### 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.

### Combined Reasoning by Automated Cooperation ⋆

"... Different reasoning systems have different strengths and weaknesses, and often it is useful to combine these systems to gain as much as possible from their strengths and retain as little as possible from their weaknesses. Of particular interest is the integration of first-order and higher-order tech ..."

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Different reasoning systems have different strengths and weaknesses, and often it is useful to combine these systems to gain as much as possible from their strengths and retain as little as possible from their weaknesses. Of particular interest is the integration of first-order and higher-order techniques. First-order reasoning systems, on the one hand, have reached considerable strength in some niches, but in many areas of mathematics they still cannot reliably solve relatively simple problems, for example, when reasoning about sets, relations, or functions. Higher-order reasoning systems, on the other hand, can solve problems of this kind automatically. But the complexity inherent in their calculi prevents them from solving a whole range of problems. However, while many problems cannot be solved by any one system alone, they can be solved by a combination of these systems. We present a general agent-based methodology for integrating different reasoning systems. It provides a generic integration framework which facilitates the cooperation between diverse reasoners, but can also be refined to enable more efficient, specialist integrations. We empirically evaluate its usefulness, effectiveness and efficiency by case studies involving the integration of first-order and higher-order automated theorem provers, computer algebra systems, and model generators.

### Resource Adaptive Agents in Interactive Theorem Proving

"... 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 e ectiveness 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

### UITP 2003 Preliminary Version Adaptable Mixed-Initiative Proof Planning for Educational Interaction

"... Abstract Today, most theorem proving systems are either used by their developers or by a (small) group of particularly trained and skilled users. In order to make theorem proving functionalities useful for a larger clientele we have to ask "What does an envisioned group of users need?&q ..."

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Abstract Today, most theorem proving systems are either used by their developers or by a (small) group of particularly trained and skilled users. In order to make theorem proving functionalities useful for a larger clientele we have to ask &quot;What does an envisioned group of users need?&quot; For educational purposes a theorem prover can be used in different scenarios and can serve students with different needs. Therefore, the user interface as well as the choice of functionalities of the underlying prover have to be adapted to the context and the learner. In this paper, we present proof planning as back-engine for interactive proof exercises as well as an interaction console, which is part of our graphical user interface. Based on the proof planning situation, the console offers suggestions for proof steps to the learner. These suggestions can dynamically be adapted, e.g., to the user and to pedagogical criteria using pedagogical knowledge on the creation and presentation of suggestions. Key words: mathematics education, adaptive GUI, adaptive theorem proving 1 Motivation So far, the main goal of developing automated theorem proving systems has been to output true/false for a statement formulated in some logic or to deliver a proof object. Interactive theorem proving systems aim to support the proof construction done by a user in different ways, they restrict the search space (the choices) by making valid suggestions for proof steps, they suggest applicable lemmas, or they produce a whole subproof automatically. These functionalities are useful, e.g., for checking a student's proof for validity or for verifying a program. They are not particularly helpful, when the goal is to This is a preliminary version. The final version will be published in

### Combining Proofs of Higher-Order and First-Order Automated Theorem Provers

"... Abstract. Ωants is an agent-oriented environment for combining inference systems. A characteristics of the Ωants approach is that a common proof object is generated by the cooperating systems. This common proof object can be inspected by verification tools to validate the correctness of the proof. Ω ..."

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Abstract. Ωants is an agent-oriented environment for combining inference systems. A characteristics of the Ωants approach is that a common proof object is generated by the cooperating systems. This common proof object can be inspected by verification tools to validate the correctness of the proof. Ωants makes use of a two layered blackboard architecture, in which the applicability of inference rules are checked on one (abstract) layer. The lower layer administrates explicit proof objects in a common language. In concrete proofs these proof objects can be quite bit, which can make communication during proof search very inefficient. As a result we had situations in which most of the resources went into the overhead of constructing explicit proof objects and communicating between different components. Therefore we have recently developed an alternative modelling of cooperating systems in Ωants which allows direct communication between related systems during proof search. This has the consequence that proof objects can no longer be directly constructed and thus the correctness-validation in this novel approach is in question. In this paper we present a pragmatic approach how this can rectified. 1

### Abstract Agent Based Mathematical Reasoning 1

"... In this contribution we propose an agent architecture for theorem proving which we intend to investigate in depth in the future. The work reported in this paper is in an early state, and by no means nished. We present and discuss our proposal in order to get feedback from the Calculemus community. 1 ..."

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In this contribution we propose an agent architecture for theorem proving which we intend to investigate in depth in the future. The work reported in this paper is in an early state, and by no means nished. We present and discuss our proposal in order to get feedback from the Calculemus community. 1

### Resource Adaptive Agents in Interactive Theorem Proving

, 1999

"... 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.

### An Interactive Proof Development Environment + Anticipation = A Mathematical Assistant?

, 2000

"... Current semi-automated theorem provers are often advertised as "mathematical assistant systems". However, these tools behave too passively and in a stereotypic way to meet this ambitious goal because they lack the capability to adequately take into account requirements on proof search cont ..."

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Current semi-automated theorem provers are often advertised as "mathematical assistant systems". However, these tools behave too passively and in a stereotypic way to meet this ambitious goal because they lack the capability to adequately take into account requirements on proof search control and user demands for their own actions. Motivated by this deficit, we have incorporated several facilities into the MEGA proof development system that anticipate a number of divergent factors, based on mathematical knowledge, proof search defaults, and expectations about users. The techniques enhance the system's functionality through proof planning by knowledge-intensive methods, proof search guidance by default suggesting agents, and proof presentation by redundancy avoidance measures. The system's behavior suggests that anticipation is without doubt a central driving force in a mathematical assistant. Keywords: Mathematical Assistant System, Automated Theorem Proving, Proof planning, Agents, ...