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Algorithms, datastructures, and other issues in efficient automated deduction
 Automated Reasoning. 1st. International Joint Conference, IJCAR 2001, number 2083 in LNAI
, 2001
"... Abstract. Algorithms and datastructures form the kernel of any efficient theorem prover. In this abstract we discuss research on algorithms and datastructures for efficient theorem proving based on our experience with the theorem prover Vampire. We also briefly overview other works related to algori ..."
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Abstract. Algorithms and datastructures form the kernel of any efficient theorem prover. In this abstract we discuss research on algorithms and datastructures for efficient theorem proving based on our experience with the theorem prover Vampire. We also briefly overview other works related to algorithms and datastructures, and to efficient theorem proving in general. 1
Using Decision Procedures With a HigherOrder Logic
 In Theorem Proving in Higher Order Logics: 14th International Conference, TPHOLs 2001
, 2001
"... In automated reasoning, there is a perceived tradeo between expressiveness and automation. Higherorder logic is typically viewed as expressive but resistant to automation, in contrast with rstorder logic and its fragments. We argue that higherorder logic and its variants actually achieve a happy ..."
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In automated reasoning, there is a perceived tradeo between expressiveness and automation. Higherorder logic is typically viewed as expressive but resistant to automation, in contrast with rstorder logic and its fragments. We argue that higherorder logic and its variants actually achieve a happy medium between expressiveness and automation, particularly when used as a frontend to a wide range of decision procedures and deductive procedures. We illustrate the discussion with examples from PVS, but some of the observations apply to other variants of higherorder logic as well.
LEO  A HigherOrder Theorem Prover
 In Proc. of CADE15, volume 1421 of LNAI
, 1998
"... this paper was supported by the Deutsche Forschungsgemeinschaft in grant HOTEL. EXT ..."
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this paper was supported by the Deutsche Forschungsgemeinschaft in grant HOTEL. EXT
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).
System description: LEO — a higherorder theorem prover
 In Proc. of CADE15, volume 1421 of LNAI
, 1998
"... Many (mathematical) problems, such as Cantor’s theorem, can be expressed very elegantly in higherorder logic, but lead to an exhaustive and unintuitive formulation when coded in firstorder logic. Thus, despite the difficulty of higherorder automated theorem proving, which has to deal with proble ..."
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Many (mathematical) problems, such as Cantor’s theorem, can be expressed very elegantly in higherorder logic, but lead to an exhaustive and unintuitive formulation when coded in firstorder logic. Thus, despite the difficulty of higherorder automated theorem proving, which has to deal with problems like the undecidability of higherorder unification (HOU) and the need for primitive substitution, there are proof problems which lie beyond the capabilities of firstorder theorem provers, but instead can be solved easily by an higherorder theorem prover (HOATP) like Leo. This is due to the expressiveness of higherorder Logic and, in the special case of Leo, due to an appropriate handling of the extensionality principles (functional extensionality and extensionality on truth values). Leo uses a higherorder Logic based upon Church’s simply typed λcalculus, so that the comprehension axioms are implicitly handled by αβηequality. Leo employs a higherorder resolution calculus ERES (see [3] in this volume for details), where the search for empty clauses and higherorder preunification [6] are
A Lost Proof
 IN TPHOLS: WORK IN PROGRESS PAPERS
, 2001
"... We reinvestigate a proof example presented by George Boolos which perspicuously illustrates Gödel’s argument for the potentially drastic increase of prooflengths in formal systems when carrying through the argument at too low a level. More concretely, restricting the order of the logic in which t ..."
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We reinvestigate a proof example presented by George Boolos which perspicuously illustrates Gödel’s argument for the potentially drastic increase of prooflengths in formal systems when carrying through the argument at too low a level. More concretely, restricting the order of the logic in which the proof is carried through to the order of the logic in which the problem is formulated in the first place can result in unmanageable long proofs, although there are short proofs in a logic of higher order. Our motivation in this paper is of practical nature and its aim is to sketch the implications of this example to current technology in automated theorem proving, to point to related questions about the foundational character of type theory (without explicit comprehension axioms) for mathematics, and to work out some challenging aspects with regard to the automation of this proof – which, as we belief, nicely illustrates the discrepancy between the creativity and intuition required in mathematics and the limitations of state of the art theorem provers.
CHRISTOPH BENZMÜLLER COMPARING APPROACHES TO RESOLUTION BASED HIGHERORDER THEOREM PROVING
"... ABSTRACT. We investigate several approaches to resolution based automated theorem proving in classical higherorder logic (based on Church’s simply typed λcalculus) and discuss their requirements with respect to Henkin completeness and full extensionality. In particular we focus on Andrews ’ higher ..."
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ABSTRACT. We investigate several approaches to resolution based automated theorem proving in classical higherorder logic (based on Church’s simply typed λcalculus) and discuss their requirements with respect to Henkin completeness and full extensionality. In particular we focus on Andrews ’ higherorder resolution (Andrews 1971), Huet’s constrained resolution (Huet 1972), higherorder Eresolution, and extensional higherorder resolution (Benzmüller and Kohlhase 1997). With the help of examples we illustrate the parallels and differences of the extensionality treatment of these approaches and demonstrate that extensional higherorder resolution is the sole approach that can completely avoid additional extensionality axioms. 1.
HigherOrder MultiValued Resolution
"... . This paper introduces a multivalued variant of higherorder resolution and proves it correct and complete with respect to a natural multivalued variant of Henkin's general model semantics. This resolution method is parametric in the number of truth values as well as in the particular choice ..."
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. This paper introduces a multivalued variant of higherorder resolution and proves it correct and complete with respect to a natural multivalued variant of Henkin's general model semantics. This resolution method is parametric in the number of truth values as well as in the particular choice of the set of connectives (given by arbitrary truth tables) and even substitutional quantifiers. In the course of the completeness proof we establish a model existence theorem for this logical system. The work reported in this paper provides a basis for developing higherorder mechanizations for many nonclassical logics. KEY WORDS: higherorder logic, resolution, multivalued, calculus 1 Introduction From the first attempts of modeling everyday reasoning within the framework of classical firstorder logic, it has been known that many relevant aspects cannot be adequately expressed in it. The attempts to cope with these aspects have lead to many specialized logics in the field of artificial in...
A Challenge for Mechanized Deduction
"... We reinvestigate a proof example presented by George Boolos which perspicuously illustrates Godel's argument for the potentially drastic increase of prooflengths in formal systems when carrying through the argument at too low a level. More concretely, restricting the order of the logic i ..."
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We reinvestigate a proof example presented by George Boolos which perspicuously illustrates Godel's argument for the potentially drastic increase of prooflengths in formal systems when carrying through the argument at too low a level. More concretely, restricting the order of the logic in which the proof is carried through to the order of the logic in which the problem is formulated in the rst place can result in unmanageable long proofs, although there are short proofs in a logic of higher order.
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