Abstract.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.

In this report we present an extension OMDoc to the OpenMath standard that allows to represent 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.We motivate and describe the OMDoc language and present an Xml document type definition for it. Furthermore, we discuss applications and tool support.

Abstract. 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.

Symbolic and logic computation systems ranging from computer algebra systems to theorem provers are finding their way into science, technology, mathematics and engineering. But such systems rely on explicitly or implicitly represented mathematical knowledge that needs to be managed to use such systems effectively.While mathematical knowledge management (MKM) "in the small " is wellstudied, scaling up to large, highly interconnected corpora remains difficult. We hold that in order to realize MKM "in the large", we need representation languages and software architectures that are designed systematically with largescale processing in mind.Therefore, we have designed and implemented the Mmt language -a module system for mathematical theories. Mmt is designed as the simplest possible language that combines a module system, a foundationally uncommitted formal semantics, and web-scalable implementations. Due to a careful choice of representational primitives, Mmt allows us to integrate existing representation languages for formal mathematical knowledge in a simple, scalable formalism. In particular, Mmt abstracts from the underlying mathematical and logical foundations so that it can serve as a standardized representation format for a formal digital library. Moreover, Mmt systematically separates logic-dependent and logic-independent concerns so that it can serve as an interface layer between computation systems and MKM systems. 1 We have already solved the integration of formal and informal mathematical knowledge in the OMDoc format, whose formal part is a predecessor of the work presented in this paper. We plan to integrate this solution with the much stronger formal basis of Mmt in the future.Corresponding to the notions of structural and semantic validation, we can define structural and semantic equivalence of theory graphs:Definition 34. Relative to a fixed foundation, two well-formed theory graphs γ and γ ′ are called structurally equivalent if the following holds:• γ > T = { } iff γ ′ > T = { }, and in that case T has meta-theory M in γ iff it does so in γ ′ ,• γ ≫ l : S → T = iff γ ′ ≫ l : S → T = ,

The Mizar Mathematical Library is one of the largest libraries of formalized mathematics. Its language is highly optimized for authoring by humans. As in natural languages, the meaning of an expression is influenced by its (mathematical) context in a way that is natural to humans, but harder to specify for machine manipulation. Thus its custom file format can make the access to the library difficult. Indeed, the Mizar system itself is currently the only system that can fully operate on the Mizar library.This paper presents a translation of the Mizar library into the OMDoc format (Open Mathematical Documents), an XML-based representation format for mathematical knowledge. OMDoc is geared towards machine support and interoperability by making formula structure and context dependencies explicit. Thus, the Mizar library becomes accessible for a wide range of OMDoc-based tools for formal mathematics and knowledge management. We exemplify interoperability by indexing the translated library in the MathWebSearch engine, which provides an "applicable theorem search" service (almost) out of the box.