Michael Kohlhase

OMDoc - An Open Markup Format for Mathematical Documents [version 1.2].

In this report we present a content markup scheme for (collections of) mathematical documents including articles, textbooks, interactive books, and courses. It can serve as the content language for agent communication of mathematical services on a mathematical software bus. We motivate and describe the OMDoc language and present an Xml document type definition for it. Furthermore, we discuss applications and tool support. This document describes version 1.1 of the OMDoc format. This version is mainly a bug-fix release that has become necessary by the experiments of encoding legacy material and theorem prover interfaces in OMDoc. The changes are relatively minor, mostly adding optional fields. Version 1.1 of OMDoc freezes the development so that version 2.0 can be started off. In contrast to the OMDoc format which has not changed much, this report is a total re-write, it closes many documentation gaps, clarifies various remaining issues. and adds a multitude of new examples.Setting the Stage for Open Mathematical Documents.- Setting the Stage for Open Mathematical Documents.- Document Markup for the Web.- Markup for Mathematical Knowledge.- OMDoc: Open Mathematical Documents.- An OMDoc Primer.- An OMDoc Primer.- Mathematical Textbooks and Articles.- OpenMath Content Dictionaries.- Structured and Parametrized Theories.- A Development Graph for Elementary Algebra.- Courseware and the Narrative/Content Distinction.- Communication with and Between Mathematical Software Systems.- The OMDoc Document Format.- The OMDoc Document Format.- OMDoc as a Modular Format.- Document Infrastructure (Module DOC).- Metadata (Modules DC and CC).- Mathematical Objects (Module MOBJ).- Mathematical Text (Modules MTXT and RT).- Mathematical Statements (Module ST).- Abstract Data Types (Module ADT).- Representing Proofs (Module PF).- Complex Theories (Modules CTH and DG).- Notation and Presentation (Module PRES).- Auxiliary Elements (Module EXT).- Exercises (Module QUIZ).- Document Models for OMDoc.- OMDoc Applications, Tools, and Projects.- OMDoc Applications, Tools, and Projects.- OMDoc Resources.- Validating OMDoc Documents.- Transforming OMDoc by XSLT Style Sheets.- OMDoc Applications and Projects.- Changes to the Specification.- Quick-Reference Table to the OMDoc Elements.- Quick-Reference Table to the OMDoc Attributes.- The RelaxNG Schema for OMDoc.- The RelaxNG Schemata for Mathematical Objects.

Omega: Towards a Mathematical Assistant

Ωmega is a mixed-initiative system with the ultimate purpose of supporting theorem proving in main-stream mathematics and mathematics education. The current system consists of a proof planner and an integrated collection of tools for formulating problems, proving subproblems, and proof presentation.

A search engine for mathematical formulae

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.

Proof Development with OMEGA

The Ωmega proof development system [2] is the core of several related and well integrated research projects of the Ωmega research group.

OMDOC: Towards an Internet Standard for the Administration, Distribution, and Teaching of Mathematical Knowledge

In this paper we present an extension OMDoc to the OPEN-MATH standard that allows the representation of the semantics and structure of various kinds of mathematical documents, including articles, textbooks, interactive books, courses. It can serve as the content language for agent communication of mathematical services on a mathematical software bus.

System Description: LEO - A Higher-Order Theorem Prover

Many (mathematical) problems, such as Cantor’s theorem, can be expressed very elegantly in higher-order logic, but lead to an exhaustive and un-intuitive formulation when coded in first-order logic. Thus, despite the difficulty of higher-order automated theorem proving, which has to deal with problems like the undecidability of higher-order unification (HOU) and the need for primitive substitution, there are proof problems which lie beyond the capabilities of first-order theorem provers, but instead can be solved easily by an higher-order theorem prover (HOATP) like Leo .T his 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). Leo uses a higher-order Logic based upon Church’s simply typed λ-calculus, so that the comprehension axioms are implicitly handled by αβη-equality. Leo employs a higher-order resolution calculus ERES (see [3] in this volume for details), where the search for empty clauses and higher-order pre-unification [6] are interleaved: the unifiability preconditions of the resolution and factoring rules are residuated as special negative equality literals that are treated by special unification rules. In contrast to other HOATP’s (such as Tps [1]) extensionality principles are build in into Leo’s unification, and hence do not have to be axiomatized in order to achieve Henkin completeness. Architecture

KI 2006: Advances in Artificial Intelligence

Session 1. Invited Talk.- Expressivity-Preserving Tempo Transformation for Music - A Case-Based Approach.- Session 2. Cognition and Emotion.- MicroPsi: Contributions to a Broad Architecture of Cognition.- Affective Cognitive Modeling for Autonomous Agents Based on Scherers Emotion Theory.- Session 3A. Semantic Web.- OWL and Qualitative Reasoning Models.- Techniques for Fast Query Relaxation in Content-Based Recommender Systems.- Session 3B. Analogy.- Solving Proportional Analogies by E-Generalization.- Building Robots with Analogy-Based Anticipation.- Session 4A. Natural Language.- Classification of Skewed and Homogenous Document Corpora with Class-Based and Corpus-Based Keywords.- Learning an Ensemble of Semantic Parsers for Building Dialog-Based Natural Language Interfaces.- Session 4B. Reasoning.- Game-Theoretic Agent Programming in Golog Under Partial Observability.- Finding Models for Blocked 3-SAT Problems in Linear Time by Systematical Refinement of a Sub-model.- Towards the Computation of Stable Probabilistic Model Semantics.- DiaWOz-II - A Tool for Wizard-of-Oz Experiments in Mathematics.- Session 5. Invited Talk.- Applications of Automated Reasoning.- Session 6A. Ontologies.- On the Scalability of Description Logic Instance Retrieval.- Relation Instantiation for Ontology Population Using the Web.- Session 6B. Spatio-temporal Reasoning.- GeTS - A Specification Language for Geo-Temporal Notions.- Active Monte Carlo Recognition.- Session 7A. Machine Learning.- Cross System Personalization and Collaborative Filtering by Learning Manifold Alignments.- A Partitioning Method for Mixed Feature-Type Symbolic Data Using a Squared Euclidean Distance.- Session 7B. Spatial Reasoning.- On Generalizing Orientation Information in .- Towards the Visualisation of Shape Features The Scope Histogram.- Session 8A. Robot Learning.- A Robot Learns to Know People-First Contacts of a Robot.- Recombinant Rule Selection in Evolutionary Algorithm for Fuzzy Path Planner of Robot Soccer.- Session 8B. Classical AI Problems.- A Framework for Quasi-exact Optimization Using Relaxed Best-First Search.- Gray Box Robustness Testing of Rule Systems.- A Unifying Framework for Hybrid Planning and Scheduling.- Session 9. Agents.- A Hybrid Time Management Approach to Agent-Based Simulation.- Adaptive Multi-agent Programming in GTGolog.- Agent Logics as Program Logics: Grounding KARO.- On the Relationship Between Playing Rationally and Knowing How to Play: A Logical Account.- Special Event. 50 Years Artificial Intelligence.- 1956-1966 How Did It All Begin? - Issues Then and Now.- Fundamental Questions.- Towards the AI Summer.- History of AI in Germany and The Third Industrial Revolution.- Three Decades of Human Language Technology in Germany.- 1996-2006 Autonomous Robots.- Projects and Vision in Robotics.- What Will Happen in Algorithm Country?.

A Mechanization of Strong Kleene Logic for Partial Functions

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truth-functional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.

System Description: The MathWeb Software Bus for Distributed Mathematical Reasoning

Automated reasoning systems have reached a high degree of maturity in the last decade. Many reasoning tasks can be delegated to an automated theorem prover (ATP) by encoding them into its interface logic, simply calling the system and waiting for a proof, which will arrive in less than a second in most cases. Despite this seemingly ideal situation, ATPs are seldom actually used by people other than their own developers. The reasons for this seem to be that it is difficult for practitioners of other fields to find information about theorem prover software, to decide which system is best suited for the problem at hand, installing it, and coping with the often idiosyncratic concrete input syntax. Of course, not only potential outside users face these problems, so that, more often than not, existing reasoning procedures are re-implemented instead of re-used.

System Description: MathWeb, an Agent-Based Communication Layer for Distributed Automated Theorem Proving

Real-world applications of theorem proving require open and modern software environments that enable modularization, distribution, inter-operability, networking, and coordination. This system description presents the MathWeb1 approach for distributed automated theorem proving that connects a wide-range of mathematical services by a common, mathematical software bus. The MathWeb system provides the functionality to turn existing theorem proving systems and tools into mathematical services that are homogeneously integrated into a networked proof development environment. The environment thus gains the services from these particular modules, but each module in turn gains from using the features of other, plugged-in components.