The MMT API: A Generic MKM System

The MMT language has been developed as a scalable representation and interchange language for formal mathematical knowledge. It permits natural representations of the syntax and semantics of virtually all declarative languages while making MMT-based MKM services easy to implement. It is foundationally unconstrained and can be instantiated with specific formal languages. The MMT API implements the MMT language along with multiple backends for persistent storage and frontends for machine and user access. Moreover, it implements a wide variety of MMT-based knowledge management services. The API and all services are generic and can be applied to any language represented in MMT. A plugin interface permits injecting syntactic and semantic idiosyncrasies of individual formal languages.Comment: Conferences on Intelligent Computer Mathematics (CICM) 2013 The final publication is available at http://link.springer.com

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

Preface to the volume Languages: Bionspired Approaches

Languages, whether they be natural or artificial, are particular cases of a symbol system. And the manipulation of symbols is the stem of formal language theory. The theory of formal languages mainly originated from mathematics and generative linguistics. It was born in the middle of the 20th century as a tool for modelling and investigating the syntax of natural languages. After 1964, it developed as a separate branch with specific problems, techniques and results and since then it has had an important role in the field of computer science. Formal language theory, due to its abstract and formal properties, has been applied to a wide range of fields (besides initial linguistic motivation): economic modelling, developmental biology, cryptography, sociology... Therefore, natural languages, computer science and formal languages had a mutual influence over the years

System Design as a Creative Mathematical Activity

This paper contributes to the understanding of rational systems design and verification. We give evidence that the rôle of mathematics in development and verification is not limited to useful calculations: Ideally, designing is a creative mathematical activity, which comprises finding a theorem, if necessary strengthening its assumptions until it can be proven. A canonical form of this ‘verification theorem’ is introduced and illustrated with informal and formal examples. Although for good reasons most systems are designed without use of formal methods it may be a source of useful insight to understand all design as an ‘approximation’ of such a mathematical activity. This leads amongst others to a taxonomy of design decisions, and it may help to relate paradigms, theories, methods, languages, and tools from different areas of computer science to each other to make optimal use of them