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A Calculus and a System Architecture for Extensional HigherOrder Resolution
, 1997
"... The first part of this paper introduces an extension for a variant of Huet's higherorder resolution calculus [Hue72, Hue73] based upon classical type theory (Church's typed calculus [Chu40]) in order to obtain a calculus which is complete with respect to Henkin models [Hen50]. The new rules connec ..."
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The first part of this paper introduces an extension for a variant of Huet's higherorder resolution calculus [Hue72, Hue73] based upon classical type theory (Church's typed calculus [Chu40]) in order to obtain a calculus which is complete with respect to Henkin models [Hen50]. The new rules connect higherorder preunification with the general refutation process in an appropriate way to establish full extensionality for the whole system. The general idea of the calculus is discussed on different examples. The second part introduces the Leo system which implements the discussed extensional higherorder resolution calculus. This part mainly focus on the embedding of the new extensionality rules into the refutation process and the treatment of higherorder unification. 1 Introduction Many mathematical problems can be expressed shortly and elegantly in higher order logic whereas they often lead to unnatural and inflated formulations in firstorder logic, e.g., when coding them into axio...
Polarized Resolution Modulo
"... Abstract. We present a restriction of Resolution modulo where the rewrite rules are such that a clause always rewrites to a clause. This way, the reduct of a clause needs not be further transformed into clause form. Restricting Resolution modulo this way requires to extend it in another way and dist ..."
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Abstract. We present a restriction of Resolution modulo where the rewrite rules are such that a clause always rewrites to a clause. This way, the reduct of a clause needs not be further transformed into clause form. Restricting Resolution modulo this way requires to extend it in another way and distinguish the rules that apply to negative and to positive atomic propositions. As an example, we show how this method applies to a firstorder presentation of Simple type theory. Finally, we show that this method can be seen as a restriction of Equational resolution that mixes clause selection restrictions and literal selection restrictions, but unlike many restrictions of Resolution, it is not an instance of Ordered resolution. 1
HigherOrder Automated Theorem Proving for Natural Language Semantics
, 1998
"... This paper describes a tableaubased higherorder theorem prover Hot and an application to natural language semantics. In this application, Hot is used to prove equivalences using world knowledge during higherorder unification (HOU). This extended form of HOU is used to compute the licensing condit ..."
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Cited by 6 (3 self)
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This paper describes a tableaubased higherorder theorem prover Hot and an application to natural language semantics. In this application, Hot is used to prove equivalences using world knowledge during higherorder unification (HOU). This extended form of HOU is used to compute the licensing conditions for corrections. 1 Introduction Mechanized reasoning systems have many applications in Computational Linguistics. Based on the observation that some phenomena of natural language can be modeled as deductive processes, firstorder theorem provers or related inference systems have been used for instance in phonology [2], generation [17] and semantic analysis [22]. [11] describes an abductive framework for natural language understanding that includes world knowledge into the semantics construction process. Other approaches use higherorder logics and in particular fireduction and higherorder unification (HOU) as inference procedures. Following Montague [18] who has used the typed calcul...
HigherOrder Automated Theorem Proving
, 1998
"... Consistency Class) Let Ñ S be a class of sets of propositions, then Ñ S is called an abstract consistency class, iff each Ñ S is closed under subsets, and satisfies conditions (1) to (8) for all sets F 2 Ñ S . If it also satisfies (9), then we call it extensional. 1. If A is atomic, then A = 2 F or ..."
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Consistency Class) Let Ñ S be a class of sets of propositions, then Ñ S is called an abstract consistency class, iff each Ñ S is closed under subsets, and satisfies conditions (1) to (8) for all sets F 2 Ñ S . If it also satisfies (9), then we call it extensional. 1. If A is atomic, then A = 2 F or :A = 2 F. 2. If A 2 F and if B is the bhnormal form of A, then B F 2 Ñ S 2 . 3. If ::A 2 F, then A F 2 Ñ S . 4. If AB2F, then F A 2 Ñ S or F B 2 Ñ S . 5. If :(AB) 2 F, then F :A :B2 Ñ S . 6. If P a A 2 F, then F AB 2 Ñ S for each closed formula B 2 wff a (S). 7. If :P a A 2 F, then F :(Aw a ) 2 Ñ S for any witness constant w a 2 W that does not occur in F. 8. If :(A = a!b B) 2 F, then F :(Aw a = Bw) 2 Ñ S for any witness constant w a 2 W that does not occur in F. 9. If :(A = o B) 2 F, then F[fA;:Bg 2 Ñ S or F[f:A;Bg 2 Ñ S . Here, we treat equality as an abbreviation for Leibniz definition. We call an abstract consistency class saturated, iff for all F 2 Ñ S and all...
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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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.
Résolution D'Équations Dans Le Système T De Gödel
, 1994
"... Contents Introduction 7 1 Le syst`eme T 13 1.1 calcul : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 1.1.1 Termes du calcul : : : : : : : : : : : : : : : : : : : : : : : : 13 1.1.2 fir'eduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 1.1.3 Fonctions de ..."
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Contents Introduction 7 1 Le syst`eme T 13 1.1 calcul : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 1.1.1 Termes du calcul : : : : : : : : : : : : : : : : : : : : : : : : 13 1.1.2 fir'eduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 1.1.3 Fonctions definissables : : : : : : : : : : : : : : : : : : : : 14 1.1.4 Repr'esentation des fonctions r'ecursives : : : : : : : : : : : : : 15 1.2 calcul typ'e : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16 1.3 Vers le Syst`eme T : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 1.3.1 Termes et Types : : : : : : : : : : : : : : : : : : : : : : : : : 18 1.3.2 Conversion : : : : : : : : : : : : : : : : : : : : : : : : : : : :
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.
HigherOrder Rigid EUnification
 5th International Conference on Logic Programming and Automated Reasoning', number 822 in `Lecture Notes in Artificial Intelligence
"... . Higherorder Eunification, i.e. the problem of finding substitutions that make two simply typed terms equal modulo fi or fij equivalence and a given equational theory, is undecidable. We propose to rigidify it, to get a resourcebounded decidable unification problem (with arbitrary high bound ..."
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. Higherorder Eunification, i.e. the problem of finding substitutions that make two simply typed terms equal modulo fi or fij equivalence and a given equational theory, is undecidable. We propose to rigidify it, to get a resourcebounded decidable unification problem (with arbitrary high bounds), providing a complete higherorder Eunification procedure. The techniques are inspired from Gallier's rigid Eunification and from Dougherty and Johann's use of combinatory logic to solve higherorder Eunification problems. We improve their results by using general equational theories, and by defining optimizations such as higherorder rigid Epreunification, where flexible terms are used, gaining much efficiency, as in the nonequational case due to Huet. 1 Introduction Higherorder Eunification is the problem of finding complete sets of unifiers of two simply typed terms modulo fi or fijequivalence, and modulo an equational theory E . This problem has applications in higherorder a...