## On the Translation of Higher-Order Problems into First-Order Logic (1994)

Venue: | Proceedings of ECAI-94 |

Citations: | 8 - 4 self |

### BibTeX

@INPROCEEDINGS{Kerber94onthe,

author = {Manfred Kerber and Manfred Kerber},

title = {On the Translation of Higher-Order Problems into First-Order Logic},

booktitle = {Proceedings of ECAI-94},

year = {1994},

pages = {145--149},

publisher = {John Wiley & Sons}

}

### Years of Citing Articles

### OpenURL

### Abstract

. In most cases higher-order logic is based on the - calculus in order to avoid the infinite set of so-called comprehension axioms. However, there is a price to be paid, namely an undecidable unification algorithm. If we do not use the -calculus, but translate higher-order expressions into firstorder expressions by standard translation techniques, we have to translate the infinite set of comprehension axioms, too. Of course, in general this is not practicable. Therefore such an approach requires some restrictions such as the choice of the necessary axioms by a human user or the restriction to certain problem classes. This paper will show how the infinite class of comprehension axioms can be represented by a finite subclass, so that an automatic translation of finite higher-order problems into finite first-order problems is possible. This translation is sound and complete with respect to a Henkin-style general model semantics. 1 Introduction First-order logic is a powerful tool for ...

### Citations

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Citation Context ...c Leon Henkin weakened the standard semantics of higher-order logic [11] and thereby showed the way to complete calculi for higherorder logic, in particular for Alonzo Church's simple theory of types =-=[6]-=-. Nevertheless, it is hard to build a corresponding theorem prover. For instance, an adequate higher-order theorem prover has to incorporate higher-order unification as introduced by G'erard Huet, whi... |

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Citation Context ...rength, in particular higherorder logic with sorts, is adequate for representing mathematics. Unfortunately one has to pay a price, namely that the notions of truth and provability no longer coincide =-=[9]-=-. In order to be able to operationalize higher-order logic Leon Henkin weakened the standard semantics of higher-order logic [11] and thereby showed the way to complete calculi for higherorder logic, ... |

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Citation Context ...ogic The syntax of our higher-order logic is essentially based on Alonzo Church's simple theory of types [6]. For details on higher-order logic the reader is referred to the excellent presentation in =-=[2]-=-. For the finite representability of the comprehension axioms the finiteness of the order of the logic is crucial, that is, we can translate problems, formulated in an n-th order logic L n , but not t... |

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Citation Context ...ce, namely that the notions of truth and provability no longer coincide [9]. In order to be able to operationalize higher-order logic Leon Henkin weakened the standard semantics of higher-order logic =-=[11]-=- and thereby showed the way to complete calculi for higherorder logic, in particular for Alonzo Church's simple theory of types [6]. Nevertheless, it is hard to build a corresponding theorem prover. F... |

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Citation Context ...ollowing we introduce the syntax and semantics of 2 For the direct operationalization of higher-order logic there is an alternative to Church's -calculus, namely the calculus of combinators, see e.g. =-=[13]-=-. 3 In another approach, first-order formulations of a class of secondorder problems can be achieved by eliminating quantified secondorder predicates with the help of the resolution calculus. Compare ... |

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Citation Context ...comprehension axioms. Since the predicate and function symbols in set theory (like 2 and ") are all known in advance, a fixed set of 18 axioms is sufficient to formalize the whole set theory (Com=-=pare [10]-=-. The axioms B1 through B8 have just the structure of comprehension axioms. Nevertheless the possibility to construct the whole theory of sets on this finite axiomatization is far from being trivial, ... |

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Citation Context ...ification is undecidable and there is no complete set of most general unifiers. Another problem of higher-order resolution theorem proving is the necessity to apply so-called splitting rules (compare =-=[14]-=-). Furthermore the treatment of the extensionality of functions is not satisfactorily solved. Perhaps the most advanced system for higher-order logic is the TPS system of Peter Andrews and his group [... |

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Citation Context ...ndness from first-order logic to higher-order logic, Leon Henkin has generalized the semantics to a weak version that allows for complete calculi and which is generally taken -- in some variant (e.g. =-=[1]) -- as the base of -=-higher-order theorem proving. Henkin's very idea is to replace the "=" in the relation of the universes above by a "`", that is the restriction is only D (ff 1 \Theta\Delta\Delta\D... |

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Citation Context ...detailed discussion on mathematical logic and set theory see [18, 21]. Perhaps the most promising approach to operationalizing set theory has been worked out by Robert Boyer et al. [5] and Art Quaife =-=[20]-=- employing the set theory of von Neumann, Bernays, and Godel, which enables a finite axiomatization in first-order logic. The advantage is the possibility to employ standard first-order theorem prover... |

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Citation Context ...In a previous work we have shown how this theorem can be proved using a comprehension axiom [15]. It has also been translated to set theory in [20], and proved in von Neumann-Bernays-Godel set theory =-=[19, 10, 4]-=- as proposed in [5]. However, in the proof in [15] as well as in that of [20] the key idea, namely the diagonalization construct, has been given by the human user. 9 In the following we give an Otter ... |

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Citation Context ...]). Furthermore the treatment of the extensionality of functions is not satisfactorily solved. Perhaps the most advanced system for higher-order logic is the TPS system of Peter Andrews and his group =-=[3]-=-. 2 A third approach 3 is based on a translation of higher-order expressions into (many-sorted) first-order logic. For secondorder logic this approach was introduced by Herbert B. Enderton [7]. For ge... |

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Citation Context ...e set theory of von Neumann, Bernays, and Godel, which enables a finite axiomatization in first-order logic. The advantage is the possibility to employ standard first-order theorem provers like Otter =-=[16]-=- for automated deduction and thereby make the reasoning power of first-order theorems available for higher-order theorems too. However, there are some severe drawbacks to using set theory. First, in s... |

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Citation Context ... unanswered. For a detailed discussion on mathematical logic and set theory see [18, 21]. Perhaps the most promising approach to operationalizing set theory has been worked out by Robert Boyer et al. =-=[5]-=- and Art Quaife [20] employing the set theory of von Neumann, Bernays, and Godel, which enables a finite axiomatization in first-order logic. The advantage is the possibility to employ standard first-... |

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Citation Context ...enschen [12] extended the first-order operationalization by constructs for handling the so-called comprehension axioms. This translation is insofar incomplete as it does not handle extensionality. In =-=[15]-=-, the author gives a translation into standard many-sorted first-order logic that is proved to be complete with respect to a Henkin-style general model semantics. This translation method cannot be use... |

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Citation Context ...s group [3]. 2 A third approach 3 is based on a translation of higher-order expressions into (many-sorted) first-order logic. For secondorder logic this approach was introduced by Herbert B. Enderton =-=[7]-=-. For general higher-order logic, Lawrence J. Henschen [12] extended the first-order operationalization by constructs for handling the so-called comprehension axioms. This translation is insofar incom... |

8 |
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Citation Context ...In a previous work we have shown how this theorem can be proved using a comprehension axiom [15]. It has also been translated to set theory in [20], and proved in von Neumann-Bernays-Godel set theory =-=[19, 10, 4]-=- as proposed in [5]. However, in the proof in [15] as well as in that of [20] the key idea, namely the diagonalization construct, has been given by the human user. 9 In the following we give an Otter ... |

8 |
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Citation Context ...eory. However, each approach has its own advantages and drawbacks and the question of which approach is better is mainly unanswered. For a detailed discussion on mathematical logic and set theory see =-=[18, 21]-=-. Perhaps the most promising approach to operationalizing set theory has been worked out by Robert Boyer et al. [5] and Art Quaife [20] employing the set theory of von Neumann, Bernays, and Godel, whi... |

6 |
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Citation Context ...eory. However, each approach has its own advantages and drawbacks and the question of which approach is better is mainly unanswered. For a detailed discussion on mathematical logic and set theory see =-=[18, 21]-=-. Perhaps the most promising approach to operationalizing set theory has been worked out by Robert Boyer et al. [5] and Art Quaife [20] employing the set theory of von Neumann, Bernays, and Godel, whi... |

1 |
Ohlbach, `Quantifier eliminiation in second-order predicate logic
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Citation Context ...n another approach, first-order formulations of a class of secondorder problems can be achieved by eliminating quantified secondorder predicates with the help of the resolution calculus. Compare e.g. =-=[17, 8]-=-. c fl 1994 M. Kerber ECAI 94. 11th European Conference on Artificial Intelligence Edited by A. Cohn Published in 1994 by John Wiley & Sons, Ltd. our higher-order logic, then we present the finite rep... |

1 |
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Citation Context ...n of higher-order expressions into (many-sorted) first-order logic. For secondorder logic this approach was introduced by Herbert B. Enderton [7]. For general higher-order logic, Lawrence J. Henschen =-=[12]-=- extended the first-order operationalization by constructs for handling the so-called comprehension axioms. This translation is insofar incomplete as it does not handle extensionality. In [15], the au... |

1 |
Proving a Subset of Second-Order Logic with First-Order Proof Procedures
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Citation Context ...n another approach, first-order formulations of a class of secondorder problems can be achieved by eliminating quantified secondorder predicates with the help of the resolution calculus. Compare e.g. =-=[17, 8]-=-. c fl 1994 M. Kerber ECAI 94. 11th European Conference on Artificial Intelligence Edited by A. Cohn Published in 1994 by John Wiley & Sons, Ltd. our higher-order logic, then we present the finite rep... |