11 research outputs found
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
Mathematical models as research data via flexiformal theory graphs
Mathematical modeling and simulation (MMS) has now been established as an essential part
of the scientific work in many disciplines. It is common to categorize the involved
numerical data and to some extent the corresponding scientific software as research
data. But both have their origin in mathematical models, therefore any holistic approach
to research data in MMS should cover all three aspects: data, software, and
models. While the problems of classifying, archiving and making accessible are largely
solved for data and first frameworks and systems are emerging for software, the question
of how to deal with mathematical models is completely open.
In this paper we propose a solution -- to cover all aspects of mathematical models: the
underlying mathematical knowledge, the equations, boundary conditions, numeric
approximations, and documents in a flexi\-formal framework, which has enough structure to
support the various uses of models in scientific and technology workflows.
Concretely we propose to use the OMDoc/MMT framework to formalize mathematical models
and show the adequacy of this approach by modeling a simple, but non-trivial model: van
Roosbroeck's drift-diffusion model for one-dimensional devices. This formalization -- and
future extensions -- allows us to support the modeler by e.g. flexibly composing models,
visualizing Model Pathway Diagrams, and annotating model equations in documents as
induced from the formalized documents by flattening. This directly solves some of the
problems in treating MMS as "research data'' and opens the way towards more MKM
services for models
Towards a Natural Representation of Mathematics in Proof Assistants
In this thesis we investigate the proof assistant Scunak in order to explore the relationship between informal mathematical texts and their Scunak counterparts. The investigation is based on a case study in which we have formalized parts of an introductory book on real analysis. Based on this case study, we illustrate significant aspects of the formal representation of mathematics in Scunak. In particular, we present the formal proof of the example lim(1/n) = 0. Moreover, we present a comparison of Scunak with two well-known systems for formalizing mathematics, the Mizar System and Isabelle/HOL. We have proved the example lim(1/n) = 0 in Mizar and Isabelle/HOL as well and we relate certain features of formal mathematics in Mizar and Isabelle/HOL to corresponding features of the Scunak type theory in light of this example
Towards Knowledge Management for HOL Light
Abstract. The libraries of deduction systems are growing constantly, so much that knowledge management concerns are becoming increasingly urgent to address. However, due to time constraints and legacy design choices, there is barely any deduction system that can keep up with the MKM state of the art. HOL Light in particular was designed as a lightweight deduction system that systematically relegates most MKM aspects to external solutions — not even the list of theorems is stored by the HOL Light kernel. We make the first and hardest step towards knowledge management for HOL Light: We provide a representation of the HOL Light library in a standard MKM format that preserves the logical semantics and notations but is independent of the system itself. This provides an interface layer at which independent MKM applications can be developed. Moreover, we develop two such applications as examples. We employ the MMT system and its interactive web browser to view and navigate the library. And w
FoCaLiZe and Dedukti to the rescue for proof interoperability
International audienceNumerous contributions have been made for some years to allow users to exchange formal proofs between different provers. The main propositions consist in ad hoc pointwise translations, e.g. between HOL Light and Isabelle in the Flyspeck project or uses of more or less complete certificates. We propose in this paper a methodology to combine proofs coming from different theorem provers. This methodology relies on the Dedukti logical framework as a common formalism in which proofs can be translated and combined. To relate the independently developed mathematical libraries used in proof assistants, we rely on the structuring features offered by FoCaLiZe, in particular parameterized modules and inheritance to build a formal library of transfer theorems called MathTransfer. We finally illustrate this methodology on the Sieve of Eratosthenes, which we prove correct using HOL and Coq in combination
Methane Hydrate Formation in Ulleung Basin Under Conditions of Variable Salinity: Reduced Model and Experiments
In this paper, we present a reduced model of methane hydrate formation in variable
salinity conditions, with details on the equilibrium phase behavior adapted to a case study
from Ulleung Basin. The model simplifies the comprehensive model considered by Liu
and Flemings using common assumptions on hydrostatic pressure, geothermal gradient, and
phase incompressibility, as well as a simplified phase equilibria model. The two-phase threecomponent model is very robust and efficient as well as amenable to various numerical
analyses, yet is capable of simulating realistic cases. We compare various thermodynamic
models for equilibria as well as attempt a quantitative explanation for anomalous spikes of
salinity observed in Ulleung Basin