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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.
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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Cited by 8 (5 self)
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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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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).
Cutsimulation in impredicate logics
 PROC. OF IJCAR 2006
, 2006
"... We investigate cutelimination and cutsimulation in impredicative (higherorder) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic — in our case a sequent calculus for classical type theory — is like adding cut. The phenomenon equally ..."
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Cited by 3 (3 self)
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We investigate cutelimination and cutsimulation in impredicative (higherorder) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic — in our case a sequent calculus for classical type theory — is like adding cut. The phenomenon equally applies to prominent axioms like Boolean and functional extensionality, induction, choice, and description. This calls for the development of calculi where these principles are builtin instead of being treated axiomatically.
MSet Models
"... In [1] Andrews studies elementary type theory, a form of Church’s type theory [12] without extensionality, descriptions, choice, and infinity. Since most of the automated search procedures implemented in Tps [4] do not build in principles of extensionality, descriptions, choice or infinity, they are ..."
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Cited by 2 (1 self)
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In [1] Andrews studies elementary type theory, a form of Church’s type theory [12] without extensionality, descriptions, choice, and infinity. Since most of the automated search procedures implemented in Tps [4] do not build in principles of extensionality, descriptions, choice or infinity, they are essentially
Proving Theorems of Type Theory Automatically with TPS
"... Reasoning plays an important role in many activities which involve intelligence, and it may be anticipated that automated reasoning will play a significant role in many applications of artificial intelligence. The importance of developing methods of automating reasoning has been recognized since the ..."
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Reasoning plays an important role in many activities which involve intelligence, and it may be anticipated that automated reasoning will play a significant role in many applications of artificial intelligence. The importance of developing methods of automating reasoning has been recognized since the inception of research on artificial intelligence. One fruitful approach to this problem is to use the language and methods of symbolic logic. Since a great variety of problems can be expressed in symbolic logic, progress in developing general purpose reasoning tools based on symbolic logic has the potential to contribute to progress in many realms of artificial intelligence. Work on automated deduction using symbolic logic has been progressing steadily, but in recent years such work has been presented primarily at conferences on automated
Complete CutFree Tableaux for Equational Simple Type Theory
, 2009
"... We present a cutfree tableau system for a version of Church’s simple type normalization operator that completely hides the details of lambda conversion. We prove completeness of the system relative to Henkin models. The proof constructs Henkin models using the novel notion of a value system. 1 ..."
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We present a cutfree tableau system for a version of Church’s simple type normalization operator that completely hides the details of lambda conversion. We prove completeness of the system relative to Henkin models. The proof constructs Henkin models using the novel notion of a value system. 1