OntoMathPROOntoMath^{PRO} Ontology: A Linked Data Hub for Mathematics

In this paper, we present an ontology of mathematical knowledge concepts that covers a wide range of the fields of mathematics and introduces a balanced representation between comprehensive and sensible models. We demonstrate the applications of this representation in information extraction, semantic search, and education. We argue that the ontology can be a core of future integration of math-aware data sets in the Web of Data and, therefore, provide mappings onto relevant datasets, such as DBpedia and ScienceWISE.Comment: 15 pages, 6 images, 1 table, Knowledge Engineering and the Semantic Web - 5th International Conferenc

User-friendly Support for Common Concepts in a Lightweight Verifier

Machine verification of formal arguments can only increase our confidence in the correctness of those arguments, but the costs of employing machine verification still outweigh the benefits for some common kinds of formal reasoning activities. As a result, usability is becoming increasingly important in the design of formal verification tools. We describe the "aartifact" lightweight verification system, designed for processing formal arguments involving basic, ubiquitous mathematical concepts. The system is a prototype for investigating potential techniques for improving the usability of formal verification systems. It leverages techniques drawn both from existing work and from our own efforts. In addition to a parser for a familiar concrete syntax and a mechanism for automated syntax lookup, the system integrates (1) a basic logical inference algorithm, (2) a database of propositions governing common mathematical concepts, and (3) a data structure that computes congruence closures of expressions involving relations found in this database. Together, these components allow the system to better accommodate the expectations of users interested in verifying formal arguments involving algebraic and logical manipulations of numbers, sets, vectors, and related operators and predicates. We demonstrate the reasonable performance of this system on typical formal arguments and briefly discuss how the system's design contributed to its usability in two case studies

Digital ecosystem ontomath: Mathematical knowledge analytics and management

© Springer International Publishing AG 2017.A mathematical knowledge management technology is discussed, its basic ideas, approaches and results are based on targeted ontologies in the field of mathematics. The solution forms the basis of the specialized digital ecosystem OntoMath which consists of a set of ontologies, text analytics tools and applications for managing mathematical knowledge. The studies are in line with the project aimed to create a World Digital Mathematical Library whose objective is to design a distributed system of interconnected repositories of digitized versions of mathematical documents

A User-friendly Interface for a Lightweight Verification System

User-friendly interfaces can play an important role in bringing the benefits of a machine-readable representation of formal arguments to a wider audience. The "aartifact" system is an easy-to-use lightweight verifier for formal arguments that involve logical and algebraic manipulations of common mathematical concepts. The system provides validation capabilities by utilizing a database of propositions governing common mathematical concepts. The "aartifact" system's multi-faceted interactive user interface combines several approaches to user-friendly interface design: (1) a familiar and natural syntax based on existing conventions in mathematical practice, (2) a real-time keyword-based lookup mechanism for interactive, context-sensitive discovery of the syntactic idioms and semantic concepts found in the system's database of propositions, and (3) immediate validation feedback in the form of reformatted raw input. The system's natural syntax and database of propositions allow it to meet a user's expectations in the formal reasoning scenarios for which it is intended. The real-time keyword-based lookup mechanism and validation feedback allow the system to teach the user about its capabilities and limitations in an immediate, interactive, and context-aware manner

Seamless composition and integration: a perspective on formal methods research

Formal methods are now a central component of computer-science education and research. However, there will always be advances in mathematical logic -- a.k.a. `formal methods' among computer scientists -- leading to advances in reliable, safe and secure computing. There are many research directions that will promote the impact of formal methods on computer science in significant and novel ways. We outline two directions, each associated with its own research challenges, that are complementary to the current state-of-the-art: one of composability and one of integration, each considered in a specific context drawn from our own recent research and teaching experience. We try to clarify why the study and ultimate resolution of these two challenges hold the promise of important breakthroughs in the accessability of formal methods and, ultimately, their applicability.National Science Foundation (CCF-0820138

A User-friendly Interface for a Lightweight Verification System

User-friendly interfaces can play an important role in bringing the benefits of a machine-readable representation of formal arguments to a wider audience. The "aartifact" system is an easy-to-use lightweight verifier for formal arguments that involve logical and algebraic manipulations of common mathematical concepts. The system provides validation capabilities by utilizing a database of propositions governing common mathematical concepts. The "aartifact" system's multi-faceted interactive user interface combines several approaches to user-friendly interface design: (1) a familiar and natural syntax based on existing conventions in mathematical practice, (2) a real-time keyword-based lookup mechanism for interactive, context-sensitive discovery of the syntactic idioms and semantic concepts found in the system's database of propositions, and (3) immediate validation feedback in the form of reformatted raw input. The system's natural syntax and database of propositions allow it to meet a user's expectations in the formal reasoning scenarios for which it is intended. The real-time keyword-based lookup mechanism and validation feedback allow the system to teach the user about its capabilities and limitations in an immediate, interactive, and context-aware manner

Improving the Accessibility of Lightweight Formal Verification Systems

In research areas involving mathematical rigor, there are numerous benefits to adopting a formal representation of models and arguments: reusability, automatic evaluation of examples, and verification of consistency and correctness. However, broad accessibility has not been a priority in the design of formal verification tools that can provide these benefits. We propose a few design criteria to address these issues: a simple, familiar, and conventional concrete syntax that is independent of any environment, application, or verification strategy, and the possibility of reducing workload and entry costs by employing features selectively. We demonstrate the feasibility of satisfying such criteria by presenting our own formal representation and verification system. Our system’s concrete syntax overlaps with English, LATEX and MediaWiki markup wherever possible, and its verifier relies on heuristic search techniques that make the formal authoring process more manageable and consistent with prevailing practices. We employ techniques and algorithms that ensure a simple, uniform, and flexible definition and design for the system, so that it easy to augment, extend, and improve

Foundations for a self-reflective, context-aware semantic representation of mathematical specifications

Das Projekt "a modeling system for mathematics" (MoSMath), das zur Zeit and der Universität Wien durchgeführt wird, hat die Erstellung eines Systems zur Spezifikation von numerischen Modellen zum Ziel, in einer Form wie sie für Mathematiker natürlich ist. Das spezifizierte Modell soll innerhalb des Systems repräsentiert und bearbeitet werden, und dann zu numerischen Solvern, die nicht Teil des System sind, übermittelt werden können. Als ein erster Schritt zu einer universal einsetzbaren Software für die Repräsentation und bearbeitung von Mathematik auf dem Computer (das FMathL Projekt) entwickeln wir eine Repräsentation von Mathematik in einem Semantischen Netz (das "semantic memory"), zusammen mit einem Typsystem das die Gültigkeit der Repräsentation prüft, und einer virtuellen Maschine, die Algorithmen ausführen kann. Der Benutzer profitiert von so einem System auf mehrfache Weise: Der offensichtlichste Vorteil ist dass der Benutzer nicht gezwungen ist eine Modellierungssprache zu erlernen und kann stattdessen die natürliche Sprache der Mathematik verwenden, welche von jedem Mathematiker, Informatiker, Physker, etc. erlernt und praktiziert wird. Zusätzlich ist diese Art der Spezifizierung eines Modells am wenigsten Fehleranfällig, und die natürlichste Art ein Modell zu kommunizieren. Einmal in dem System repräsentiert, können ohne zusätzlichen Aufwand Ausgaben des Modells in verschiedenen Modellierungssprachen, und verschiednenen natürlichen Sprachen erzeugt werden, vorausgesetzt dass passende Transformationsroutinen verfügbar sind.The project "a modeling system for mathematics" (MoSMath), currently carried out at the University of Vienna, aims to create a modeling system for the specification of models for the numerical work in optimization in a form that is natural for the working mathematician. The specified model is represented and processed inside a framework and can then be communicated to numerical solvers or other systems. As a first step towards a general purpose tool for representing and interfacing general mathematics on the computer (the FMathL project), we developed a representation of mathematics in a semantic network called the semantic memory, together with a type system that checks validity of the representation and a virtual machine that can execute algorithms. The user benefits from this input format in multiple ways: The most obvious advantage is that a user is not forced to learn an algebraic modeling language and can use the usual natural mathematical language, which is learned and practiced by every mathematician, computer scientist, physicist, and engineer. In addition, this kind of specification of a model is the least error prone, and the most natural way to communicate a model. Once represented in the framework, multiple outputs in different modeling languages (or even descriptions in different natural languages) would not mean extra work for the user if appropriate transformation modules are available

Students´ language in computer-assisted tutoring of mathematical proofs

Truth and proof are central to mathematics. Proving (or disproving) seemingly simple statements often turns out to be one of the hardest mathematical tasks. Yet, doing proofs is rarely taught in the classroom. Studies on cognitive difficulties in learning to do proofs have shown that pupils and students not only often do not understand or cannot apply basic formal reasoning techniques and do not know how to use formal mathematical language, but, at a far more fundamental level, they also do not understand what it means to prove a statement or even do not see the purpose of proof at all. Since insight into the importance of proof and doing proofs as such cannot be learnt other than by practice, learning support through individualised tutoring is in demand. This volume presents a part of an interdisciplinary project, set at the intersection of pedagogical science, artificial intelligence, and (computational) linguistics, which investigated issues involved in provisioning computer-based tutoring of mathematical proofs through dialogue in natural language. The ultimate goal in this context, addressing the above-mentioned need for learning support, is to build intelligent automated tutoring systems for mathematical proofs. The research presented here has been focused on the language that students use while interacting with such a system: its linguistic propeties and computational modelling. Contribution is made at three levels: first, an analysis of language phenomena found in students´ input to a (simulated) proof tutoring system is conducted and the variety of students´ verbalisations is quantitatively assessed, second, a general computational processing strategy for informal mathematical language and methods of modelling prominent language phenomena are proposed, and third, the prospects for natural language as an input modality for proof tutoring systems is evaluated based on collected corpora