Florian Rabe

THF0 --- The Core of the TPTP Language for Higher-Order Logic

One of the keys to the success of the Thousands of Problems for Theorem Provers (TPTP) problem library and related infrastructure is the consistent use of the TPTP language. This paper introduces the core of the TPTP language for higher-order logic --- THF0, based on Churchs simple type theory. THF0 is a syntactically conservative extension of the untyped first-order TPTP language.

Project abstract: logic atlas and integrator (LATIN)

LATIN aims at developing methods, techniques, and tools for interfacing logics and related formal systems. These systems are at the core of mathematics and computer science and are implemented in systems like (semi-)automated theorem provers, model checkers, computer algebra systems, constraint solvers, or concept classifiers. Unfortunately, these systems have differing domains of applications, foundational assumptions, and input languages, which makes them non-interoperable and difficult to compare and evaluate in practice.

Publishing math lecture notes as linked data

We mark up a corpus of

The MMT API: a generic MKM system

{\mbox{\LaTeX}}

A practical module system for LF

lecture notes semantically and expose them as Linked Data in XHTML+MathML+RDFa. Our application makes the resulting documents interactively browsable for students. Our ontology helps to answer queries from students and lecturers, and paves the path towards an integration of our corpus with external sites.

Notations for Living Mathematical Documents

The MMT language has been developed as a scalable representation and interchange language for formal mathematical knowledge. It permits natural representations of the syntax and semantics of virtually all declarative languages while making Mmt-based MKM services easy to implement. It is foundationally unconstrained and can be instantiated with specific formal languages. The MMT API implements the MMT language along with multiple backends for persistent storage and frontends for machine and user access. Moreover, it implements a wide variety of Mmt-based knowledge management services. The API and all services are generic and can be applied to any language represented in MMT. A plugin interface permits injecting syntactic and semantic idiosyncrasies of individual formal languages.

The Mizar Mathematical Library in OMDoc: Translation and Applications

Module systems for proof assistants provide administrative support for large developments when mechanizing the meta-theory of programming languages and logics. We describe a module system for the logical framework LF that is based on two main primitives: signatures and signature morphisms. Signatures are defined as collections of constant declarations, and signature morphisms as homo-morphism in between them. Our design is semantically transparent in the sense that it is always possible to elaborate modules into the module free version of LF. We have implemented our design as part of the Twelf system and rewritten parts of the Twelf example library to take advantage of the module system.

Integrating Web Services into Active Mathematical Documents

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.

Towards logical frameworks in the heterogeneous tool set hets

The Mizar Mathematical Library is one of the largest libraries of formalized and mechanically verified 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.

A logical framework combining model and proof theory

Active mathematical documents are distinguished from traditional paper-oriented ones by their ability to interactively adapt to a readers inputs. This includes changes in the presentation of the content of the document as well as changes of that content itself. We have developed the JOBAD architecture, a client/server infrastructure for active mathematical documents. A server-side module generates active documents, which a client-side JavaScript library makes accessible for user interaction. Further server-side modules --- in the same backend, or distributed web services --- dynamically respond to callbacks invoked when the user interacts with the client. These three components are tied together by the JOBAD active document format, which backwards-compatibly enhances MathML by information about interactivity. JOBAD is designed to be modular in the specific web services offered. As examples, we present folding and elision in mathematical expressions, type and definition lookup of symbols, as well as conversion of physical units. We evaluate our framework with a case study where a large collection of lecture notes is served as an active document.