## Can a higher-order and a first-order theorem prover cooperate? (2005)

Venue: | IN FRANZ BAADER AND ANDREI VORONKOV, EDITORS, LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE, AND REASONING — 11TH INTERNATIONAL WORKSHOP, LPAR 2004, LNAI 3452 |

Citations: | 11 - 8 self |

### BibTeX

@INPROCEEDINGS{Benzmüller05cana,

author = {Christoph Benzmüller and Volker Sorge and Mateja Jamnik and Manfred Kerber},

title = {Can a higher-order and a first-order theorem prover cooperate?},

booktitle = {IN FRANZ BAADER AND ANDREI VORONKOV, EDITORS, LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE, AND REASONING — 11TH INTERNATIONAL WORKSHOP, LPAR 2004, LNAI 3452},

year = {2005},

pages = {415--431},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

State-of-the-art first-order 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, first-order systems still exhibit serious weaknesses. While it has been shown in the past that higher-order 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 higher-order and a first-order 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 non-trivial for many first-order theorem provers.

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Citation Context ...g of 1.00, saying that they cannot be solved by any TPTP prover to date. Technically, the combination — described in more detail in Sec. 3 — has been realised in the concurrent reasoning system Oants =-=[22,8]-=- which enables the cooperation of hybrid reasoning systems to construct a common proof object. In our past experiments, Oants has been successfully employed to check the validity of set equations usin... |

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Citation Context ...) representation. For the experiments with Leo and the cooperation of Leo with the first-order theorem prover Bliksem, λ-abstraction as well as the extensionality treatment inherent in Leo’s calculus =-=[4]-=- is used. This enables a theoretically 4 Henkincomplete proof system for set theory. In the above example SET171+3, Leo generally uses the application of functional extensionality to push extensional ... |

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Citation Context ... the Saturate system. The extended Saturate system can solve some problems from the SET domain in the TPTP [24] which Vampire [21] and E-Setheo’s [23] cannot solve. While it has already been shown in =-=[6,2]-=- that many problems of this nature can be easily proved from first principles using a concise higher-order representation and the higher-order resolution ATP Leo, the combinatorial explosion inherent ... |

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Citation Context ...culi and systems, as for example presented in [14] for the Saturate system. The extended Saturate system can solve some problems from the SET domain in the TPTP [24] which Vampire [21] and E-Setheo’s =-=[23]-=- cannot solve. While it has already been shown in [6,2] that many problems of this nature can be easily proved from first principles using a concise higher-order representation and the higher-order re... |

2 |
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Citation Context ...mmon proof object. In our past experiments, Oants has been successfully employed to check the validity of set equations using higher-order and first-order ATPs, model generation, and computer algebra =-=[5]-=-. While this already enabled a cooperation between Leo and a first-order ATP, the proposed solution could not be classified as a general purpose approach. A major shortcoming was that all communicatio... |

1 | Comparing approaches to resolution based higher-order theorem proving - unknown authors - 1998 |