Dennis Müller

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.

Mathematical Models as Research Data via Flexiformal Theory Graphs

Mathematical modeling and simulation (MMS) has now been established as an essential part of the scientific work in many disciplines. It is common to categorize the involved numerical data and to some extent the corresponding scientific software as research data. But both have their origin in mathematical models, therefore any holistic approach to research data in MMS should cover all three aspects: data, software, and models. While the problems of classifying, archiving and making accessible are largely solved for data and first frameworks and systems are emerging for software, the question of how to deal with mathematical models is completely open.

Classification of Alignments Between Concepts of Formal Mathematical Systems

Mathematical knowledge is publicly available in dozens of different formats and languages, ranging from informal (e.g. Wikipedia) to formal corpora (e.g., Mizar). Despite an enormous amount of overlap between these corpora, only few machine-actionable connections exist. We speak of alignment if the same concept occurs in different libraries, possibly with slightly different names, notations, or formal definitions. Leveraging these alignments creates a huge potential for knowledge sharing and transfer, e.g., integrating theorem provers or reusing services across systems. Notably, even imperfect alignments, i.e. concepts that are very similar rather than identical, can often play very important roles. Specifically, in machine learning techniques for theorem proving and in automation techniques that use these, they allow learning-reasoning based automation for theorem provers to take inspiration from proofs from different formal proof libraries or semi-formal libraries even if the latter is based on a different mathematical foundation. We present a classification of alignments and design a simple format for describing alignments, as well as an infrastructure for sharing them. We propose these as a centralized standard for the community. Finally, we present an initial collection of (approx )12000 alignments from the different kinds of mathematical corpora, including proof assistant libraries and semi-formal corpora as a public resource.

Making PVS Accessible to Generic Services by Interpretation in a Universal Format

PVS is one of the most powerful proof assistant systems and its libraries of formalized mathematics are among the most comprehensive albeit under-appreciated ones. A characteristic feature of PVS is the use of a very rich mathematical and logical foundation, including e.g., record types, undecidable subtyping, and a deep integration of decision procedures. That makes it particularly difficult to develop integrations of PVS with other systems such as other reasoning tools or library management periphery.

Knowledge-based interoperability for mathematical software systems

There is a large ecosystem of mathematical software systems. Individually, these are optimized for particular domains and functionalities, and together they cover many needs of practical and theoretical mathematics. However, each system specializes on one particular area, and it remains very difficult to solve problems that need to involve multiple systems. Some integrations exist, but the are ad-hoc and have scalability and maintainability issues. In particular, there is not yet an interoperability layer that combines the various systems into a virtual research environment (VRE) for mathematics.

Alignment-based Translations Across Formal Systems Using Interface Theories.

Translating expressions between different logics and theorem provers is notoriously and often prohibitively difficult, due to the large differences between the logical foundations, the implementations of the systems, and the structure of the respective libraries. Practical solutions for exchanging theorems across theorem provers have remained both weak and brittle. Consequently, libraries are not easily reusable across systems, and substantial effort must be spent on reformalizing and proving basic results in each system. Notably, this problem exists already if we only try to exchange theorem statements and forgo exchanging proofs. nIn previous work we introduced alignments as a lightweight standard for relating concepts across libraries and conjectured that it would provide a good base for translating expressions. In this paper, we demonstrate the feasibility of this approach. We use a foundationally uncommitted framework to write interface theories that abstract from logical foundation, implementation, and library structure. Then we use alignments to record how the concepts in the interface theories are realized in several major proof assistant libraries, and we use that information to translate expressions across libraries. Concretely, we present exemplary interface theories for several areas of mathematics and - in total - several hundred alignments that were found manually.

Theories as Types

Theories are an essential structuring principle that enable modularity, encapsulation, and reuse in formal libraries and programs (called classes there). Similar effects can be achieved by dependent record types. While the former form a separate language layer, the latter are a normal part of the type theory. This overlap in functionality can render different systems non-interoperable and lead to duplication of work.

Automatically Finding Theory Morphisms for Knowledge Management

We present a method for finding morphisms between formal theories, both within as well as across libraries based on different logical foundations. As they induce new theorems in the target theory for any of the source theory, theory morphisms are high-value elements of a modular formal library. Usually, theory morphisms are manually encoded, but this practice requires authors who are familiar with source and target theories at the same time, which limits the scalability of the manual approach.

Mathematical models as research data in numerical simulation of opto-electronic devices

Mathematical models are the foundation of numerical simulation of optoelectronic devices. We present a concept for a machine-actionable as well as human-understandable representation of the mathematical knowledge they contain and the domain-specific knowledge they are based on. We propose to use theory graphs to formalize mathematical models and model pathway diagrams to visualize them. We illustrate our approach by application to the stationary one-dimensional drift-diffusion equations (van Roosbroeck system).