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21
A Proof Markup Language for Semantic Web Services
, 2004
"... Web services propose that they provide the means for remote interoperable access of components and software systems. However, successful interoperation between components that do anything more than the simplest information retrieval is dependent upon those components having a shared understandi ..."
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Cited by 71 (31 self)
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Web services propose that they provide the means for remote interoperable access of components and software systems. However, successful interoperation between components that do anything more than the simplest information retrieval is dependent upon those components having a shared understanding of the results that have passed between them. In this paper, we address the issue of understanding and trusting results generated by web services. We introduce a proof markup language (PML) that provides an interlingua for capturing the information agents need to understand results and to justify why they should believe the results. We also introduce our Inference Web infrastructure that uses PML as the foundation for providing explanations of web services to end users. We additionally show how PML is critical for and provides the foundation for hybrid reasoning. Our contributions in this paper focus on technological foundations for capturing formal representations of term meaning and justification descriptions thereby facilitating trust and reuse of answers from web agents.
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 34 (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.
A Generic Modular Data Structure for Proof Attempts Alternating on Ideas and Granularity
 Proceedings of MKM’05, volume 3863 of LNAI, IUB
, 2006
"... Abstract. A practically useful mathematical assistant system requires the sophisticated combination of interaction and automation. Central in such a system is the proof data structure, which has to maintain the current proof state and which has to allow the flexible interplay of various components i ..."
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Cited by 15 (6 self)
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Abstract. A practically useful mathematical assistant system requires the sophisticated combination of interaction and automation. Central in such a system is the proof data structure, which has to maintain the current proof state and which has to allow the flexible interplay of various components including the human user. We describe a parameterized proof data structure for the management of proofs, which includes our experience with the development of two proof assistants. It supports and bridges the gap between abstract level proof explanation and lowlevel proof verification. The proof data structure enables, in particular, the flexible handling of lemmas, the maintenance of different proof alternatives, and the representation of different granularities of proof attempts. 1
A Proof Planning Framework for Isabelle
, 2005
"... Proof planning is a paradigm for the automation of proof that focuses on encoding intelligence to guide the proof process. The idea is to capture common patterns of reasoning which can be used to derive abstract descriptions of proofs known as proof plans. These can then be executed to provide fully ..."
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Cited by 13 (9 self)
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Proof planning is a paradigm for the automation of proof that focuses on encoding intelligence to guide the proof process. The idea is to capture common patterns of reasoning which can be used to derive abstract descriptions of proofs known as proof plans. These can then be executed to provide fully formal proofs. This thesis concerns the development and analysis of a novel approach to proof planning that focuses on an explicit representation of choices during search. We embody our approach as a proof planner for the generic proof assistant Isabelle and use the Isar language, which is humanreadable and machinecheckable, to represent proof plans. Within this framework we develop an inductive theorem prover as a case study of our approach to proof planning. Our prover uses the difference reduction heuristic known as rippling to automate the step cases of the inductive proofs. The development of a flexible approach to rippling that supports its various modifications and extensions is the second major focus of this thesis. Here, our inductive theorem prover provides a context in which to evaluate rippling experimentally. This work results in an efficient and powerful inductive theorem prover for Isabelle as well as proposals for further improving the efficiency of rippling. We also draw observations in order
Experiments with an Agentoriented Reasoning System
 In In Proc. of KI 2001, volume 2174 of LNAI
, 2001
"... Abstract. This paper discusses experiments with an agent oriented approach to automated and interactive reasoning. The approach combines ideas from two subfields of AI (theorem proving/proof planning and multiagent systems) and makes use of state of the art distribution techniques to decentralise a ..."
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Cited by 12 (8 self)
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Abstract. This paper discusses experiments with an agent oriented approach to automated and interactive reasoning. The approach combines ideas from two subfields of AI (theorem proving/proof planning and multiagent systems) and makes use of state of the art distribution techniques to decentralise and spread its reasoning agents over the internet. It particularly supports cooperative proofs between reasoning systems which are strong in different application areas, e.g., higherorder and firstorder theorem provers and computer algebra systems. 1
Proof Development with ΩMEGA
 PROCEEDINGS OF THE 18TH CONFERENCE ON AUTOMATED DEDUCTION (CADE–18), VOLUME 2392 OF LNAI
, 2002
"... ..."
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.
Nontrivial Symbolic Computations in Proof Planning
 In Proc. of FroCoS 2000, LNCS 1794
, 2000
"... We discuss a pragmatic approach to integrate computer algebra into proof planning. It is based on the idea to separate computation and verification and can thereby exploit the fact that many elaborate symbolic computations are trivially checked. In proof planning the separation is realized by using ..."
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Cited by 6 (3 self)
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We discuss a pragmatic approach to integrate computer algebra into proof planning. It is based on the idea to separate computation and verification and can thereby exploit the fact that many elaborate symbolic computations are trivially checked. In proof planning the separation is realized by using a powerful computer algebra system during the planning process to do nontrivial symbolic computations. Results of these computations are checked during the refinement of a proof plan to a calculus level proof using a small, selfimplemented, system that gives us protocol information on its calculation. This protocol can be easily expanded into a checkable lowlevel calculus proof ensuring the correctness of the computation. We demonstrate our approach with the concrete implementation in the Omega system.
Randomization and HeavyTailed Behavior in Proof Planning
, 2000
"... Proof planning is the application of Artificial Intelligence planning techniques to prove mathematical theorems. While exploring the domain of the residue classes over the integers with the multistrategy proof planner Multi we found a class of hard problems on which proof planning showed a remarkab ..."
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Cited by 5 (4 self)
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Proof planning is the application of Artificial Intelligence planning techniques to prove mathematical theorems. While exploring the domain of the residue classes over the integers with the multistrategy proof planner Multi we found a class of hard problems on which proof planning showed a remarkable high degree of variance. On problems of the same complexity we either succeeded very quickly with short proofs or the proof planning process took significantly longer and resulted in a large proof. Recent work in Artificial Intelligence points out that the unpredictability in the running time of heuristic search procedures can often be explained by the phenomenon of heavytailed cost distributions. Because of the nonstandard nature of these heavytailed cost distributions the controled introduction of randomization into the search procedures and quick restarts of the randomized procedure can eliminate heavytailed behavior and can take advantage of short runs. In this report,...
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