A Universal Machine for Biform Theory Graphs

Broadly speaking, there are two kinds of semantics-aware assistant systems for mathematics: proof assistants express the semantic in logic and emphasize deduction, and computer algebra systems express the semantics in programming languages and emphasize computation. Combining the complementary strengths of both approaches while mending their complementary weaknesses has been an important goal of the mechanized mathematics community for some time. We pick up on the idea of biform theories and interpret it in the MMTt/OMDoc framework which introduced the foundations-as-theories approach, and can thus represent both logics and programming languages as theories. This yields a formal, modular framework of biform theory graphs which mixes specifications and implementations sharing the module system and typing information. We present automated knowledge management work flows that interface to existing specification/programming tools and enable an OpenMath Machine, that operationalizes biform theories, evaluating expressions by exhaustively applying the implementations of the respective operators. We evaluate the new biform framework by adding implementations to the OpenMath standard content dictionaries.Comment: Conferences on Intelligent Computer Mathematics, CICM 2013 The final publication is available at http://link.springer.com

Integrating multiple sources to answer questions in Algebraic Topology

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 front-end 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.Comment: To appear in The 9th International Conference on Mathematical Knowledge Management: MKM 201

Knowledge-based interoperability for mathematical software systems

Funding: OpenDreamKit Horizon 2020 European Research Infrastructures project (#676541) and DFG project RA-18723-1 OAF.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 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. The OpenDreamKit project aims at building a toolkit for such VREs. It suggests using a central system-agnostic formalization of mathematics (Math-in-the-Middle, MitM) as the needed interoperability layer. In this paper, we conduct the first major case study that instantiates the MitM paradigm for a concrete domain as well as a concrete set of systems. Specifically, we integrate GAP, Sage, and Singular to perform computation in group and ring theory. Our work involves massive practical efforts, including a novel formalization of computational group theory, improvements to the involved software systems, and a novel mediating system that sits at the center of a star-shaped integration layout between mathematical software systems.Postprin

Linking HOL Light to Mathematica using OpenMath

One of the most important benefits of using a theorem prover system is the absolute accuracy of the obtained result. However, solving mathematical problems often requires both deductive reasoning and algebraic computation. This issue is due to the fact that many real-life problems can be described with equations for which we cannot find easily symbolic (or closed-form) solutions and therefore we are not able to formalize them using the theorem prover. In other cases, some applications require well developed libraries and a deep knowledge of the theories to formalize simple expressions. A straightforward way to overcome these issues is the use of computer algebra systems or numerical approaches which are known to be the most efficient tools in symbolic computation. However, to preserve the soundness of the computation, the results of these systems should be formally verified. In this thesis, we present a general architecture to connect HOL Light, a higher-order logic theorem prover, to any mechanized mathematical system that supports the mathematical standard OpenMath. We implemented a prototype, called HolMatica, which links HOL Light to the computer algebra system Mathematica through OpenMath. We describe our implementation of a HOL Light translator which converts HOL Light statements into OpenMath object and vice-versa

The Planetary System: Web 3.0 Active Documents for STEM

AbstractIn this paper we present the Active Documents Paradigm (semantically annotated documents associated with a content commons that holds the corresponding background ontologies) and the Planetary system (as an active document player). We show that the current Planetary system gives a solid foundation and can be extended modularly to address most of the criteria of the Executable Papers Challenge