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Translating HigherOrder Clauses to FirstOrder Clauses
"... Abstract. Interactive provers typically use higherorder logic, while automatic provers typically use firstorder logic. In order to integrate interactive provers with automatic ones, it is necessary to translate higherorder formulae to firstorder form. The translation should ideally be both sound ..."
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Abstract. Interactive provers typically use higherorder logic, while automatic provers typically use firstorder logic. In order to integrate interactive provers with automatic ones, it is necessary to translate higherorder formulae to firstorder form. The translation should ideally be both sound and practical. We have investigated several methods of translating function applications, types and λabstractions. Omitting some type information improves the success rate, but can be unsound, so the interactive prover must verify the proofs. This paper presents experimental data that compares the translations in respect of their success rates for three automatic provers. 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
A Structured Set of HigherOrder Problems
 Theorem Proving in Higher Order Logics: TPHOLs 2005, LNCS 3603
, 2005
"... Abstract. We present a set of problems that may support the development of calculi and theorem provers for classical higherorder logic. We propose to employ these test problems as quick and easy criteria preceding the formal soundness and completeness analysis of proof systems under development. Ou ..."
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Abstract. We present a set of problems that may support the development of calculi and theorem provers for classical higherorder logic. We propose to employ these test problems as quick and easy criteria preceding the formal soundness and completeness analysis of proof systems under development. Our set of problems is structured according to different technical issues and along different notions of semantics (including Henkin semantics) for higherorder logic. Many examples are either theorems or nontheorems depending on the choice of semantics. The examples can thus indicate the deductive strength of a proof system. 1 Motivation: Test Problems for HigherOrder Reasoning Systems Test problems are important for the practical implementation of theorem provers as well as for the preceding theoretical development of calculi, strategies and heuristics. If the test theorems can be proven (resp. the nontheorems cannot) then they ideally provide a strong indication for completeness (resp. soundness). Examples for early publications providing firstorder test problems are [21,29,23]. For more than decade now the TPTP library [28] has been developed as a systematically structured electronic repository of
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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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).
Progress Report on LeoII, anAutomatic Theorem Prover for HigherOrder Logic ⋆
"... 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
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 firstorder and higherorder 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 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 firstorder and higherorder automated theorem provers, computer algebra systems, and model generators.
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"... LEOII, a resolution based theorem prover for classical higherorder logic, is currently being developed in a one year ..."
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LEOII, a resolution based theorem prover for classical higherorder logic, is currently being developed in a one year
LEO II: An Effective HigherOrder Theorem Prover
"... is developing proof tools. His early work made fundamental contributions to Prof. M. J. C. Gordon’s proof assistant, HOL. In 1986, Paulson introduced Isabelle, a generic proof assistant. Isabelle supports higherorder logic (HOL), ZermeloFraenkel set theory (ZF) and other formalisms. Many developme ..."
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is developing proof tools. His early work made fundamental contributions to Prof. M. J. C. Gordon’s proof assistant, HOL. In 1986, Paulson introduced Isabelle, a generic proof assistant. Isabelle supports higherorder logic (HOL), ZermeloFraenkel set theory (ZF) and other formalisms. Many developments are due to Prof. Tobias Nipkow’s group at the Technical University of Munich. Automatic proof search, one of Isabelle’s particular strengths, is however due to Paulson [17]. The designated Visiting Researcher, Dr. Christoph Benzmüller, is indispensable for this project. He is the principal architect of LEO, the only higherorder theorem prover to incorporate modern techniques. Benzmüller’s previous work [11] is the starting point for the current proposal, which is to develop a new automatic theorem prover for higherorder logic. More generally, Benzmüller has an outstanding reputation in the field of automated reasoning. He heads the research group at Saarland University that is developing OMEGA, an integrated mathematics assistance environment. The work will be done within the Cambridge Automated Reasoning Group. Hardware verification was pioneered here by Prof. Gordon and his students. They introduced what have become standard techniques, such as the use of higherorder logic to model hardware and software systems. The group’s work continues to attract worldwide attention. Former members such as Dr. John Harrison have taken formal verification to Intel and other companies. The group has built two of the world’s leading proof environments, namely HOL and Isabelle. Institutes using Isabelle as a basis for their research include the University of Edinburgh, Carnegie