## Exploring Properties of Residue Classes (2000)

Citations: | 18 - 11 self |

### BibTeX

@INPROCEEDINGS{Meier00exploringproperties,

author = {Andreas Meier and Volker Sorge},

title = {Exploring Properties of Residue Classes},

booktitle = {},

year = {2000},

pages = {175--190},

publisher = {A K Peters}

}

### Years of Citing Articles

### OpenURL

### Abstract

We report on an experiment in exploring properties of residue classes over the integers with the combined effort of a multi-strategy proof planner and two computer algebra systems. An exploration module classifies a given set and a given operation in terms of the algebraic structure they form. It then calls the proof planner to prove or refute simple properties of the operation. Moreover, we use different proof planning strategies to implement various proving techniques: from naive testing of all possible cases to elaborate techniques of equational reasoning and reduction to known cases.

### Citations

265 | The use of explicit plans to guide inductive proofs
- Bundy
- 1988
(Show Context)
Citation Context ...ture it forms, i.e., whether it is a magma (property 1 holds), semi-group (1+2), monoid (1+2+3), quasi-group (1+5), loop (1+5+3), or group (1+2+3+4) 1 . 3 Multi-Strategy Proof Planning Proof planning =-=[2]-=- considers mathematical theorems as planning problems where an initial partial plan is composed of the proof assumptions and the theorem as open goal. A proof is then constructed with the help of abst... |

80 | Knowledge-based proof planning
- Melis, Siekmann
- 1999
(Show Context)
Citation Context .... 1 Naturally property 5 holds for a group as well. 4 In the mega system [6] the traditional proof planning approach is enriched by incorporating mathematical knowledge into the planning process (see =-=[13]-=- for details). That is, methods can encode general proving steps as well as knowledge particular to a mathematical domain. Moreover, control rules specify how to traverse the search space by preferrin... |

56 | Automated reasoning and exhaustive search: Quasigroup existence problems
- Slaney, Fujita, et al.
- 1993
(Show Context)
Citation Context ...peration between the deduction system Nuprl and the computer algebra system Weyl in the domain of abstract algebra is sketched in [7]. More concrete work in exploration insnite algebra is reported in =-=[3, 11, 15]-=- where model generation techniques are used to tackle quasi-group existence problems. In particular, some open problems in quasi-group theory were solved. Our work has neither the claim to advance the... |

53 | Proof-Planning with multiple strategies
- Meier
- 2003
(Show Context)
Citation Context ... proof obligations are theorems of the form: the set is closed under the operation, or the operation is not associative, etc. Proof obligations are passed to mega's multi-strategy proof planner Multi =-=[12]-=- that constructs a proof with the help of the two computer algebra systems Maple [14] and Gap [4]. Although the proofs are not very pretentious in their own right they need to be constructed in a way ... |

43 |
Transforming Matings into Natural Deduction Proofs
- Andrews
- 1980
(Show Context)
Citation Context ...he overall planning behavior. Proof plans in mega result in proof objects in a variant of the natural deduction calculus [5]. We will present them in this paper in a linearized style as introduced in [1]. A proof line is of the form L: `F (R), where L is a unique label, `F a sequent denoting that the formula F can be derived from the set of hypotheses , and (R) is a justication expressing how the... |

41 | Integrating computer algebra into proof planning
- Kerber, Kohlhase, et al.
- 1998
(Show Context)
Citation Context ...l not be concerned with the technical side and the soundness issues of integrating computer algebra with proof planning in general and with mega in particular since this has already been discussed in =-=[16, 9]-=-. We rather concentrate on the cooperation between the systems in the context of exploring residue class properties. We use symbolic calculations in two ways: (1) in control rules hints are computed t... |

31 | Middle-out reasoning for logic program synthesis
- Kraan, Basin, et al.
- 1993
(Show Context)
Citation Context .... However, if the method is applied in an early stage of the proof, the planner generally has no knowledge on the true nature of the witness term. Therefore, the method invokes a middle-out-reasoning =-=[10]-=- process to postpone the actual instantiation, that is, a meta-variable is used as placeholder for the actual witness term which will be determined at a later point in the planning process and subsequ... |

23 |
The Maple Handbook: Maple V Release 5
- Redfern
- 1998
(Show Context)
Citation Context ...r the operation is not associative, etc. Proof obligations are passed to mega's multi-strategy proof planner Multi [12] that constructs a proof with the help of the two computer algebra systems Maple =-=[14-=-] and Gap [4]. Although the proofs are not very pretentious in their own right they need to be constructed in a way that dierent proving techniques can be taught to a user of the interactive course. T... |

22 |
Exploring abstract algebra in constructive type theory
- Jackson
- 1994
(Show Context)
Citation Context ...e combined system to a specic problem domain. A possibly fruitful cooperation between the deduction system Nuprl and the computer algebra system Weyl in the domain of abstract algebra is sketched in [=-=7]-=-. More concrete work in exploration insnite algebra is reported in [3, 11, 15] where model generation techniques are used to tackle quasi-group existence problems. In particular, some open problems in... |

13 |
A Davis-Putnam program and its application to model search
- McCune
- 1994
(Show Context)
Citation Context ...peration between the deduction system Nuprl and the computer algebra system Weyl in the domain of abstract algebra is sketched in [7]. More concrete work in exploration insnite algebra is reported in =-=[3, 11, 15]-=- where model generation techniques are used to tackle quasi-group existence problems. In particular, some open problems in quasi-group theory were solved. Our work has neither the claim to advance the... |

12 |
Untersuchungen über das logische
- Gentzen
- 1935
(Show Context)
Citation Context ..., allows also for the specication of dierent planning strategies to control the overall planning behavior. Proof plans in mega result in proof objects in a variant of the natural deduction calculus [5]. We will present them in this paper in a linearized style as introduced in [1]. A proof line is of the form L: `F (R), where L is a unique label, `F a sequent denoting that the formula F can be der... |

12 |
Non-Trivial Computations in Proof Planning
- Sorge
- 2000
(Show Context)
Citation Context ...l not be concerned with the technical side and the soundness issues of integrating computer algebra with proof planning in general and with mega in particular since this has already been discussed in =-=[16, 9]-=-. We rather concentrate on the cooperation between the systems in the context of exploring residue class properties. We use symbolic calculations in two ways: (1) in control rules hints are computed t... |

6 |
Automatic generation of some results in algebra
- Fujita, Slaney, et al.
- 1993
(Show Context)
Citation Context ...peration between the deduction system Nuprl and the computer algebra system Weyl in the domain of abstract algebra is sketched in [7]. More concrete work in exploration insnite algebra is reported in =-=[3, 11, 15]-=- where model generation techniques are used to tackle quasi-group existence problems. In particular, some open problems in quasi-group theory were solved. Our work has neither the claim to advance the... |