Armin Fiedler

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.

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.

P.rex: An Interactive Proof Explainer

This paper outlines the interactive proof explanation system P.rex, which adapts its explanation to the user and allows him anytime to utter questions or requests, to which it reacts flexibly. As a generic system, it can be connected to different theorem provers. The distribution is available via the P.rex home page at http://www.ags.uni-sb.de/~prex.

Proof Development with Ωmega: The Irrationality of \sqrt 2

The well-known theorem asserting the irrationality of \(\sqrt 2\) was proposed as a case study for a comparison of fifteen (interactive) theorem proving systems [Wiedijk, 2002]. This represents an important shift of emphasis in the field of automated deduction away from the somehow artificial problems of the past back to real mathematical challenges.

Presenting Machine-Found Proofs

This paper outlines an implemented system named PROVERB that transforms and abstracts machine-found proofs to natural deduction style proofs at an adequate level of abstraction and then verbalizes them in natural language. The abstracted proofs, originally employed only as an intermediate representation, also prove to be useful for proof planning and proving by analogy.

LΩUI: Lovely ΩMEGA User Interface

Abstract. The capabilities of a automated theorem provers interface are essential for the effective use of (interactive) proof systems. LΩUI is the multi-modal interface that combines several features: a graphical display of information in a proof graph, a selective term browser with hypertext facilities, proof and proof plan presentation in natural language, and an editor for adding and maintaining the knowledge base. LΩUI is realized in an agent-based client-server architecture and implemented in the concurrent constraint programming language Oz.

Assertion-level Proof Representation with Under-Specification

Abstract We propose a proof representation format for human-oriented proofs at the assertion level with under-specification. This work aims at providing a possible solution to challenging phenomena worked out in em-pirical studies in the Dialog project at Saarland University. A particular challenge in this project is to bridge the gap between the human-oriented proof representation format with under-specification used in the proof manager of the tutorial dialogue system and the calculus- and machine-oriented representation format of the domain reasoner.

A multi-dimensional taxonomy for automating hinting

Hints are an important ingredient of natural language tutorial dialogues. Existing models of hints, however, are limited in capturing their various underlying functions, since hints are typically treated as a unit directly associated with some problem solving script or discourse situation. Putting emphasis on making cognitive functions of hints explicit and allowing for automatic incorporation in a natural dialogue context, we present a multi-dimensional hint taxonomy where each dimension defines a decision point for the associated function. Hint categories are then conceived as convergent points of the dimensions. So far, we have elaborated four dimensions: (1) domain knowledge, (2) inferential role, (3) elicitation status, (4) problem referential perspective. These fine-grained distinctions support the constructive generation of hint specifications from modular knowledge sources.

LOUI: Lovely OMEGA User Interface

Abstract. The capabilities of a automated theorem provers interface are essential for the effective use of (interactive) proof systems. LΩUI is the multi-modal interface that combines several features: a graphical display of information in a proof graph, a selective term browser with hypertext facilities, proof and proof plan presentation in natural language, and an editor for adding and maintaining the knowledge base. LΩUI is realized in an agent-based client-server architecture and implemented in the concurrent constraint programming language Oz.

Argumentation within deductive reasoning

Deductive reasoning is an area related to argumentation where machine‐based techniques, notably theorem proving, can contribute substantially to the formation of arguments. However, making use of the functionality of theorem provers for this issue is associated with a number of difficulties and, as we will demonstrate, requires considerable effort for obtaining reasonable results. Aiming at the exploitation of machine‐oriented reasoning for human‐adequate argumentation in a broader sense, we present our model for producing proof presentations from machine‐oriented inference structures. Capabilities of the model include adaptation to human‐adequate degrees of granularity and explicitness in the underlying argumentation and interactive exploration of proofs. Enhancing capabilities in all these respects, even just those we have addressed so far, does not only improve the interactive use of theorem provers, but shows they are essential ingredients to support the functionality of dialog‐oriented tutorial systems in formal domains.