Mihnea Iancu

The Mizar Mathematical Library in OMDoc: Translation and Applications

The Mizar Mathematical Library is one of the largest libraries of formalized and mechanically verified mathematics. Its language is highly optimized for authoring by humans. As in natural languages, the meaning of an expression is influenced by its (mathematical) context in a way that is natural to humans, but harder to specify for machine manipulation. Thus its custom file format can make the access to the library difficult. Indeed, the Mizar system itself is currently the only system that can fully operate on the Mizar library. This paper presents a translation of the Mizar library into the OMDoc format (Open Mathematical Documents), an XML-based representation format for mathematical knowledge. OMDoc is geared towards machine support and interoperability by making formula structure and context dependencies explicit. Thus, the Mizar library becomes accessible for a wide range of OMDoc-based tools for formal mathematics and knowledge management. We exemplify interoperability by indexing the translated library in the MathWebSearch engine, which provides an “applicable theorem search” service (almost) out of the box.

Formalising foundations of mathematics

Over recent decades there has been a trend towards formalised mathematics, and a number of sophisticated systems have been developed both to support the formalisation process and to verify the results mechanically. However, each tool is based on a specific foundation of mathematics, and formalisations in different systems are not necessarily compatible. Therefore, the integration of these foundations has received growing interest. We contribute to this goal by using LF as a foundational framework in which the mathematical foundations themselves can be formalised and therefore also the relations between them. We represent three of the most important foundations – Isabelle/HOL, Mizar and ZFC set theory – as well as relations between them. The relations are formalised in such a way that the framework permits the extraction of translation functions, which are guaranteed to be well defined and sound. Our work provides the starting point for a systematic study of formalised foundations in order to compare, relate and integrate them.

Interoperability in the OpenDreamKit Project: The Math-in-the-Middle Approach

OpenDreamKit – “Open Digital Research Environment Toolkit for the Advancement of Mathematics” – is an H2020 EU Research Infrastructure project that aims at supporting, over the period 2015–2019, the ecosystem of open-source mathematical software systems. OpenDreamKit will deliver a flexible toolkit enabling research groups to set up Virtual Research Environments, customised to meet the varied needs of research projects in pure mathematics and applications.

System Description: MathHub.info

We present the MathHub.info system, a development environment for active mathematical documents and an archive for flexiformal mathematics. It offers a rich interface for reading, writing, and interacting with mathematical documents and knowledge. The core of the MathHub.info system is an archive for flexiformal mathematical documents and libraries in the OMDoc/MMT format. Content can be authored or archived in the source format of the respective system, is versioned in GIT repositories, and transformed into OMDoc/MMT for machine-support and further into HTML5 for reading and interaction.

Management of change in declarative languages

Due to the high degree of interconnectedness of formal mathematical statements and theories, human authors often have difficulties anticipating and tracking the effects of a change in large bodies of symbolic mathematical knowledge. Therefore, the automation of change management is often desirable. But while computers can in principle detect and propagate changes automatically, this process must take the semantics of the underlying mathematical formalism into account. Therefore, concrete management of change solutions are difficult to realize. The Mmt language was designed as a generic declarative language that captures universal structural features while avoiding a commitment to a particular formalism. Therefore, it provides a promising framework for the systematic study of changes in declarative languages. We leverage this framework by providing a generic change management solution at the Mmt level, which can be instantiated for arbitrary specific languages.

Representing, Archiving, and Searching the Space of Mathematical Knowledge

There is an interesting duality between the forms and extents of mathematical knowledge that is verbally expressed (published in articles, scribbled on blackboards, or presented in talks/discussions) and the forms that are needed to successfully extend and apply mathematics. To “do mathematics”, we need to judge the veracity, extract the relevant structures, and reconcile them with the context of our existing knowledge - recognizing parts as already known and identifying those that are new to us. In this process we may abstract from syntactic differences, and even employ interpretations via non-trivial mappings as long as they are meaning-preserving.

Math Literate Knowledge Management via Induced Material

Mathematicians integrate acquired knowledge into a mental model. For trained mathematicians, the mental model seems to include not just the bare facts, but various induced forms of knowledge, and the amount of this and the ability to perform all reasoning and knowledge operations taking that into account can be seen as a measure of mathematical training and literacy. Current MKM systems only act on the bare facts given to them; we contend that they --- their users actuallyi¾?--- would profit from a good dose of mathematical literacy so that they can better complement the abilities of human mathematicians and thus enhance their productivity. In this paper we discuss how we can model induced knowledge naturally in highly modular, theory-graph based, mathematical libraries and establish how to access it to make it available for applications, creating a form of mathematical literacy. We show two examples of math-literate MKM systems --- searching for induced statements and accessing a knowledge via induced theories --- to show the utility of the approach.

The SMGloM Project and System: Towards a Terminology and Ontology for Mathematics

Mathematical vernacular – the everyday language we use to communicate about mathematics is characterized by a special vocabulary. If we want to support humans with mathematical documents, we need to extract their semantics and for that we need a resource that captures the terminological, linguistic, and ontological aspects of the mathematical vocabulary. In the SMGloM project and system, we aim to do just this. We present the glossary system prototype, the content organization, and the envisioned community aspects.

Discourse-Level Parallel Markup and Meaning Adoption in Flexiformal Theory Graphs

Representation formats based on theory graphs have been successful in formalized mathematics as they provide valuable logic-compatible modularity and foster reuse. Theories - sets of symbols and axioms – serve as modules and theory morphisms - truth-preserving mappings from the (language of the) source theory to the target theory – formalize inheritance and applicability of theorems. The MMT [4] system re-developed the formal part of the OMDoc theory graph into a foundation-independent meta-system for formal mathematics and implemented it in the MMT API.