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COMPARING APPROACHES TO RESOLUTION BASED HIGHERORDER THEOREM PROVING
, 2002
"... 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 res ..."
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Cited by 18 (12 self)
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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.
Corrections and higher–order unification
, 1996
"... We propose an analysis of corrections which models some of the requirements corrections place on context. We then show that this analysis naturally extends to the interaction of corrections with pronominal anaphora on the one hand, and (in)definiteness on the other. The analysis builds on previous u ..."
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Cited by 15 (11 self)
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We propose an analysis of corrections which models some of the requirements corrections place on context. We then show that this analysis naturally extends to the interaction of corrections with pronominal anaphora on the one hand, and (in)definiteness on the other. The analysis builds on previous unification–based approaches to NL semantics and relies on Higher–Order Unification with Equivalences, a form of unification which takes into account not only syntactic βηidentity but also denotational equivalence. Wir schlagen eine Analyse vor, die einige der Anforderungen von Korrekturen an den Kontext modelliert und sich natürlich auf die Interaktion
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 14 (9 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
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 ru ..."
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Cited by 10 (7 self)
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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...
Sloppy identity
 Logical Aspects of Computational Linguistics
, 1997
"... Abstract. Although sloppy interpretation is usually accounted for by theories of ellipsis, it often arises in nonelliptical contexts. In this paper, a theory of sloppy interpretation is provided which captures this fact. The underlying idea is that sloppy interpretation results from a semantic cons ..."
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Cited by 8 (4 self)
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Abstract. Although sloppy interpretation is usually accounted for by theories of ellipsis, it often arises in nonelliptical contexts. In this paper, a theory of sloppy interpretation is provided which captures this fact. The underlying idea is that sloppy interpretation results from a semantic constraint on parallel structures and the theory is shown to predict sloppy readings for deaccented and paycheck sentences as well as relational, event, and oneanaphora. It is further shown to capture the interaction of sloppy/strict ambiguity with quantification and binding. Finally, it is compared with other approaches to sloppy identity, in particular [4,12] and [5]. 1
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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Cited by 7 (2 self)
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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...
HOT: A Concurrent Automated Theorem Prover based on HigherOrder Tableaux
, 1998
"... Hot is an automated higherorder theorem prover based on HTE, an extensional higherorder tableaux calculus. The first part of this paper introduces an improved variant of the calculus which closely corresponds to the proof procedure implemented in Hot. The second part discusses Hot's design th ..."
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Cited by 5 (1 self)
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Hot is an automated higherorder theorem prover based on HTE, an extensional higherorder tableaux calculus. The first part of this paper introduces an improved variant of the calculus which closely corresponds to the proof procedure implemented in Hot. The second part discusses Hot's design that can be characterized as a concurrent blackboard architecture. We show the usefulness of the implementation by including benchmark results for over one hundred solved problems from logic and set theory.
Rijke. A resolution calculus for dynamic semantics
 Logics in Artificial Intelligence (JELIA'98), Lecture Notes in Artificial Intelligence 1489
, 1998
"... Abstract. This paper applies resolution theorem proving to natural language semantics. The aim is to circumvent the computational complexity triggered by natural language ambiguities like pronoun binding, by interleaving pronoun binding with resolution deduction. To this end, disambiguation is only ..."
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Cited by 4 (4 self)
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Abstract. This paper applies resolution theorem proving to natural language semantics. The aim is to circumvent the computational complexity triggered by natural language ambiguities like pronoun binding, by interleaving pronoun binding with resolution deduction. To this end, disambiguation is only applied to expressions that actually occur during derivations. Given a set of premises and a conclusion, our resolution method only delivers pronoun bindings that are needed to derive the conclusion. 1
Terminating Tableaux for the Basic Fragment of Simple Type Theory
, 2009
"... We consider the basic fragment of simple type theory, which restricts equations to base types and disallows lambda abstractions and quantifiers. We show that this fragment has the finite model property and that satisfiability can be decided with a terminating tableau system. Both results are with re ..."
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Cited by 3 (2 self)
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We consider the basic fragment of simple type theory, which restricts equations to base types and disallows lambda abstractions and quantifiers. We show that this fragment has the finite model property and that satisfiability can be decided with a terminating tableau system. Both results are with respect to standard models. 1
Ramified HigherOrder Unification
, 1996
"... While unification in the simple theory of types (a.k.a. higherorder logic) is undecidable, we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable uni ..."
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Cited by 1 (0 self)
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While unification in the simple theory of types (a.k.a. higherorder logic) is undecidable, we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable unification problem even at order 2. However, the decidability result for the pure subsystem indicates that unification terminates more often than general higherorder unification. We present an application to ACA 0 and other expressive subsystems of secondorder Peano arithmetic.