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

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

Integrated semantic math I/O in ActiveMath: an evaluation

The ActiveMath system is a web-based learning environment that integrates static mathematical content and interactive exercises with evaluated mathematical input from learners. Mathematical formulæ in ActiveMath are encoded in OpenMath and presented with regional notations. Users can input formulæ using the same notations via a formula editor or using plain-text input. Input to the editor is assisted by allowing users to copy formulæ from other parts of ActiveMath. In this paper we will describe how all these components are integrated and work within the system. We will then discuss recent evaluations of the formulæ input methods run within the LeActiveMath project in Malaga and Edinburgh. The results indicate that, even though the assisted input methods provided by the Formula Editor and copy-andpaste are appreciated by users the most popular input method remains the plain text input fields. Proposals are made for how direct input of text can be facilitated and assisted in future formulæ input systems

The GF Mathematical Grammar Library: from OpenMath to natural languages

Postprint (published version

Integrated Semantic Math I/O in ActiveMath: an Evaluation

The ActiveMath system is a web-based learning environment that integrates static mathematical content and interactive exercises with evaluated mathematical input from learners. Mathematical formulæ in ActiveMath are encoded in OpenMath and presented with regional notations. Users can input formulæ using the same notations via a formula editor or using plain-text input. Input to the editor is assisted by allowing users to copy formulæ from other parts of ActiveMath. In this paper we will describe how all these components are integrated and work within the system. We will then discuss recent evaluations of the formulæ input methods run within the LeActiveMath project in Malaga and Edinburgh. The results indicate that, even though the assisted input methods provided by the Formula Editor and copy-andpaste are appreciated by users the most popular input method remains the plain text input fields. Proposals are made for how direct input of text can be facilitated and assisted in future formulæ input systems

Functional declarative language design and predicate calculus: A practical approach

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