Towards MKM in the Large: Modular Representation and Scalable Software Architecture

Kohlhase, Michael; Rabe, Florian; Zholudev, Vyacheslav

doi:10.1007/978-3-642-14128-7_32

Cited by 13 publications

(12 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The next step in our investigation will be to realize and test such support in the OMDoc/MMT [MMT;KRZ10] and the MathScheme [CFO11] systems, fully develop the examples sketched in this paper, and test the interactions on developers, students, and practitioners (see section 3).…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Realms: A Structure for Consolidating Knowledge about Mathematical Theories

Carette

Farmer

Kohlhase

2014

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract.Since there are different ways of axiomatizing and developing a mathematical theory, knowledge about a such a theory may reside in many places and in many forms within a library of formalized mathematics. We introduce the notion of a realm as a structure for consolidating knowledge about a mathematical theory. A realm contains several axiomatizations of a theory that are separately developed. Views interconnect these developments and establish that the axiomatizations are equivalent in the sense of being mutually interpretable. A realm also contains an external interface that is convenient for users of the library who want to apply the concepts and facts of the theory without delving into the details of how the concepts and facts were developed. We illustrate the utility of realms through a series of examples. We also give an outline of the mechanisms that are needed to create and maintain realms.

show abstract

Section: Resultsmentioning

confidence: 99%

“…Our work on OMDoc/MMT [MMT;KRZ10] and the MathScheme [CFO11] systems have given us a decent intuition (or so we feel) regarding the services that a theory-graph based system should provide. We now extent this to realms.…”

Section: Representing and Growing Realms In A Udlmmentioning

confidence: 99%

Realms: A Structure for Consolidating Knowledge about Mathematical Theories

Carette

Farmer

Kohlhase

2014

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…The latter include both deduction systems, such as theorem provers, proof assistants and type checkers, and management systems, such as databases, browsers, IDEs and search engines. MMT combines a modular representation format with a scalable knowledge management infrastructure (Kohlhase et al 2010) to provide an exchange format for logics, signatures and theories, as well as their morphisms, for foundations, and for expressions, proofs and models.…”

Section: Discussionmentioning

confidence: 99%

A logical framework combining model and proof theory

Rabe¹

2013

Math. Struct. Comp. Sci.

Self Cite

View full text Add to dashboard Cite

Mathematical logic and computer science have driven the design of a growing number of logics and related formalisms such as set theories and type theories. In response to this population explosion, logical frameworks have been developed as formal meta-languages in which to represent, structure, relate and reason about logics. Research on logical frameworks has diverged into separate communities, often with conflicting backgrounds and philosophies. In particular, two of the most important logical frameworks are the framework of institutions, from the area of model theory based on category theory, and the Edinburgh Logical Framework LF, from the area of proof theory based on dependent type theory. Even though their ultimate motivations overlap -for example in applications to software verification -they have fundamentally different perspectives on logic. In the current paper, we design a logical framework that integrates the frameworks of institutions and LF in a way that combines their complementary advantages while retaining the elegance of each of them. In particular, our framework takes a balanced approach between model theory and proof theory, and permits the representation of logics in a way that comprises all major ingredients of a logic: syntax, models, satisfaction, judgments and proofs. This provides a theoretical basis for the systematic study of logics in a comprehensive logical framework. Our framework has been applied to obtain a large library of structured and machine-verified encodings of logics and logic translations. † This work was developed over the course of five years, during F. Rabe 948 A similar difference in primacy is found when looking at families of logics. For example, the model-theoretical view takes precedence in description logic (see, for example, Brachman and Schmolze (1985) and Baader et al. (2003)), where the model theory is fixed and proof theory is studied chiefly to provide reasoning tools. This view is arguably paramount as it dominates accounts of (classical) first-order logic and most uses of logic in mathematics. On the other hand, the proof-theoretical view takes precedence in, for example, higher-order logic (see Church (1940) for proof theory and Henkin (1950) for model theory); and some logics such as intuitionistic logics (Brouwer 1907) are explicitly defined by their proof theory.At the level of logical frameworks, proof-theoretical principles have been integrated into model-theoretical frameworks in several ways. Parchments are used in Goguen and Burstall (1986) and Mossakowski et al. (1997) to express the syntax of a logic as a formal language. And, for example, in Meseguer (1989), Fiadeiro andSernadas (1988) and Mossakowski et al. (2005), proof theory is formalised in the abstract style of category theory. These approaches are very elegant, but often fail to exploit a major advantage of proof theory -the constructive reasoning over concrete syntax.Model-theoretical principles have also been integrated into proof-theoretical frameworks, albeit to a lesser extent. Fo...

show abstract

“…Supported knowledge repositories are file systems, SVN working copies and repositories, and TNTBase databases [12]. The latter also supports Mmt-specific indexing and querying functions [8] permitting, e.g., the efficient retrieval of the dependency closure of an Mmt knowledge item.…”

Section: Knowledge Management Servicesmentioning

confidence: 99%

The MMT API: A Generic MKM System

Rabe

2013

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract. 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.The Mmt Language Content-oriented representation languages for mathematical knowledge are usually designed to focus on either of two goals: (i) the automation potential offered by mechanically verifiable representations, as pursued in semi-automated proof assistants like Isabelle and (ii) the universal applicability offered by a generic meta-language, as pursued in XML-based content markup languages like OMDoc. The Mmt language [11] (Module system for Mathematical Theories) was designed to realize both goals in one coherent system. It uses a minimal number of primitives with a precise semantics chosen to permit natural and adequate representations of many individual languages.A key feature is foundation-independence: Mmt systematically avoids a commitment to a particular type theory or logic. Instead, it represents every formal system as an Mmt theory: domain theories (like the theory Group), logics (like first-order logic FOL), and logical frameworks (like LF [4]) are represented uniformly as Mmt theories. These theories are related by the meta-theory relation, e.g., LF is the meta-theory of FOL, which in turn is the meta-theory of Group. Mmt uses this relation to obtain the semantics of a theory from that of its metatheory; thus, an external semantics (called the foundation), e.g., a research article or an implementation, only has to be supplied for the topmost meta-theories. For example, a foundation for LF can be given in the form of a type system. Theories contain typed symbol declarations, which permit the uniform representation of constants, functions, and predicates as well as -via the CurryHoward correspondence -judgments, inference rules, axioms, and theorems. Theories are related via theory morphisms, which subsume translations, functors, and models. Finally, Mmt provides a module system for building large theories and morphisms via reuse and inheritance.

show abstract

Towards MKM in the Large: Modular Representation and Scalable Software Architecture

Cited by 13 publications

References 23 publications

Realms: A Structure for Consolidating Knowledge about Mathematical Theories

Realms: A Structure for Consolidating Knowledge about Mathematical Theories

A logical framework combining model and proof theory

The MMT API: A Generic MKM System

Contact Info

Product

Resources

About