A case study on formalizing algebra in a module system

Dumbrava, Stefania; Horozal, Fulya; Sojakova, Kristina

doi:10.1145/1735813.1735816

Cited by 3 publications

(4 citation statements)

References 8 publications

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“…However, it does not support theory expressions. Apart from inclusions, all morphisms need to be given manually - [25] shows some examples. Many of the scaling problems that we have identified are still present.…”

Section: Discussionmentioning

confidence: 99%

Building on the Diamonds between Theories: Theory Presentation Combinators

Carette,

O'Connor,

Sharoda

2018

Preprint

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To build a large library of mathematics, it seems more efficient to take advantage of the inherent structure of mathematical theories. Various theory presentation combinators have been proposed, and some have been implemented, in both legacy and current systems. Surprisingly, the "standard library" of most systems do not make pervasive use of these combinators.We present a set of combinators optimized for reuse, via the tiny theories approach. Our combinators draw their power from the inherent structure already present in the category of contexts associated to a dependently typed language. The current work builds on ideas originating in CLEAR and Specware and their descendents (both direct and intellectual). Driven by some design criteria for user-centric library design, our library-building experience via the systematic use of combinators has fed back into the semantics of these combinators, and later into an updated syntax for them.

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Section: Discussionmentioning

confidence: 99%

Building on the Diamonds between Theories: Theory Presentation Combinators

Carette,

O'Connor,

Sharoda

2018

Preprint

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“…A major advantage of this approach is that authors can benefit from the advanced Twelf features, in particular infix parsing, type reconstruction, and implicit arguments. This implementation was used successfully to generate large case studies of Mmt theory graphs in [DHS09], [HR11], and [IR11].…”

Section: Twelf -An Mmt-compliant Logical Frameworkmentioning

confidence: 99%

A scalable module system

Rabe

Kohlhase

2013

Information and Computation

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Symbolic and logic computation systems ranging from computer algebra systems to theorem provers are finding their way into science, technology, mathematics and engineering. But such systems rely on explicitly or implicitly represented mathematical knowledge that needs to be managed to use such systems effectively.While mathematical knowledge management (MKM) "in the small " is wellstudied, scaling up to large, highly interconnected corpora remains difficult. We hold that in order to realize MKM "in the large", we need representation languages and software architectures that are designed systematically with largescale processing in mind.Therefore, we have designed and implemented the Mmt language -a module system for mathematical theories. Mmt is designed as the simplest possible language that combines a module system, a foundationally uncommitted formal semantics, and web-scalable implementations. Due to a careful choice of representational primitives, Mmt allows us to integrate existing representation languages for formal mathematical knowledge in a simple, scalable formalism. In particular, Mmt abstracts from the underlying mathematical and logical foundations so that it can serve as a standardized representation format for a formal digital library. Moreover, Mmt systematically separates logic-dependent and logic-independent concerns so that it can serve as an interface layer between computation systems and MKM systems. 1 We have already solved the integration of formal and informal mathematical knowledge in the OMDoc format, whose formal part is a predecessor of the work presented in this paper. We plan to integrate this solution with the much stronger formal basis of Mmt in the future.Corresponding to the notions of structural and semantic validation, we can define structural and semantic equivalence of theory graphs:Definition 34. Relative to a fixed foundation, two well-formed theory graphs γ and γ ′ are called structurally equivalent if the following holds:• γ > T = { } iff γ ′ > T = { }, and in that case T has meta-theory M in γ iff it does so in γ ′ ,• γ ≫ l : S → T = iff γ ′ ≫ l : S → T = ,

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“…For example, consider the theory graph in Fig. 1 for a portion of algebra, which was formalized in MMT in [9]. It defines the theory of magmas (A magma has a binary operation without axioms.)…”

Section: Module Systemmentioning

confidence: 99%

“…Case Study 1: The formalization in [9] uses the Twelf module system ( [31]), which is a special case of MMT. Twelf automatically computes the flattened theory graph.…”

Section: Module Systemmentioning

confidence: 99%

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

Kohlhase

Rabe

Zholudev

2010

Lecture Notes in Computer Science

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MKM has been defined as the quest for technologies to manage mathematical knowledge. MKM "in the small" is well-studied, so the real problem is to scale up to large, highly interconnected corpora: "MKM in the large". We contend that advances in two areas are needed to reach this goal. We need representation languages that support incremental processing of all primitive MKM operations, and we need software architectures and implementations that implement these operations scalably on large knowledge bases. We present instances of both in this paper: the MMT framework for modular theory-graphs that integrates meta-logical foundations, which forms the base of the next OMDOC version; and TNTBase, a versioned storage system for XMLbased document formats. TNTBase becomes an MMT database by instantiating it with special MKM operations for MMT.

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A case study on formalizing algebra in a module system

Cited by 3 publications

References 8 publications

Building on the Diamonds between Theories: Theory Presentation Combinators

Building on the Diamonds between Theories: Theory Presentation Combinators

A scalable module system

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

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