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14
ΩANTS  An open approach at combining Interactive and Automated Theorem Proving
 IN PROC. OF CALCULEMUS2000. AK PETERS
, 2000
"... We present the ΩAnts theorem prover that is built on top of an agentbased command suggestion mechanism. The theorem prover inherits beneficial properties from the underlying suggestion mechanism such as runtime extendibility and resource adaptability. Moreover, it supports the distributed integ ..."
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Cited by 35 (23 self)
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We present the ΩAnts theorem prover that is built on top of an agentbased command suggestion mechanism. The theorem prover inherits beneficial properties from the underlying suggestion mechanism such as runtime extendibility and resource adaptability. Moreover, it supports the distributed integration of external reasoning systems. We also introduce some notions that need to be considered to check completeness and soundness of such a system with respect to an underlying calculus.
Critical Agents Supporting Interactive Theorem Proving
 PROC. OF EPIA99, LNAI 1695
, 1999
"... We introduce a resource adaptive agent mechanism which supports the user of an interactive theorem proving system. The mechanism, an extension of [4], uses a two layered architecture of agent societies to suggest applicable commands together with appropriate command argument instantiations. Exp ..."
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Cited by 14 (12 self)
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We introduce a resource adaptive agent mechanism which supports the user of an interactive theorem proving system. The mechanism, an extension of [4], uses a two layered architecture of agent societies to suggest applicable commands together with appropriate 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.
Can a higherorder and a firstorder theorem prover cooperate?
 IN FRANZ BAADER AND ANDREI VORONKOV, EDITORS, LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE, AND REASONING — 11TH INTERNATIONAL WORKSHOP, LPAR 2004, LNAI 3452
, 2005
"... Stateoftheart firstorder automated theorem proving systems have reached considerable strength over recent years. However, in many areas of mathematics they are still a long way from reliably proving theorems that would be considered relatively simple by humans. For example, when reasoning about ..."
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Cited by 11 (8 self)
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Stateoftheart firstorder automated theorem proving systems have reached considerable strength over recent years. However, in many areas of mathematics they are still a long way from reliably proving theorems that would be considered relatively simple by humans. For example, when reasoning about sets, relations, or functions, firstorder systems still exhibit serious weaknesses. While it has been shown in the past that higherorder reasoning systems can solve problems of this kind automatically, the complexity inherent in their calculi and their inefficiency in dealing with large numbers of clauses prevent these systems from solving a whole range of problems. We present a solution to this challenge by combining a higherorder and a firstorder automated theorem prover, both based on the resolution principle, in a flexible and distributed environment. By this we can exploit concise problem formulations without forgoing efficient reasoning on firstorder subproblems. We demonstrate the effectiveness of our approach on a set of problems still considered nontrivial for many firstorder theorem provers.
Combined reasoning by automated cooperation
 JOURNAL OF APPLIED LOGIC
, 2008
"... 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 firstorder and higherorder tech ..."
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Cited by 11 (7 self)
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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 firstorder and higherorder techniques. Firstorder 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. Higherorder 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 agentbased 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 higherorder automated theorem provers, computer algebra systems, and model generators.
Integrating TPS and ΩMEGA
 JOURNAL OF UNIVERSAL COMPUTER SCIENCE
, 1999
"... This paper reports on the integration of the higherorder theorem proving environment Tps [Andrews et al., 1996] into the mathematical assistant Ωmega [Benzmuller et al., 1997]. Tps can be called from mega either as a black box or as an interactive system. In black box mode, the user has control ov ..."
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Cited by 7 (4 self)
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This paper reports on the integration of the higherorder theorem proving environment Tps [Andrews et al., 1996] into the mathematical assistant Ωmega [Benzmuller et al., 1997]. Tps can be called from mega either as a black box or as an interactive system. In black box mode, the user has control over the parameters which control proof search in Tps; in interactive mode, all features of the Tpssystem are available to the user. If the subproblem which is passed to Tps contains concepts defined in Ωmega's database of mathematical theories, these definitions are not instantiated but are also passed to Tps. Using a special theory which contains proof tactics that model the NDcalculus variant of Tps within mega, any complete or partial proof generated in Tps can be translated one to one into an mega proof plan. Proof transformation is realised by proof plan expansion in Ωmega's 3dimensional proof data structure, and remains transparent to the user.
ΩMEGA: Computer supported mathematics
 IN: PROCEEDINGS OF THE 27TH GERMAN CONFERENCE ON ARTIFICIAL INTELLIGENCE (KI 2004)
, 2004
"... The year 2004 marks the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number” (with Martin Davis’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated dedu ..."
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Cited by 3 (3 self)
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The year 2004 marks the fiftieth birthday of the first computer generated proof of a mathematical theorem: “the sum of two even numbers is again an even number” (with Martin Davis’ implementation of Presburger Arithmetic in 1954). While Martin Davis and later the research community of automated deduction used machine oriented calculi to find the proof for a theorem by automatic means, the Automath project of N.G. de Bruijn – more modest in its aims with respect to automation – showed in the late 1960s and early 70s that a complete mathematical textbook could be coded and proofchecked by a computer. Classical theorem proving procedures of today are based on ingenious search techniques to find a proof for a given theorem in very large search spaces – often in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited. The shift
Towards Concurrent Resource Managed Deduction
 UNIVERSITY OF BIRMINGHAM, SCHOOL OF COMPUTER SCIENCE. URL
, 1999
"... In this paper, we describe an architecture for resource guided concurrent mechanised deduction which is motivated by some findings in cognitive science. Its benefits are illustrated by comparing it with traditional proof search techniques. In particular, we introduce the notion of focused search ..."
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Cited by 3 (2 self)
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In this paper, we describe an architecture for resource guided concurrent mechanised deduction which is motivated by some findings in cognitive science. Its benefits are illustrated by comparing it with traditional proof search techniques. In particular, we introduce the notion of focused search and show that a reasoning system can be built as the cooperative collection of concurrently acting specialised problem solvers. These reasoners typically perform well in a particular problem domain. The system architecture that we describe assesses the subgoals of a theorem and distributes them to the specialised solvers that look the most promising. Furthermore it allocates resources (above all computation time and memory) to the specialised reasoners. This technique is referred to as resource management. Each reasoner terminates its search for a solution of a given subgoal when the solution is found or when it runs out of its assigned resources. We argue that the effect of resource ma...
Towards FineGrained Proof Planning with Critical Agents
, 1999
"... ,u:L 1 ) (s:L 2 ,u:L 1 ,pl:(1)) : : : goal is HO goal is HO goal is HO message: goal is HO Classif. Agent Figure 1: The two layered suggestion mechanism. We have developed a mechanism that suggests commands, applicable in the current proof state together with a suitable argument instanti ..."
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Cited by 2 (1 self)
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,u:L 1 ) (s:L 2 ,u:L 1 ,pl:(1)) : : : goal is HO goal is HO goal is HO message: goal is HO Classif. Agent Figure 1: The two layered suggestion mechanism. We have developed a mechanism that suggests commands, applicable in the current proof state together with a suitable argument instantiations [1]. It is based on two layers of societies of autonomous, concurrent agents which steadily work in the background of a system and dynamically update their computational behavior to the state of the proof and/or specific user queries to the suggestion mechanism (cf. Fig. 1). By exchanging relevant results via blackboards the agents cooperatively accumulate useful command suggestions which can be heuristically sorted and presented to the user. The architecture has been prototypically implemented in the\Omega mega mathematical assistant system [4] for interactive tactical theorem provi
Adaptable mixedinitiative proof planning for educational interaction
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2004
"... 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?” For educational purpos ..."
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Cited by 1 (1 self)
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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?” 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 backengine 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.
Agent based mathematical reasoning
 IN PROCEEDINGS OF THE CALCULEMUS WORKSHOP: SYSTEMS FOR INTEGRATED COMPUTATION AND DEDUCTION
, 1999
"... 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 finished. We present and discuss our proposal in order to get feedback from the Calculemus community. ..."
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Cited by 1 (1 self)
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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 finished. We present and discuss our proposal in order to get feedback from the Calculemus community.