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Three Years of Experience with Sledgehammer, a Practical Link between Automatic and Interactive Theorem Provers
"... Sledgehammer is a highly successful subsystem of Isabelle/HOL that calls automatic theorem provers to assist with interactive proof construction. It requires no user configuration: it can be invoked with a single mouse gesture at any point in a proof. It automatically finds relevant lemmas from all ..."
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Sledgehammer is a highly successful subsystem of Isabelle/HOL that calls automatic theorem provers to assist with interactive proof construction. It requires no user configuration: it can be invoked with a single mouse gesture at any point in a proof. It automatically finds relevant lemmas from all those currently available. An unusual aspect of its architecture is its use of unsound translations, coupled with its delivery of results as Isabelle/HOL proof scripts: its output cannot be trusted, but it does not need to be trusted. Sledgehammer works well with Isar structured proofs and allows beginners to prove challenging theorems.
Assertion application in theorem proving and proof planning
 In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI
, 2003
"... Our work addresses assertion retrieval and application in theorem proving systems or proof planning systems for classical firstorder logic. We propose a distributed mediator M between a mathematical knowledge base KB and a theorem proving system TP which is independent of the particular proof and k ..."
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Our work addresses assertion retrieval and application in theorem proving systems or proof planning systems for classical firstorder logic. We propose a distributed mediator M between a mathematical knowledge base KB and a theorem proving system TP which is independent of the particular proof and knowledge representation formats of TP and KB and which applies generalized resolution in order to analyze the logical consequences of arbitrary assertions for a proof context at hand. We discuss the connection to proof planning and motivate an application in a project aiming at a tutorial dialogue system for mathematics. This paper is a short version of [9]. 1 Proof planning at the assertion level Due to Huang [6], the notion of assertion comprises mathematical knowledge from a mathematical knowledge base KB such as axioms, definitions, and theorems. Huang argues that an assertionbased representation, i.e. assertion level, is just
Proof Development with ΩMEGA
 PROCEEDINGS OF THE 18TH CONFERENCE ON AUTOMATED DEDUCTION (CADE–18), VOLUME 2392 OF LNAI
, 2002
"... ..."
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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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
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.
Progress report on LEOII, an automatic theorem prover for higherorder logic
, 2007
"... Abstract. LeoII, a resolution based theorem prover for classical higherorder logic, is currently being developed in a one year research project at the University of Cambridge, UK, with support from Saarland University, Germany. We report on the current stage of development of LeoII. In particular, ..."
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Abstract. LeoII, a resolution based theorem prover for classical higherorder logic, is currently being developed in a one year research project at the University of Cambridge, UK, with support from Saarland University, Germany. We report on the current stage of development of LeoII. In particular, we sketch some main aspects of LeoII’s automated proof search procedure, discuss its cooperation with firstorder specialist provers, show that LeoII is also an interactive proof assistant, and explain its shared term data structure and its term indexing mechanism. 1
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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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.
Cooperating reasoning processes: More than just the sum of their parts. Research Excellence Award Acceptance Speech at IJCAI07
, 2007
"... Using the achievements of my research group over the last 30+ years, I provide evidence to support the following hypothesis: By complementing each other, cooperating reasoning process can achieve much more than they could if they only acted individually. Most of the work of my group has been on pro ..."
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Cited by 8 (0 self)
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Using the achievements of my research group over the last 30+ years, I provide evidence to support the following hypothesis: By complementing each other, cooperating reasoning process can achieve much more than they could if they only acted individually. Most of the work of my group has been on processes for mathematical reasoning and its applications, e.g. to formal methods. The reasoning processes we have studied include: Proof Search: by metalevel inference, proof planning, abstraction, analogy, symmetry, and reasoning with diagrams. Representation Discovery, Formation and Evolution: by analysing, diagnosing and repairing failed proof and planning attempts, forming and repairing new concepts and conjectures, and forming logical representations of informally stated problems. ¤I would like to thank the many colleagues with whom I have worked over the last 30+ years on the research reported in this paper:
PLATΩ: A mediator between texteditors and proof assistance systems
, 2007
"... We present a generic mediator, called PlatΩ, between texteditors and proof assistants. PlatΩ aims at integrated support for the development, publication, formalization, and verification of mathematical documents in a natural way as possible: The user authors his mathematical documents with a scient ..."
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We present a generic mediator, called PlatΩ, between texteditors and proof assistants. PlatΩ aims at integrated support for the development, publication, formalization, and verification of mathematical documents in a natural way as possible: The user authors his mathematical documents with a scientific WYSIWYG texteditor in the informal language he is used to, that is a mixture of natural language and formulas. These documents are then semantically annotated preserving the textual structure by using the flexible, parameterized proof language which we present. From this informal semantic representation PlatΩ automatically generates the corresponding formal representation for a proof assistant, in our case Ωmega. The primary task of PlatΩ is the maintenance of consistent formal and informal representations during the interactive development of the document.
System description: LEO – a resolution based higherorder theorem prover
 IN PROC. OF LPAR05 WORKSHOP: EMPIRICALLY SUCCESSFULL AUTOMATED REASONING IN HIGHERORDER LOGIC (ESHOL), MONTEGO
, 2005
"... We present Leo, a resolution based theorem prover for classical higherorder logic. It can be employed as both an fully automated theorem prover and an interactive theorem prover. Leo has been implemented as part of the Ωmega environment [23] and has been integrated with the Ωmega proof assistant. ..."
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Cited by 4 (4 self)
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We present Leo, a resolution based theorem prover for classical higherorder logic. It can be employed as both an fully automated theorem prover and an interactive theorem prover. Leo has been implemented as part of the Ωmega environment [23] and has been integrated with the Ωmega proof assistant. Higherorder resolution proofs developed with Leo can be displayed and communicated to the user via Ωmega’s graphical user interface Loui. The Leo system has recently been successfully coupled with a firstorder resolution theorem prover (Bliksem).