now: Different
mathematical theorems and theories have been
taken into consideration and included in
mathematical repositories. Applications more
distant from pure mathematics, however
— though based on these theories
— often need more detailed knowledge
about the underlying theories. In this paper
we present an example Mizar formalization
from the area of electrical engineering
focusing on stability theory which is based
on complex analysis. We discuss what kind of
special knowledge is necessary and which
amount of this knowledge is included in
existing repositories.
[Preprint] [Springerlink]

10:0010:30 
On Duplication in Mathematical Repositories
Adam Grabowski, Christoph Schwarzweller
Abstract: Building a
repository of proofchecked mathematical
knowledge is without any doubt a lot of
work, and besides the actual formalization
process there is also the task of
maintaining the repository. Thus it seems
obvious to keep a repository as small as
possible, in particular each piece of
mathematical knowledge should be formalized
only once. In this paper, however, we claim
that it might be reasonable or even
necessary to duplicate knowledge in a
mathematical repository. We analyze
different situations and reasons for doing
so, provide a number of examples supporting
our thesis and discuss some implications for
building mathematical repositories.
[Preprint] [Springerlink]


Coffee Break (10:3011:00)

Session 4 (11:0012:30, Chair: Volker Sorge)
11:0011:30 
An OpenMath Content Dictionary for Tensor Concepts
Joseph Collins
Abstract: We introduce a new
OpenMath content dictionary named
“tensor1” containing symbols for the
expression of tensor formulas. These symbols
support the expression of nonCartesian
coordinates and invariant, multilinear
expressions in the context of coordinate
transformations. While current OpenMath symbols
support the expression of linear algebra
formulas using matrices and vectors, we find
that there is an underlying assumption of
Cartesian, or standard, coordinates that makes
the expression of general tensor formulas
difficult, if not impossible. In introducing
these new OpenMath symbols for the expression of
tensor formulas, we attempt to maintain, as much
as possible, consistency with prior OpenMath
symbol definitions for linear algebra.
[Preprint] [Springerlink]

11:3012:00 
Smart matching
Andrea Asperti, Enrico Tassi
Abstract: One of the most
annoying aspects in the formalization of
mathematics is the need of transforming notions to
match a given, existing result. This kind of
transformations, often based on a conspicuous
background knowledge in the given scientific
domain (mostly expressed in the form of equalities
or isomorphisms), are usually implicit in the
mathematical discourse, and it would be highly
desirable to obtain a similar behaviour in
interactive provers. The paper describes the
superpositionbased implementation of this feature
inside the Matita interactive theorem prover,
focusing in particular on the so called smart
application tactic, supporting smart matching
between a goal and a given result.
[Preprint] [Springerlink]

12:0012:30 
Proviola: a Tool for Proof Reanimation
Carst Tankink, James McKinna, Herman Geuvers, Freek Wiedijk
Abstract: To improve on existing
models of interaction with a proof assistant (PA),
in particular for storage and replay of proofs, we
introduce three related concepts, those of: a
proof movie, consisting of frames which record
both user input and the corresponding PA response;
a camera, which films a user’s interactive
session with a PA as a movie; and a proviola,
which replays a movie framebyframe to a third
party. In this paper we describe the movie data
structure and we discuss a prototype
implementation of the camera and proviola based on
the ProofWeb system. ProofWeb uncouples the
interaction with a PA via a webinterface (the
client) from the actual PA that resides on the
server. Our camera films a movie by
“listening” to the ProofWeb
communication.
The first reason for developing movies is to
uncouple the reviewing of a formal proof from the
PA used to develop it: the movie concept enables
users to discuss small code fragments without the
need to install the PA or to load a whole library
into it.
Other advantages include the possibility to
develop a separate commentary track to discuss or
explain the PA interaction. We assert that a
combined camera+proviola provides a generic layer
between a client (user) and a server (PA). Finally
we claim that movies are the right type of data to
be stored in an encyclopedia of formalized
mathematics, based on our experience in filming
the Coq standard library.
[Preprint] [Springerlink]


Lunch (12:3014:00)

Invited Talk (14:0015:00, Chair: Alan Sexton)
14:0015:00 
Against Rigor
Doron Zeilberger
Abstract:The ancient
Greeks gave (western) civilization quite a
few gifts, but we should beware of Greeks
bearing gifts. The gifts of theatre and
democracy were definitely good ones, and
perhaps even the gift of philosophy, but
the "gift" of the socalled "axiomatic
method" and the notion of "rigorous" proof
did much more harm than good. If we want
to maximize Mathematical Knowledge, and
its Management, we have to return to
Euclid this dubious gift, and giveup our
fanatical insistence on perfect rigor. Of
course, we should not go to the other
extreme, of demanding that everything
should be nonrigorous. We should
encourage diversity of proofstyles and
rigor levels, and remember that nothing is
absolutely sure in this world, and there
does not exist an absolutely rigorous
proof, nor absolute certainty, and "truth"
has many shades and levels.


Break (15:0015:20)

Session 5 (15:2016:30, Chair: Andrea Asperti)
15:2015:40 
Towards Automatic Formalization of Informal Mathematics with MathNat (presentation only)
Muhammad Humayoun and Christophe Raffalli
Abstract:Can we build a
program that understands informal
mathematical text and can we mechanically
verify it's correctness? MathNat project
aims at being a first step towards answering
this question.
We develop a controlled language for
mathematics (CLM), which is a precisely
defined subset of English with restricted
grammar and dictionary. Like textbook
mathematics, CLM supports some complex
linguistic features to make it natural and
expressive.
For formalisation, we automatically
translate CLM into a systemindependent
formal language (MathAbs). MathAbs can
represent most of informal mathematical
text. It also preserves the logical
structure, proof steps and line of reasoning
of a given math text. For verification,
which is the second part of the question, we
have done very limited work so far.
Therefore we answer it partially and it
needs to be explored further.
In this paper, we mainly focus on
formalisation by giving a formal definition
of MathAbs and by describing the algorithm
which supports proofbycase method in CLM.
Further, we validate our work with two
examples.

15:4016:00 
Integrating multiple sources to answer questions in Algebraic Topology (short paper)
Jönathan Heras, Vico Pascual, Ana Romero, Julio Rubio
Abstract: We present in
this paper an evolution of a tool from a
user interface for a concrete Computer
Algebra system for Algebraic Topology (the
Kenzo system), to a frontend allowing the
interoperability among different sources for
computation and deduction. The architecture
allows the system not only to interface
several systems, but also to make them
cooperate in shared calculations.
[Preprint] [Springerlink]

16:0016:30 
Dimensions of Formality: A Case Study for MKM in Software Engineering
Andrea Kohlhase, Michael Kohlhase, Christoph Lange
Abstract: We study the
formalization of a collection of documents
created for a Software Engineering project
from an MKM perspective. We analyze how
document and collection markup formats can
cope with an openended, multidimensional
space of primary and secondary
classifications and relationships.We show
that RDFabased extensions of MKM formats,
employing flexible “metadata”
relationships referencing specific
vocabularies for distinct dimensions, are
wellsuited to encode this and to put it
into service. This formalized knowledge can
be used for enriching interactive document
browsing, for enabling multidimensional
metadata queries over documents and
collections, and for exporting Linked Data
to the Semantic Web and thus enabling
further reuse.
[Preprint] [Springerlink]


Coffee Break (16:3017:00)

Session 6 (17:0018:30, Chair: William Farmer)
17:0017:30 
Adapting Mathematical Domain Reasoners
Bastiaan Heeren, Johan Jeuring
Abstract: Mathematical
learning environments help students in
mastering mathematical knowledge. Mature
environments typically offer thousands of
interactive exercises. Providing feedback to
students solving interactive exercises
requires domain reasoners for doing the
exercisespecific calculations. Since a
domain reasoner has to solve an exercise in
the same way a student should solve it, the
structure of domain reasoners should follow
the layered structure of the mathematical
domains. Furthermore, learners, teachers,
and environment builders have different
requirements for adapting domain reasoners,
such as providing more details, disallowing
or enforcing certain solutions, and
combining multiple mathematical domains in a
new domain. In previous work we have shown
how domain reasoners for solving interactive
exercises can be expressed in terms of
rewrite strategies, rewrite rules, and
views. This paper shows how users can adapt
and configure such domain reasoners to their
own needs. This is achieved by enabling
users to explicitly communicate the
components that are used for solving an
exercise.
[Preprint] [Springerlink]

17:3018:00 
sTeXIDE: An Integrated Development Environment for sTeX Collections
Constantin Jucovschi, Michael Kohlhase
Abstract: Authoring
documents in MKM formats like OMDoc is a
very tedious task. After years of working on
a semantically annotated corpus of STEX
documents (GenCS), we identified a set of
common, timeconsuming subtasks, which can
be supported in an integrated authoring
environment. We have adapted the
modular Eclipse IDE into STEXIDE, an
authoring solution for enhancing
productivity in contributing to STEX based
corpora. STEXIDE supports contextaware
command completion, module management,
semantic macro retrieval, and theory graph
navigation.
[Preprint] [Springerlink]

18:0018:30 
Notations Around the World: Census and Exploitation
Paul Libbrecht
Abstract: Mathematical
notations around the world are diverse. Not
as much as requiring computing machines'
makers to adapt to each culture, but as much
as to disorient a person landing on a
webpage with a text in mathematics.
In order to understand better this
diversity, we are building a census of
notations: it should allow any content
creator or mathematician to grasp which
mathematical notation is used in which
language and culture. The census is built
collaboratively, collected in pages with a
given semantic and presenting observations
of the widespread notations being used in
existing materials by a graphical extract.
We contend that our approach should
dissipate the fallacies found here and there
about the notations in “other
cultures” so that a better
understanding of the cultures can be
realized. The exploitation of the
census in the mathbridge project is also
presented: this project aims at taking
learners “where they are in their
mathknowledge” and bring them to a
level ready to start engineering studies.
The census serves as definitive reference
for the transformation elements that
generate the rendering of formulae in
webbrowsers.
[Preprint] [Springerlink]


Closing Remarks (18:3018:40)

MKM Business Meeting (18:4019:30)


