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...tings method of Peter [3], which extend their first-order counterparts in much the same way. Since higher-order calculi cannot be complete for the standard semantics by Gödel’s incompleteness theorem =-=[11]-=-, only the weaker notion of Henkin models [12] leads to a meaningful notion of completeness in higher-order logic. It turns out that the calculi in [1, 13, 3, 19] are not Henkin-complete, since they f...

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...ther with further model existence theorems in connection with additional notions of models can be found in [5]. Such theorems were first introduced by Smullyan (who calls them unifying principles) in =-=[24]-=- based on work by Hintikka and Beth and later generalized to higher-order logic by Andrews in [1]. Since there is no simple Herbrand theorem in higher-order logic, Andrews “unifying principle for type...

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...first-order counterparts in much the same way. Since higher-order calculi cannot be complete for the standard semantics by Gödel’s incompleteness theorem [11], only the weaker notion of Henkin models =-=[12]-=- leads to a meaningful notion of completeness in higher-order logic. It turns out that the calculi in [1, 13, 3, 19] are not Henkin-complete, since they fail to capture the extensionality principles o...

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...ackboard architecture for higher-order tableaux where multiple tableau agents work together in order to construct a proof. 26We can conceptualize tableaux implementations as blackboard architectures =-=[9]-=- where tableau agents, equipped with abilities from their underlying calculus, manipulate a Blackboard-like data structure, the tableau. The proof search space is defined by each agent’s possible nond...

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...l quantifications in the scope of a universal quantifier ∀X may not occur in any formula instantiated for X) in form of an explicit relation R which is similar, but not identical to the approaches in =-=[22, 8]-=-.. Definition 3.1 (Variable Condition, R-Substitution) We will call a partial function R that maps witness constants to sets of variables a variable condition. We will call a witness constant w ∈ W R-...

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...t and instantiation rules in HT to be treated by the unification rule from . Note that the rules from figure 3 together with HT (subst) directly correspond to the rules of higher-order preunification =-=[14]-=-. With these rules we use the tableau mechanism to construct unification trees (see Theorem 3.10 or Chapter I.2.12, Section 9). All of these rules are used with the understanding that all formulae are...

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...essful attempts to mechanize and implement higher-order logic were those of Huet [13] and Jensen and Pietrzykowski [17]. They combine the resolution principle for higher-order logic (first studied in =-=[1]-=-) with higher-order unification. The unification problem in typed λ-calculi is much more complex than that for first-order terms, since it has to take the theory of αβη-equality into account. As a con...

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...r higherorder logic. It is possible to prove an instance of the axiom of choice that is known to be independent of HOL using naive Skolemization. Dale Miller has investigated this problem in depth in =-=[22]-=-. For HT , we use a direct encoding of dependencies induced by the sequencing of quantifiers (witnesses for existential quantifications in the scope of a universal quantifier ∀X may not occur in any f...

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...omated theorem provers for higher-order logic is almost as old as the field of deduction systems itself. The first successful attempts to mechanize and implement higher-order logic were those of Huet =-=[13]-=- and Jensen and Pietrzykowski [17]. They combine the resolution principle for higher-order logic (first studied in [1]) with higher-order unification. The unification problem in typed λ-calculi is muc...

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...nce the generalization of the colored unification algorithms needed to manipulate the search restrictions has already worked out [16]. Acknowledgments While most of the work reported here is based on =-=[18, 19]-=-, important parts of the treatment of extensionality [5, 6] have been developed in collaboration with Christoph Benzmüller and was funded by the DFG in Project HOTEL. Karsten Konrad has implemented of...

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...pter to first-order calculi with equality. Since reduction orderings tend to be very weak for higher-order logic, it seems advantageous to focus on difference-reducing calculi like those discussed in =-=[15]-=-, particularly, since the generalization of the colored unification algorithms needed to manipulate the search restrictions has already worked out [16]. Acknowledgments While most of the work reported...

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...cuss the higherorder tableau calculi HT and HTE first presented in [19]. The methods in this paper also apply to higher-order resolution calculi [1, 13, 6] or the higher-order matings method of Peter =-=[3]-=-, which extend their first-order counterparts in much the same way. Since higher-order calculi cannot be complete for the standard semantics by Gödel’s incompleteness theorem [11], only the weaker not...

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...r the sake of simplicity take the tableau method as a basis (for a general introduction to first-order tableaux see part I.1) and discuss the higherorder tableau calculi HT and HTE first presented in =-=[19]-=-. The methods in this paper also apply to higher-order resolution calculi [1, 13, 6] or the higher-order matings method of Peter [3], which extend their first-order counterparts in much the same way. ...

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...n Konrad has implemented of the Hot theorem prover [21] in the LISA project of the Sonderforschungsbereich 378. The earlier implementation experiments and a higher-order resolution theorem prover Leo =-=[7]-=-, that have yielded valuable intuition for the course of the proofs have been carried out in the deduction tool-box KEIM which already provides most of the primitives needed for higher-order theorem p...

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...ment of functional extensionality. In this respect, the calculi from [19] were not complete with respect to Σ-models Christoph Benzmüller has noted this shortcoming and proposed the rule HT (leib) in =-=[4]-=-. The case of standard tableaux for higher-order logic is a simple extension of first-order tableau methods to the higher-order language. The only significant 8difference is that αβη-equality has to ...

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...order logic is almost as old as the field of deduction systems itself. The first successful attempts to mechanize and implement higher-order logic were those of Huet [13] and Jensen and Pietrzykowski =-=[17]-=-. They combine the resolution principle for higher-order logic (first studied in [1]) with higher-order unification. The unification problem in typed λ-calculi is much more complex than that for first...

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...abstract consistency class. For conditions 1. and 3.– 5., and 7. in the definition of abstract consistency class (cf. definition 2.11) this can be achieved with the usual techniques (see for instance =-=[10]-=-). Condition 2. is trivially met, since all formulae are kept in βη-normal form and we have treated condition 6. in the tableau lifting theorem above. For 8. let Φ = Φ ′ ∗ ¬(A = α→β B) and let R.T be ...

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...degenerate do not occur, if suffices to employ calculi without rules such as HTE(ext). 5 Implementation: Concurrent Higher-Order Tableaux The HTE-calculus has been used as the basis of the Hot system =-=[21]-=-, a concurrent higher-order tableau theorem prover that is intended for applications in mathematics and computational linguistics [20]. The system is implemented in Oz [23], a constraint programming l...

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...roduction to first-order tableaux see part I.1) and discuss the higherorder tableau calculi HT and HTE first presented in [19]. The methods in this paper also apply to higher-order resolution calculi =-=[1, 13, 6]-=- or the higher-order matings method of Peter [3], which extend their first-order counterparts in much the same way. Since higher-order calculi cannot be complete for the standard semantics by Gödel’s ...

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...ference-reducing calculi like those discussed in [15], particularly, since the generalization of the colored unification algorithms needed to manipulate the search restrictions has already worked out =-=[16]-=-. Acknowledgments While most of the work reported here is based on [18, 19], important parts of the treatment of extensionality [5, 6] have been developed in collaboration with Christoph Benzmüller an...