MBase: Representing Knowledge and Context for the Integration of Mathematical Software Systems

Abstract

In this article we describe the data model of the MBase system, a web-based, distributed mathematical knowledge base. This system is a mathematical service in MathWeb that offers a universal repository of formalized mathematics where the formal representation allows semantics-based retrieval of distributed mathematical facts.We classify the data necessary to represent mathematical knowledge and analyze its structure. For the logical formulation of mathematical concepts, we propose a methodology for developing representation formalisms for mathematical knowledge bases. We propose to concretely equip knowledge bases with a hierarchy of logical systems that are linked by logic morphisms. These mappings relativize formulae and proofs and thus support translation of the knowledge to the various formats currently in use in deduction systems. On the other hand they define higher language features from simpler ones and ultimately serve as a means to found the whole knowledge base in axiomatic set theory.The viability of this approach is proven by developing a sorted record-λ-calculus with dependent sorts and labeled abstraction that is well-suited both for formalizing mathematical practice and supporting efficient inference services. This "mathematical vernacular" is an extension of a sorted λ-calculus by records, dependent record sorts and selection sorts.

OMDoc – An Open Markup Format for Mathematical Documents [version 1.2]

“…• an interface language of a mathematical knowledge base like the MBase system [FK00,KF00]. The system offers a service that allows to store and (flexibly) reproduce (parts of) OMDoc documents.…”

A Search Engine for Mathematical Formulae

“…The second approach is taken by the MBase system [KF01], which applies the pattern matching of the underlying programming language to search for OMDoc-encoded [Koh06] mathematical documents in the knowledge base. The search engine for the Helm project indexes structural meta-data gleaned from Content MathML representations for efficient retrieval [AS04].…”

“…As we will see, this poses some problems in associating presentation (for the human reader) with the content representation. Other repositories include the ActiveMath repository [MBG + 03], or the MBase system [KF01].…”

Towards Collaborative Content Management and Version Control for Structured Mathematical Knowledge

“…As an effect, MBase/Maya support theory management on the fragment of OMDoc that corresponds to the Maya development graph. In fact, in [KF01] we have proposed to distinguish two kinds of MBases, different in their data changing policies.…”