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**11 - 14**of**14**### Proof Development with Ωmega: √ 2 Is Irrational

"... Abstract. Freek Wiedijk proposed the well-known theorem about the irrationality of √ 2 as a case study and used this theorem for a comparison of fifteen (interactive) theorem proving systems, which were asked to present their solution (see [48]). This represents an important shift of emphasis in the ..."

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Abstract. Freek Wiedijk proposed the well-known theorem about the irrationality of √ 2 as a case study and used this theorem for a comparison of fifteen (interactive) theorem proving systems, which were asked to present their solution (see [48]). This represents an important shift of emphasis in the field of automated deduction away from the somehow artificial problems of the past as represented, for example, in the test set of the TPTP library [45] back to real mathematical challenges. In this paper we present an overview of the Ωmega system as far as it is relevant for the purpose of this paper and show the development of a proof for this theorem. 1 Ωmega The Ωmega proof development system [40] is at the core of several related and well-integrated research projects of the Ωmega research group, whose aim is to develop system support for the working mathematician.

### Coordination of Mathematical Agents

, 2001

"... Mathematical Services . . . . . . . . . . . . . . . . . . . 10 2.3.2 Autonomy and Decentralization . . . . . . . . . . . . . . . . . . . 11 2.3.3 Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Distributed Articial Intelligence 12 3.1 Agent-Oriented Programming . . . . . ..."

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Mathematical Services . . . . . . . . . . . . . . . . . . . 10 2.3.2 Autonomy and Decentralization . . . . . . . . . . . . . . . . . . . 11 2.3.3 Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Distributed Articial Intelligence 12 3.1 Agent-Oriented Programming . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 The Knowledge Query and Manipulation Language . . . . . . . . . . . . 13 3.3 Coordination in Multi-Agent Systems . . . . . . . . . . . . . . . . . . . 13 4 Agent Technology for Distributed Mathematical Reasoning 15 4.1 MathWeb Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Communication between MathWeb agents . . . . . . . . . . . . . . . . . 18 4.2.1 Technical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2.2 Characterization of Reasoning Capabilities . . . . . . . . . . . . 18 4.2.3 Context in Mathematical Communication . . . . . . . . . . . . . 19 4.3 Coordination of MathWeb Agents . . . . . . . . . . . . . . . . . . . . . 20 5 Summary and Work Plan 22 5.1 Work Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1

### Organization, Transformation, and Propagation of Mathematical Knowledge in Ωmega

"... Abstract. Mathematical assistance systems and proof assistance systems in general have traditionally been developed as large, monolithic systems which are often hard to maintain and extend. In this article we propose a component network architecture as a means to design and implement such systems. U ..."

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Abstract. Mathematical assistance systems and proof assistance systems in general have traditionally been developed as large, monolithic systems which are often hard to maintain and extend. In this article we propose a component network architecture as a means to design and implement such systems. Under this view a mathematical assistance system is an integrated knowledge-based system composed as a network of individual, specialized components. These components manipulate and mutually exchange different kinds of mathematical knowledge encoded within different document formats. Consequently, several units of mathematical knowledge coexist throughout the system within these components and this knowledge changes non-monotonically over time. Our approach has resulted in a lean and maintainable system code and makes the system open for extensions. Moreover, it naturally decomposes the global and complex reasoning and truth maintenance task into local reasoning and truth maintenance tasks inside the system components. The interplay between neighboring components in the network is thereby realized by nonmonotonic updates over agreed interface representations encoding different kinds of mathematical knowledge. 1.

### AGS 2004 This SEKI Report was internally reviewed by:

, 901

"... We introduce a resource adaptive agent mechanism which supports the user in interactive theorem proving. The mechanism, an extension of [4], uses a two layered architecture of agent societies to suggest appropriate commands together with possible command argument instantiations. Experiments with thi ..."

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We introduce a resource adaptive agent mechanism which supports the user in interactive theorem proving. The mechanism, an extension of [4], uses a two layered architecture of agent societies to suggest appropriate commands together with possible command argument instantiations. Experiments with this approach show that its effectiveness can be further improved by introducing a resource concept. In this paper we provide an abstract view on the overall mechanism, motivate the necessity of an appropriate resource concept and discuss its realization within the agent architecture. 1