8 research outputs found
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
Higher-Order Tarski Grothendieck as a Foundation for Formal Proof
We formally introduce a foundation for computer verified proofs based on higher-order Tarski-Grothendieck set theory. We show that this theory has a model if a 2-inaccessible cardinal exists. This assumption is the same as the one needed for a model of plain Tarski-Grothendieck set theory. The foundation allows the co-existence of proofs based on two major competing foundations for formal proofs: higher-order logic and TG set theory. We align two co-existing Isabelle libraries, Isabelle/HOL and Isabelle/Mizar, in a single foundation in the Isabelle logical framework. We do this by defining isomorphisms between the basic concepts, including integers, functions, lists, and algebraic structures that preserve the important operations. With this we can transfer theorems proved in higher-order logic to TG set theory and vice versa. We practically show this by formally transferring Lagrange\u27s four-square theorem, Fermat 3-4, and other theorems between the foundations in the Isabelle framework
A Distributed and Trusted Web of Formal Proofs
International audienceMost computer checked proofs are tied to the particular technology of a prover's software. While sharing results between proof assistants is a recognized and desirable goal, the current organization of theorem proving tools makes such sharing an exception instead of the rule. In this talk, I argue that we need to turn the current architecture of proof assistants and formal proofs inside-out. That is, instead of having a few mature theorem provers include within them their formally checked theorems and proofs, I propose that proof assistants should sit on the edge of a web of formal proofs and that proof assistant should be exporting their proofs so that they can exist independently of any theorem prover. While it is necessary to maintain the dependencies between definitions, theories, and theorems, no explicit library structure should be imposed on this web of formal proofs. Thus a theorem and its proofs should not necessarily be located at a particular URL or within a particular prover's library. While the world of symbolic logic and proof theory certainly allows for proofs to be seen as global and permanent objects, there is a lot of research and engineering work that is needed to make this possible. I describe some of the required research and development that must be done to achieve this goal
Toward a formal theory for computing machines made out of whatever physics offers: extended version
Approaching limitations of digital computing technologies have spurred
research in neuromorphic and other unconventional approaches to computing. Here
we argue that if we want to systematically engineer computing systems that are
based on unconventional physical effects, we need guidance from a formal theory
that is different from the symbolic-algorithmic theory of today's computer
science textbooks. We propose a general strategy for developing such a theory,
and within that general view, a specific approach that we call "fluent
computing". In contrast to Turing, who modeled computing processes from a
top-down perspective as symbolic reasoning, we adopt the scientific paradigm of
physics and model physical computing systems bottom-up by formalizing what can
ultimately be measured in any physical substrate. This leads to an
understanding of computing as the structuring of processes, while classical
models of computing systems describe the processing of structures.Comment: 76 pages. This is an extended version of a perspective article with
the same title that will appear in Nature Communications soon after this
manuscript goes public on arxi