A scalable module system

Abstract

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 = ,

Theory Presentation Combinators

“…This should allow us to be able to generalize our work in ways similar to Kohlhase and Rabe's MMT [16]. The careful reader might have notice that in the syntax of section 2, our combine had an over keyword.…”

“…While great for humans, they are in part at fault in the failure of being able to infer arrows. If, like in MMT [16], we used long names, might we be able to build a robust system? Maybe so, but it would immediately fall afoul of our second requirement: irrelevant information such as choices made by developers regarding the order in which to build theories, would leak into the long names, and thus be seen by users.…”

“…From the mathematical knowledge management side, it should be clear that MMT [16] is closely related. The main differences are that they are quite explicit about being foundations-independent (it is implicit in our work), they use long names, and their theory operations are mostly theory-internal, while ours are external.…”

On algebraic hierarchies in mathematical repository of Mizar

“…Interesting categorical motivation of such an approach is presented in [7]. Modules can be treated globally for all proof assistants, and a kind of interface allowing for information interchange is proposed as MMT -a module system for mathematical theories with scalable formalism [34]. All axiomatic theories can be viewed from the metalevel, using the concept of realms -which could also enable reasonings via consolidating knowledge about theories.…”

Alignment-based Translations Across Formal Systems Using Interface Theories

“…This approach has been used by the authors' research group in [IKR11] for Mizar, in [KR14] for HOL Light and in [KMOR17] for PVS, using the MMT framework [RK13]. A similar approach is currently underway using Dedukti [BCH12] as the framework system.…”