## 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.

### Citations

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Citation Context ...he universally quantified variable Xoα denotes a mapping from objects of type α to objects of type o. Weuse Church’s notation oα, which stands for the functional type α → o. The reader is referred to =-=[1]-=- for a more detailed introduction. In the remainder, o will denote the type of truth values, and small Greek letters will denote arbitrary types. Thus, Xoα (resp. its η-longform λyα Xy) is actually a ... |

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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 ...enzmüller et al. calculi 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... |

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Citation Context ...ntly from the proof generation. In particular, reasoning steps of ATPs have to be translated into Oants’s natural deduction calculus via the Tramp proofs426 C. Benzmüller et al. transformation system =-=[17]-=- to be machine-checkable. Since the cooperative proof result of Leo-Bliksem cannot yet be directly inserted into the centralised proof object, the generation of a machine-checkable proof object is not... |

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Citation Context ...Order Cooperation via Oants The cooperation between higher-oder and first-order reasoners, which we investigate in this paper, is realised in the concurrent hierarchical blackboard architecture Oants =-=[7]-=-. We first describe in Sec. 3.1 the existing Oants architecture. In order to overcome some of its problems, in particular efficiency problems, we devised within Oants a new and improved cooperation me... |

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Citation Context ... = (λxα ⊥)) ⇔ (λxα (Ax ∧¬Bx)) = (λxα Ax)] F which it normalises into: [(λxα (Ax ∧ Bx)) = ? (λxα ⊥)] ∨ [(λxα (Ax ∧¬Bx)) = ? (λxα Ax)] (9) [(λxα (Ax ∧ Bx)) = (λxα ⊥)] T ∨ [(λxα (Ax ∧¬Bx)) = (λxα Ax)] T =-=(10)-=- As mentioned before, the unification constraint (9) corresponds to: [(λxα (Ax ∧ Bx)) = (λxα ⊥)] F ∨ [(λxα (Ax ∧¬Bx)) = (λxα Ax)] F (11) Leo has to apply to each of these clauses and to each of their ... |

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Citation Context ...re needs 108 seconds. The Saturate system [14] (which extends Vampire with Boolean extensionality rules that are a one-to-one correspondence to Leo’s rules for Extensional Higher-Order Paramodulation =-=[3]-=-) can solve the problem in 2.9 seconds while generating 159 clauses. The significance of such comparisons is clearly limited since different systems are optimised to a different degree. One noted diff... |

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Citation Context ...ion, Leo is first given the input clause: [∀Aoα,Boα (λxα (Ax ∧ Bx)) = (λxα ⊥)) ⇔ (λxα (Ax ∧¬Bx)) = (λxα Ax)] F which it normalises into: [(λxα (Ax ∧ Bx)) = ? (λxα ⊥)] ∨ [(λxα (Ax ∧¬Bx)) = ? (λxα Ax)] =-=(9)-=- [(λxα (Ax ∧ Bx)) = (λxα ⊥)] T ∨ [(λxα (Ax ∧¬Bx)) = (λxα Ax)] T (10) As mentioned before, the unification constraint (9) corresponds to: [(λxα (Ax ∧ Bx)) = (λxα ⊥)] F ∨ [(λxα (Ax ∧¬Bx)) = (λxα Ax)] F ... |

15 |
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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 ... |

14 |
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Citation Context ...rst-order problems for several reasons. Unlike in the case of first-order provers, for which sophisticated calculi and strategies, as well as advanced implementation techniques, such as term indexing =-=[19]-=-, have been developed, fully mechanisable higher-order calculi are still at a comparably early stage of development. Some problems are much harder in higher-order, for instance, unification is undecid... |

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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 ...concepts, such as sets, functions, or relations, today’s state-of-the-art first-order automated theorem provers (ATPs) still exhibit weaknesses on problems considered relatively simple by humans (cf. =-=[14]-=-). One reason is that the problem formulations use an encoding in a first-order set theory, which makes it particularly challenging when trying to prove theorems from first principles, that is, basic ... |

10 |
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(Show Context)
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... |

10 | Universität des Saarlandes - thesis - 2006 |

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(Show Context)
Citation Context ...ur approach, partial results in Techs are exchanged between the different theorem provers in form of clauses. The main difference to the work of Denzinger et al. (and other related architectures like =-=[13]-=-) is that our system bridges between higher-order and first-order automated theorem proving. Also, unlike in Techs, we provide a declarative specification framework for modelling external systems as c... |

7 |
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Citation Context ... in Leo. It is therefore not surprising that most first-order ATPs still fail to prove this problem. In fact, very few TPTP provers were successful in proving SET171+3. Amongst them are Muscadet 2.4. =-=[20]-=-, Vampire 7.0, and Saturate. The natural deduction system Muscadet uses special inference rules for sets and needs 0.2 seconds to prove this problem. Vampire needs 108 seconds. The Saturate system [14... |

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(Show Context)
Citation Context ...[14], in Sec. 4 to show the effectiveness of our approach. While many of the considered problems can be proved by Leo alone with some strategy, the combination of Leo with the first-order ATP Bliksem =-=[11]-=- is not only able to show more problems, but also needs only a single strategy to solve them. Several of our problems are considered very challenging by the first-order community and five of them (of ... |

3 |
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(Show Context)
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 |
Experiments with an AgentOriented Reasoning
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(Show Context)
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 |