The effort to align MathML 3 and OpenMath has led to a realisation that (pragmatic) MathML’s condition and domainofapplication elements, when used with quantifiers, do not have a neat expression in OpenMath. This paper analyzes the situation focusing on quantifiers and proposes a solution, via six new symbols. Two of them fit completely within the existing OpenMath structure, and we place them in the associated quant3 CD. The others require a generalization of OMBIND. We also propose, logically separately but in (...) the same area, a quant2 CD with existsuniquely, commonly written ∃!, and the ‘fusion’ symbol existsuniquelyin. In a second step we generalize the solution to the phenomenon of big operators that MathML 2 implicitly provides but which do not have a direct counterpart in the OpenMath CDs. (shrink)
In this paper, we show that Higher{Order Coloured Uni cation { a form of uni cation developed for automated theorem proving { provides a general theory for modeling the interface between the interpretation process and other sources of linguistic, non semantic information. In particular, it provides the general theory for the Primary Occurrence Restriction which (Dalrymple et al., 1991)'s analysis called for.
Although knowledge is a central topic for MKM there is little explicit discussion on what ‘knowledge’ might actually be. There are specific intuitions about form and content of knowledge, about its structure, and epistemological nature that shape the MKM systems, but a conceptual model is missing. In this paper we try to rationalize this discussion to give MKM a firmer footing, to start a discussion among MKM researchers and help relate the MKM intuitions and discourses to other communities. Starting from (...) the observation that many concrete realizations of mathematical knowledge objects are considered equivalent, we propose a conceptual model of the space of (mathematical) knowledge objects graded by levels of abstraction and presentational explicitness and draw conclusions for MKM markup formats. (shrink)
The integration of reasoning and computation services across system and language boundaries has been mostly treated from an engineering perspective. In this paper we take a foundational point of view. We identify the following form of integration problems: an informal (mathematical; i.e, logically underspecified) specification has multiple concrete formal implementations between which queries and results have to be transported. The integration challenge consists in dealing with the implementation-specific details such as additional constants and properties. We pinpoint their role in safe (...) and unsafe integration schemes and propose a proof-theoretic solution based on modular theory-graphs that include the meta-logical foundations. This also gives a clean conceptual basis for earlier attempts that explain integration via “content/semantic markup”. (shrink)
Notations are central for understanding mathematical discourse. Readers would like to read notations that transport the meaning well and prefer notations that are familiar to them. Therefore, authors optimize the choice of notations with respect to these two criteria, while at the same time trying to remain consistent over the document and their own prior publications. In print media where notations are fixed at publication time, this is an over-constrained problem. In living documents notations can be adapted at reading time, (...) taking reader preferences into account. We present a representational infrastructure for notations in living mathematical documents. Mathematical notations can be defined declaratively. Author and reader can extensionally define the set of available notation definitions at arbitrary document levels, and they can guide the notation selection function via intensional annotations. We give an abstract specification of notation definitions and the flexible rendering algorithms and show their coverage on paradigmatic examples. We show how to use this framework to render OpenMath and Content-MathML to Presentation-MathML, but the approach extends to arbitrary content and presentation formats. We discuss prototypical implementations of all aspects of the rendering pipeline. (shrink)
We present a search engine for mathematical formulae. The MathWebSearch system harvests the web for content representations (currently MathML and OpenMath) of formulae and indexes them with substitution tree indexing, a technique originally developed for accessing intermediate results in automated theorem provers. For querying, we present a generic language extension approach that allows constructing queries by minimally annotating existing representations. First experiments show that this architecture results in a scalable application.
In the last two decades, the World Wide Web has become the universal, and — for many users — main information source. Search engines can efficiently serve daily life information needs due to the enormous redundancy of relevant resources on the web. For educational — and even more so for scientific information needs, the web functions much less efficiently: Scientific publishing is built on a culture of unique reference publications, and moreover abounds with specialized structures, such as technical nomenclature, notational (...) conventions, references, tables, or graphs. Moreover, many of these structures are peculiar to specialized communities determined by nationality, research group membership, or adherence to a special school of thought. To keep the much-lamented “digital divide” from becoming a “cultural divide”, we have to make online material more accessible and adaptable to individual users. In this paper we attack this goal for the field of mathematics where knowledge is abstract, highly structured, and extraordinarily interlinked. Modern, content-based representation formats like OpenMath or content MathML allow us to capture, model, relate, and represent mathematical knowledge objects and thus make them context-aware and machine-adaptable to the respective user contexts. Building on previous work which can make mathematical notations adaptable we employ user modeling techniques to make them adaptive to relieve the reader of configuration tasks. We present a comprehensive framework for adaptive notation management and evaluate it on an implementation integrated in the e-learning platform panta rhei. (shrink)
Automated knowledge management techniques critically depend on the availability of semantically enhanced documents which are hard to come by in practice. Starting from a detailed look at the motivations of users to produce semantic data, we argue that the authoring problem experienced by MKM is actually an author’s dilemma. An analysis of the content authoring process suggests that the dilemma can partially be overcome by providing authoring tools like invasive editors aimed specifically at supporting the content creator. We present the (...) CPoint application, a semantic, invasive editor for Microsoft PowerPoint, geared towards the OMDoc MKM format. (shrink)
We explore the social context of mathematical knowledge: Even though, the community of mathematicians may look homogeneous from the outside, it is actually structured into various sub-communities that differ in preferred notations, the choice of basic assumptions, or e.g. in the choice of motivating examples. We contend that we cannot manage mathematical knowledge for human recipients if we do not take these factors into account. As a basis for a future extension of MKM systems, we analyze the social context of (...) information in terms of Communities of Practice (CoP; a concept from learning theory) and present a concrete extensional model for CoPs in mathematics. (shrink)
Spreadsheets are mathematical documents that are heavily employed in administration, financial forecasting, education, and science because of their intuitive, flexible, and direct approach to computation. In this paper we show that spreadsheets are interesting applications for MKM techniques which can alleviate usability and maintenance problems as spreadsheet-based applications grow evermore complex and longlived. We present the software and information architecture of a semantic enhancement of MS Excel spreadsheets that aims at compensating the computational bias in spreadsheets.
Spreadsheets are heavily employed in administration, financial forecasting, education, and science because of their intuitive, flexible, and direct approach to computation. They represent examples for “active documents” which are considered very worthwhile in the field of Knowledge Management. But these activity traits also lead to usability and maintenance problems, as spreadsheet-based applications grow evermore complex and longlived. We argue that these difficulties come from a fundamental bias of spreadsheets towards computational aspects of the conceptual models underlying the applications, i.e., a (...) semantic bias. Concretely, we present the software architecture of the SACHS project geared towards a semantic enhancement of MS Excel spreadsheets that aims at compensating this semantic bias. (shrink)
The OMDoc (Open Mathematical Documents) format is a content markup scheme for (collections of) mathematical documents including articles, textbooks, interactive books, and courses. OMDoc also serves as the content language for agent communication of mathematical services on a mathematical software bus.
We present a search engine for mathematical formulae. The MathWebSearch system harvests the web for content representations of formulae and indexes them with substitution tree indexing. In version 0.4 we have parallelized and distributed the search server and augmented the web interface with a new JavaScript-based visual editor for content math formulae. Furthermore, we have extended the query language by generalization, variants, unification, and text search facilities, which can also be mixed. Our experiments show that this architecture results in a (...) scalable application. (shrink)
Notations are central for understanding mathematical discourse. Readers would like to read notations that transport the meaning well and prefer notations that are familiar to them. Therefore, authors optimize the choice of notations with respect to these two criteria, while at the same time trying to remain consistent over the document and their own prior publications. In print media where notations are fixed at publication time, this is an over-constrained problem. In living documents notations can be adapted at reading time, (...) taking reader preferences into account. We present a representational infrastructure for notations in living mathematical documents. Mathematical notations can be defined declaratively. Author and reader can extensionally define the set of available notation definitions at arbitrary document levels, and they can guide the notation selection function via intensional annotations. We give an abstract specification of notation definitions and the flexible rendering algorithms and show their coverage on paradigmatic examples. We show how to use this framework to render OpenMath and Content-MathML to Presentation-MathML, but the approach extends to arbitrary content and presentation formats. We discuss prototypical implementations of all aspects of the rendering pipeline. (shrink)
The Analytica system is a theorem proving system for 19 th century mathematics written on top of the Mathematica computer algebra system. It was developed in the early 1990’s by X. Zhao and E. Clarke and has since been dormant. We describe recent work to resurrect the theorem prover and port it to newer versions of Mathematica. The new system Analytica 2 can still prove the same theorems, but has been significantly cleaned up. The code has been restructured and documented, (...) the declarative knowledge has been separated from a logical kernel, and the system is being made available as a MathWeb service. (shrink)
This paper introduces the concept of Virtual Documents and its prototypical realization in our TNTBase system, a versioned XML database. VDs integrate XQuery-based computational facilities into documents like JSP/PHP do for relational queries. We view the integration of computation in documents as an enabling technology and evaluate it on a handfull of real-world use cases.
‘Semantic technologies’ are touted as the next big wave in Educational Technology and as the solution to many problems in this arena. Interdisciplinary work between the fields of Knowledge Management (KM) and Educational Technology (ET) is booming. But the crop of actual systems and semantically enhanced learning objects is still meager, maybe KM and EL they are lacking a consensus on the underlying notions e.g. of ‘semantics’, yielding specific problems in their interplay. In this paper we take a look at (...) semantic educational technologies and draw conclusions for their approach in KM. In particular, we (re)-evaluate the notions of ‘semantics’, ‘knowledge’, ‘learning’, their role for learning materials in ET, and how they interact with the contexts involved in the learning/teaching process. Based on this, we distill a list of conditions the underlying knowledge representation format must fulfil to support these. As these conditions are still rather abstract, we show how they can be realized in a concrete language design, taking in our OMDoc (Open Mathematical Documents) format as a point of departure. (shrink)
We present a collection of TEX macro packages that allow to markup TEX/L ATEX documents semantically without leaving the document format, essentially turning TEX/L ATEX into a document format for mathematical knowledge management (MKM).
transparency as a user interface property that enables giving appropriate help. We explicate this notion in document player applications found in office suites, for example. Moreover, we show how semantic transparency can be strengthened when the underlying software is complemented by a semantic ally system. The approach consists in illustrating existing software semantically. We present some semantic extensions of office applications as examples. We also deaction task. scribe how the semantic transparency approach allows the..
This Note discusses the facilities that are available in the MathML 2.0 Recommendation to facilitate the capturing of mathematical type information. It demonstrates how a combination of these features can be systematically used to provide support for general mathematical types.
We describe an experiment of transforming large collections of L ATEX documents to more machine-understandable representations. Concretely, we are translating the collection of scientific publications of..
Spreadsheets are heavily employed in administration, financial forecasting, education, and science because of their intuitive, flexible, and direct approach to computation. In previous work we have studied how an explicit representation of the background knowledge associated with the spreadsheet can be exploited to alleviate usability problems with spreadsheet-based applications. The SACHS system implements this approach to provide a semantic help system for DCS, an Excelbased financial controlling system.
LF has been designed as a meta-logical framework to represent logics, and has become a standard tool for studying properties of logics. Building on the newly introduced module system for LF, we present the nucleus of an integrated and structured development of the syntax, semantics, and proof theory of logics, and of the relations between those logics. The methodology is chosen so that it will scale to an atlas for the zoo of logics currently used in reasoning systems, and the (...) modular nature of this development aids the practical integration of systems because shared features of the logics are reused directly. Finally we show how these encodings in LF are imported into the Hets system, which provides automated proof support on the modular level and integrates various automated theorem provers for the represented object logics. (shrink)
One of the fundamental and seemingly simple aims of mathematical knowledge management (MKM) is to develop and standardize formats that allow to “represent the meaning of the objects of mathematics”. The open formats OpenMath and MathML address this, but differ subtly in syntax, rigor, and structural viewpoints (notably over calculus). To avoid fragmentation and smooth out interoperability obstacles, effort is under way to align them into a joint format OpenMath/MathML 3. We illustrate the issues that come up in such an (...) alignment by looking at three main areas: bound variables and conditions, calculus (which relates to the previous) and “lifted” n-ary operators. (shrink)
One of the major issues for user assistance systems consists of “providing help at an appropriate level”. In this paper we analyze the problem of modeling task experience — a prerequisite for provisioning adequate help. In contrast to level-based approaches we propose an ontology-based model, which allows fine-grained modeling of task experience using the concepts of the task domain as granules. The model is semantic in the sense that it allows to take advantage of the relations between concepts to provide (...) novel semantic services and interactions. We present the SACHS (Semantic Annotations for a Controlling Help System, a semantic help system for a spreadsheet-based financial controlling system) software as an exemplary application of the proposed task experience model. (shrink)
OpenMath is a standard for the representation and communication of mathematical objects, which are built up from symbols and variables using applications, binding expressions, and key-value attributions. OpenMath2 introduced a set of symbol roles that can be specified in content dictionaries to restrict the occurrences of the respective symbols. This yields a simple, high-level notion of well-formed objects. While this system is appealing in its simplicity, the definition of wellformedness is purely extensional without an intuitive or formal condition that distinguishes (...) well-formed objects from ill-formed ones. Moreover, some well-formed objects should arguably rather be ill-formed. We try to remedy that with a refined role system while preserving the simplicity of the existing one. In particular, by distinguishing syntactic and semantic roles, we can capture the intuitive notion of well-formedness better. (shrink)
The semantic web ontology languages RDFS and OWL are widely used but limited in both their expressivity and their support for modularity and integrated documentation. Expressivity, modularity, and documentation of formal knowledge have always been important issues in the MKM community. Therefore, we try to improve these ontology languages by well-tried MKM techniques. Concretely, we propose OM- Doc as an alternative. We show how OMDoc can be made compatible with semantic web ontology languages, focusing on knowledge representation, modular design, documentation, (...) and metadata. We evaluate our technology by re-implementing the Friend-of-a-friend (FOAF) ontology and applying it in a novel metadata framework for technical documents (including ontologies). (shrink)
The usage of sorts in first-order automated deduction has brought greater conciseness of representation and a considerable gain in efficiency by reducing the search spaces involved. This suggests that sort information can be employed in higher-order theorem proving with similar results.
Both Higher-Order Uni cation (HOU) approaches to In linguistic theories on discourse coherence Kehler, discourse semantics Dalrymple et al., 1991; Shieber et..
We present a content markup language for physics realized by extending the OMDoc format by an infrastructure for the principal concepts of physics: observables, physical systems, and experiments.
The history of building automated theorem provers for higher-order logic is almost as old as the field of deduction systems itself. The first successful attempts to mechanize and implement higher-order logic were those of Huet [13] and Jensen and Pietrzykowski [17]. They combine the resolution principle for higher-order logic (first studied in [1]) with higher-order unification. The unification problem in typed λ-calculi is much more complex than that for first-order terms, since it has to take the theory of αβη-equality into (...) account. As a consequence, the higher-order unification problem is undecidable and sets of solutions need not even always have most general elements that represent them. Thus the mentioned calculi for higher-order logic have take special measures to circumvent the problems posed by the theoretical complexity of higher-order unification. In this paper, we will exemplify the methods and proof- and model-theoretic tools needed for extending first-order automated theorem proving to higherorder logic. For the sake of simplicity take the tableau method as a basis (for a general introduction to first-order tableaux see part I.1) and discuss the higherorder tableau calculi HT and HTE first presented in [19]. The methods in this paper also apply to higher-order resolution calculi [1, 13, 6] or the higher-order matings method of Peter [3], which extend their first-order counterparts in much the same way. Since higher-order calculi cannot be complete for the standard semantics by Gödel’s incompleteness theorem [11], only the weaker notion of Henkin models [12] leads to a meaningful notion of completeness in higher-order logic. It turns out that the calculi in [1, 13, 3, 19] are not Henkin-complete, since they fail to capture the extensionality principles of higher-order logic. We will characterize the deductive power of our calculus HT (which is roughly equivalent to these calculi) by the semantics of functional Σ-models. To arrive at a calculus that is complete with respect to Henkin models, we build on ideas from [6] and augment HT with tableau construction rules that use the extensionality principles in a goal-oriented way.. (shrink)
This paper introduces a multi-valued variant of higher-order resolution and proves it correct and complete with respect to a variant of Henkin’s general model semantics. This resolution method is parametric in the number of truth values as well as in the particular choice of the set of connectives (given by arbitrary truth tables) and even substitutional quantifiers. In the course of the completeness proof we establish a model existence theorem for this logical system. The work reported in this paper provides (...) a basis for developing higherorder mechanizations for many non-classical logics. (shrink)
Semantic analysis, – inference on the basis of semantic information and world knowledge – still is largely uncharted territory in dy- (3) namic semantics. It is needed, among other things, for the reconstruction of linguistically unspecified parts of the discourse or for restricting ambiguities introduced by prior analysis processes, i.e.
This package supplies the infrastructure for extending STEX macros with OMDoc metadata. This package is mainly intended for authors of STEX extension packages.
Listed buildings, even if they are not top landmarks, are increasingly attracting visitors. People express interest in hidden gems in their neighborhood or along their travel itinerary, and in the history of the building they live in. All required data has been meticulously collected by the offices for historical monuments but is not flexibly accessible. In Bremen, the database of buildings (with location, map of the estate, construction history, architect, photos) is searchable and browsable online3, but that only helps users (...) who know how to use a rigidly structured database search form. Our beginning BauDenkMalNetz effort (“listed building web”) aims at a wider purpose: modeling the semantic structure of these data, starting in Bremen but open for other data, and exposing them via a semantic web interface with enhanced querying and presentation capabilities. Requirements beyond interactive browsing comprise auto-generation of customized printed guides (e. g. “Bauhaus villas in my neighborhood”), on-demand presentation on mobile devices (e. g. “medieval churches along my travel itinerary”), and serving linked data for usage by other services. (shrink)
This package provides an infrastructure for semantically enhanced requirements specifications used in software engineering. This allows to embed structural information into documents that can be used by semantic document managment systems e.g. for management of change and requirements tracing.
Thus, despite the di culty of higher-order automated theorem proving, which has to deal with problems like the undecidability of higher-order uni - cation (HOU) and the need for primitive substitution, there are proof problems which lie beyond the capabilities of rst-order theorem provers, but instead can be solved easily by an higher-order theorem prover (HOATP) like Leo. This is due to the expressiveness of higher-order Logic and, in the special case of Leo, due to an appropriate handling of the (...) extensionality principles (functional extensionality and extensionality on truth values). (shrink)
Even though OpenMath has been around for more than 10 years, there is still confusion about the “semantics of OpenMath”. As the upcoming MathML3 recommendation will semantically base Content MathML on OpenMath Objects, this question becomes more pressing. One source of confusions about OpenMath semantics is that it is given on two levels: a very weak algebraic semantics for expression trees, which..
Over the past few years, there have been a series of attempts Zee89, GS90, EK95, Mus94, KKP95] to combine the Montagovian type theoretic framework Mon74] with dynamic approaches, such as DRT Kam81]. The motivation for these developments is to obtain a general logical framework for discourse semantics that combines compositionality and dynamic binding.
We propose an infrastructure for collaborative content management and version control for structured mathematical knowledge. This will enable multiple users to work jointly on mathematical theories with minimal interference. We describe the API and the functionality needed to realize a cvs-like version control and distribution model. This architecture extends the cvs architecture in two ways, motivated by the specific needs of distributed management of structured mathematical knowledge on the Internet. On the one hand the one-level client/server model of cvs is (...) generalized to a multi-level graph of client/server relations, and on the other hand the underlying change-detection tools take the math-specific structure of the data into account. (shrink)
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 XML- based document formats. TNTBase becomes an MMT database by instantiating it with special MKM operations for MMT. (shrink)
The interest of the field of Mathematical Knowledge Management is predicated on the assumption that by investing into markup or formalization of mathematical knowledge, we can reap benefits in managing (creating, classifying, reusing, verifying, and finding) mathematical theories, statements, and objects. This global value proposition has been used to motivate the pursuit of technologies that can add machine support to these knowledge management tasks. But this (rather naive) technology-centered motivation takes a view merely from the global (macro) perspective, and almost (...) totally disregards the user’s point of view and motivations for using it, the local (micro) perspective. In this paper we go a first step into a more principled analysis of the MKM value proposition by focusing on motivations for mathematical search engines from the micro perspective. We will use a table-based method called the “Added-Value Analysis” (AVA) developed by one of the authors. Even though we apply the AVA only to mathematical search engines, the method quickly leads to value considerations that are relevant for the whole field of MKM. (shrink)
Since Mathematics really is about what mathematicians do, in this paper, we will look at the mathematical practice of framing , in which an object of interest is viewed in terms of well-understood mathematical structures. The new perspective not only allows to deepen the understanding of e resp. object, it also facilitates new insights. We propose a model for framing in the context of theory graphs, and show how framing can be exploited to enhance the interaction with MKM systems. We (...) use the framing extension of our SACHS system — a semantic help system for MS Excel — as a concrete example. (shrink)
We propose related algorithms for unification and constraint simplification in }F’&, a refinement of the simply-typed A-calculus with subtypes and bounded intersection types. }F""’ is intended as the basis of a logical framework in order to achieve more succinct and declarative axiomatiza-.
(2012). Reasoning without believing: on the mechanisation of presuppositions and partiality. Journal of Applied Non-Classical Logics: Vol. 22, No. 4, pp. 295-317. doi: 10.1080/11663081.2012.705962.
In this paper we re-examine the semantics of classical higher-order logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of higher-order models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we develop a methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of (machine-oriented) higher-order calculi with respect to these model classes.
Version Control systems like CVS and Subversion have transformed collaboration workflows in software engineering, and made possible the globally distributed project teams we know from the Open Source Phenomenon. On the other hand, XML is coming of age as a basis for document formats, and even though XML as a text-based format is amenable to version control in principle, the fact that version control systems work on files makes difficult the integration of fragment access techniques like XPath, XQuery that are (...) currently revolutionizing XML workflows. In this paper we present the TNTBase system, an open-source versioned XML database obtained by integrating Berkeley DB XML into the Subversion Server. The.. (shrink)
