## Certifying solutions to permutation group problems (2003)

Venue: | In F. Baader, ed, CADE-19, LNAI 2741 |

Citations: | 12 - 0 self |

### BibTeX

@INPROCEEDINGS{Cohen03certifyingsolutions,

author = {Arjeh Cohen and Scott H. Murray and Martin Pollet and Volker Sorge},

title = {Certifying solutions to permutation group problems},

booktitle = {In F. Baader, ed, CADE-19, LNAI 2741},

year = {2003},

pages = {258--273},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. We describe the integration of permutation group algorithms with proof planning. We consider eight basic questions arising in computational permutation group theory, for which our code provides both answers and a set of certificates enabling a user, or an intelligent software system, to provide a full proof of correctness of the answer. To guarantee correctness we use proof planning techniques, which construct proofs in a human-oriented reasoning style. This gives the human mathematician the necessary insight into the computed solution, as well as making it feasible to check the solution for relatively large groups. 1

### Citations

267 | The use of explicit plans to guide inductive proofs
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- 1988
(Show Context)
Citation Context ...he stabiliser of xi in 〈Ai−1〉. Query output Since 〈Ak〉 = 1, we conclude that B is a base with stabiliser chain the groups 〈Ai〉 for i =1,...,k. Example 7. The Mathieu group on 11 points, M, has a base =-=[1, 2, 3, 4]-=-.s264 Arjeh Cohen et al. 2.7 Nonmembership Here we deal with the query that is complementary to the first one treated: Prove that the permutation g does not belong to G. Suppose we have a base B =[x1,... |

80 | Knowledge-based proof planning
- Melis, Siekmann
- 1999
(Show Context)
Citation Context ... introducing mathematical knowledge on how to proceed in the proof planning process by influencing the planner’s behaviour at choice points (e.g., which goal to tackle next or which method to prefer) =-=[12]-=-. 4.1 Hierarchical Proof Planning Omega also allows for hierarchical proof planning. The basic idea is to generate proofs of different granularity by postponing the planning process for certain subgoa... |

41 | Integrating computer algebra into proof planning
- Kerber, Kohlhase, et al.
- 1998
(Show Context)
Citation Context ...mega can produce a proof from a certificate also ensures that the certificates have indeed supplied sufficient information. We emphasise that this work extends the boundaries of what is feasible (cf. =-=[7,9,13,2]-=-). Clearly, the eight permutation group queries we are dealing with can all be handled by simple enumeration. For instance, in order to decide if a given permutation g belongs to a permutation group G... |

31 | Middle-out reasoning for logic program synthesis
- Kraan, Basin, et al.
- 1993
(Show Context)
Citation Context ... can again contain calls to GAP, which 3 Meta-variables are place-holders for terms whose actual form is computed at a later stage in the proof search. This is an example of middle-out reasoning (see =-=[10]-=-).s268 Arjeh Cohen et al. have to be in turn verified by execution of the calling tactics themselves. The overall process is extensively described in [13]. 5 Planning the Subproblems In this section, ... |

27 | P.rex: An Interactive Proof Explainer
- Fiedler
- 2001
(Show Context)
Citation Context ... turning answers and certificates into natural language proofs using simple templates. Similarly, the proofs produced by Omega can be turned into natural language by means of the related P.Rex system =-=[5]-=-, which employs elaborate linguistic techniques. Notice that the natural language proof from GAP does not satisfy a correctness criterion, as it is implemented by humans and so a mistake in the proof ... |

27 |
A skeptic’s approach to combining hol and Maple
- Harrison, Théry
- 1998
(Show Context)
Citation Context ...mega can produce a proof from a certificate also ensures that the certificates have indeed supplied sufficient information. We emphasise that this work extends the boundaries of what is feasible (cf. =-=[7,9,13,2]-=-). Clearly, the eight permutation group queries we are dealing with can all be handled by simple enumeration. For instance, in order to decide if a given permutation g belongs to a permutation group G... |

23 | Comparing approaches to the exploration of the domain of residue classes
- Meier, Pollet, et al.
(Show Context)
Citation Context ... proof, select-instance keeps a record of already conducted computations and their results. select-instance is a generic control rule that has also been used in other contexts and with other CAS (see =-=[11]-=- for example). The use of calculations for (2) is realised by three methods: two to simplify terms, by computing permutation applications or permutation compositions, and one to solve equations. In th... |

6 |
Formal and efficient primality proofs by use of computer algebra oracles
- Caprotti, Oostdijk
(Show Context)
Citation Context ...mega can produce a proof from a certificate also ensures that the certificates have indeed supplied sufficient information. We emphasise that this work extends the boundaries of what is feasible (cf. =-=[7,9,13,2]-=-). Clearly, the eight permutation group queries we are dealing with can all be handled by simple enumeration. For instance, in order to decide if a given permutation g belongs to a permutation group G... |

6 | Non-trivial symbolic computations in proof planning, in
- Sorge
(Show Context)
Citation Context |

5 |
et al. Proof development with OMEGA
- Siekmann
- 2002
(Show Context)
Citation Context ...he same certificates to formally guarantee correctness with proof planning. The experiments were carried out by combining the computer algebra system GAP [6] and the proof planner of the Omega system =-=[4]-=-. We chose GAP, since it is particularly good in group theory and has a rich collection of permutation group algorithms (we could equally well have chosen Magma, however). The choice of Omega was moti... |

1 |
An automated proof theory approach to computation with permutation groups
- Cohen, Murray
- 2002
(Show Context)
Citation Context ...In other words, the elements of A belong to the symmetric group Sym n. We often write G = 〈A〉 to denote that G is generated by A. Our permutations act on the right. In this section, which is based on =-=[3]-=-, we give efficient solutions of eight basic queries. GAP already has efficient built-in functions which answer these queries. We have written functions in the GAP language that provide certificates f... |

1 | Solutions to Permutation Group Problems 273 - Springer |