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

A Foundational View on Integration Problems

The integration of reasoning and computation services across system and language boundaries is a challenging problem of computer science. In this paper, we use integration for the scenario where we have two systems that we integrate by moving problems and solutions between them. While this scenario is often approached from an engineering perspective, we take a foundational view. Based on the generic declarative language MMT, we develop a theoretical framework for system integration using theories and partial theory morphisms. Because MMT permits representations of the meta-logical foundations themselves, this includes integration across logics. We discuss safe and unsafe integration schemes and devise a general form of safe integration

Using dialogue to learn math in the LeActiveMath project

We describe a tutorial dialogue system under development that assists students in learning how to differentiate equations. The system uses deep natural language understanding and generation to both interpret students ’ utterances and automatically generate a response that is both mathematically correct and adapted pedagogically and linguistically to the local dialogue context. A domain reasoner provides the necessary knowledge about how students should approach math problems as well as their (in)correctness, while a dialogue manager directs pedagogical strategies and keeps track of what needs to be done to keep the dialogue moving along.

Technologies for teaching mathematics via the world wide web

Published ArticleThis paper tries to find answers to the question concerning the availability of suitable technologies to accommodate the teaching and learning of mathematics by means of the World Wide Web. It addresses three standards for the presentation of content mark-up and touches on the importance of adequate browser applicability with respect to MathML as one of the standards. Various tools for rendering MathML on the web, as well as plug-ins and extensions and other combinations of technologies, are discussed. The paper concludes with the introduction of a dynamic mathematics object model (DMOM) by Robert Miner from Design Science Inc. Requirements for a DMOM are formulated and its implementation is discussed

OpenMath and SMT-LIB

OpenMath and SMT-LIB are languages with very different origins, but both "represent mathematics". We describe SMT-LIB for the OpenMath community and consider adaptations for both languages to support the growing SC-Square initiative.Comment: Presented in the OpenMath 2017 Workshop, at CICM 2017, Edinburgh, U

Generic access to symbolic computing services

Symbolic computation is one of the computational domains that requires large computational resources. Computer Algebra Systems (CAS), the main tools used for symbolic computations, are mainly designed to be used as software tools installed on standalone machines that do not provide the required resources for solving large symbolic computation problems. In order to support symbolic computations an infrastructure built upon massively distributed computational environments must be developed. Building an infrastructure for symbolic computations requires a thorough analysis of the most important requirements raised by the symbolic computation world and must be built based on the most suitable architectural styles and technologies. The architecture that we propose is composed of several main components: the Computer Algebra System (CAS) Server that exposes the functionality implemented by one or more supporting CASs through generic interfaces of Grid Services; the Architecture for Grid Symbolic Services Orchestration (AGSSO) Server that allows seamless composition of CAS Server capabilities; and client side libraries to assist the users in describing workflows for symbolic computations directly within the CAS environment. We have also designed and developed a framework for automatic data management of mathematical content that relies on OpenMath encoding. To support the validation and fine tuning of the system we have developed a simulation platform that mimics the environment on which the architecture is deployed