## Reusing Proofs (1994)

Venue: | in Proceedings of ECAI-94 |

Citations: | 34 - 8 self |

### BibTeX

@INPROCEEDINGS{Kolbe94reusingproofs,

author = {Thomas Kolbe and Christoph Walther},

title = {Reusing Proofs},

booktitle = {in Proceedings of ECAI-94},

year = {1994},

pages = {80--84},

publisher = {John Wiley & Sons, Ltd}

}

### OpenURL

### Abstract

. 1 We develop a learning component for a theorem prover designed for verifying statements by mathematical induction. If the prover has found a proof, it is analyzed yielding a so-called catch. The catch provides the features of the proof which are relevant for reusing it in subsequent verification tasks and may also suggest useful lemmata. Proof analysis techniques for computing the catch are presented. A catch is generalized in a certain sense for increasing the reusability of proofs. We discuss problems arising when learning from proofs and illustrate our method by several examples. 1 INTRODUCTION The improvement of problem solvers by reusing previously computed solutions is an active research area of Artificial Intelligence, emerging in the methodologies of explanationbased learning (EBL) [11, 4, 5] and analogical reasoning (AR) [2, 7, 12]. In EBL a problem's solution is analyzed, yielding an explanation why the solution succeeds. After generalization, the explanation is used for...

### Citations

506 |
Explanation-Based Generalization: A Unifying View
- Mitchell, Keller, et al.
- 1986
(Show Context)
Citation Context ...1 , cf. [10]. 5 CONCLUSION Our method for reusing proofs is related to abstraction techniquessin problem solving [13, 6] as well as to the machine learning methodologies of explanation-based learning =-=[11, 4, 5]-=- and reasoning by analogy [2, 7, 12], see [10] for a comparison with our method. The usefulness of our proposal depends on the frequency and the costs of proof reuses in realistic applications, and th... |

358 | Explanation-Based Learning: An alternative view
- DeJong, Mooney
- 1986
(Show Context)
Citation Context ...ION The improvement of problem solvers by reusing previously computed solutions is an active research area of Artificial Intelligence, emerging in the methodologies of explanationbased learning (EBL) =-=[11, 4, 5]-=- and analogical reasoning (AR) [2, 7, 12]. In EBL a problem's solution is analyzed, yielding an explanation why the solution succeeds. After generalization, the explanation is used for solving (simila... |

188 | A theory of abstraction
- Giunchiglia, Walsh
- 1992
(Show Context)
Citation Context ...olution of (similar) new problems by suggesting corresponding inference steps. We present an approach for reusing proofs that combines ideas of EBL and AR as well as ideas from abstraction techniques =-=[13, 6]-=-. The domain of a problem solver must exhibit a certain regularity, i.e. solutions have to be similar in some sense, because otherwise nothing can be learned for subsequent problem solving. Of course,... |

162 | A.: Rippling: A Heuristic for Guiding Inductive Proofs
- Bundy, Stevens, et al.
- 1993
(Show Context)
Citation Context ...uction hypothesis should be applied such that a tautology eventually is obtained. This problem is one of the main challenges in automated mathematical induction and therefore studied intensively, cf. =-=[3, 9, 14]-=-. The work presented here is based on the following scenario: We assume an automated theorem prover which shall be supplemented by a learning component. Once the prover has computed an induction proof... |

151 |
Proving and applying program transformations expressed with secondorder patterns
- Huet, Lang
(Show Context)
Citation Context ... reuse only. Formally, solutions for free function variables are computed by an algorithm for second-order matching "modulo evaluation " [10] which is based on the matching algorithm of Huet=-= and Lang [8]-=-. Note that the heart of success is the free function variable F 2 for which the right solution can be computed by this second-order matching. In the simple analysis approach, the bound function varia... |

79 | Computational approaches to analogical reasoning: a comparative analysis
- Hall
- 1989
(Show Context)
Citation Context ... reusing previously computed solutions is an active research area of Artificial Intelligence, emerging in the methodologies of explanationbased learning (EBL) [11, 4, 5] and analogical reasoning (AR) =-=[2, 7, 12]-=-. In EBL a problem's solution is analyzed, yielding an explanation why the solution succeeds. After generalization, the explanation is used for solving (similar) new problems. In AR a problem's soluti... |

69 |
Theorem proving with abstraction
- Plaisted
- 1981
(Show Context)
Citation Context ...olution of (similar) new problems by suggesting corresponding inference steps. We present an approach for reusing proofs that combines ideas of EBL and AR as well as ideas from abstraction techniques =-=[13, 6]-=-. The domain of a problem solver must exhibit a certain regularity, i.e. solutions have to be similar in some sense, because otherwise nothing can be learned for subsequent problem solving. Of course,... |

61 |
Guiding inductive proofs
- Hutter
- 1990
(Show Context)
Citation Context ...uction hypothesis should be applied such that a tautology eventually is obtained. This problem is one of the main challenges in automated mathematical induction and therefore studied intensively, cf. =-=[3, 9, 14]-=-. The work presented here is based on the following scenario: We assume an automated theorem prover which shall be supplemented by a learning component. Once the prover has computed an induction proof... |

56 |
Explanation-based learning: A survey of programs and perspectives
- Ellman
- 1989
(Show Context)
Citation Context ...ION The improvement of problem solvers by reusing previously computed solutions is an active research area of Artificial Intelligence, emerging in the methodologies of explanationbased learning (EBL) =-=[11, 4, 5]-=- and analogical reasoning (AR) [2, 7, 12]. In EBL a problem's solution is analyzed, yielding an explanation why the solution succeeds. After generalization, the explanation is used for solving (simila... |

41 |
Analogy for Automated Reasoning
- Owen
- 1990
(Show Context)
Citation Context ... reusing previously computed solutions is an active research area of Artificial Intelligence, emerging in the methodologies of explanationbased learning (EBL) [11, 4, 5] and analogical reasoning (AR) =-=[2, 7, 12]-=-. In EBL a problem's solution is analyzed, yielding an explanation why the solution succeeds. After generalization, the explanation is used for solving (similar) new problems. In AR a problem's soluti... |

37 |
Mathematical induction
- Walther
- 1994
(Show Context)
Citation Context ...fore a learning component can be developed. Theorem proving by (mathematical) induction is an area of automated reasoning where proofs often are similar and can be obtained in a uniform way, see e.g. =-=[14]-=- for a survey. 2 For proving some statement ' an induction theorem prover computes a set f'0 ; : : : ; 'ng of induction formulas for ' such that ['0s: : :s'n ! '] is an induction axiom. Then the syste... |

21 |
Analogical reasoning and proof discovery
- Brock, Cooper, et al.
- 1988
(Show Context)
Citation Context ... reusing previously computed solutions is an active research area of Artificial Intelligence, emerging in the methodologies of explanationbased learning (EBL) [11, 4, 5] and analogical reasoning (AR) =-=[2, 7, 12]-=-. In EBL a problem's solution is analyzed, yielding an explanation why the solution succeeds. After generalization, the explanation is used for solving (similar) new problems. In AR a problem's soluti... |

2 |
Plagiator: Entwurf und Implementierung eines lernenden Beweisers. Diploma Thesis
- Brauburger
- 1994
(Show Context)
Citation Context ...euses in realistic applications, and therefore can only be evaluated after experiments of appropriate size have been carried out. We have just finished the implementation of a system called Plagiator =-=[1]-=-. This system consists of a device for analyzing, generalizing and managing proofs and is based on the techniques discussed above. If a statement cannot be verified by reuse, the user must support the... |

1 |
Reusing Proofs --- A first Report
- Kolbe, Walther
- 1994
(Show Context)
Citation Context ...e proved (which may necessitate further proof reuses), / is verified by reuse since the truth of an instantiated schematic catch implies the truth of its instantiated schematic induction formula, cf. =-=[10]-=-. 6 E.g. assume that the new conjecture 8x; y /1 [x; y] shall be proved, where /1 [x; y] :j times(prod(x); prod(y)) = prod(app(x; y)) and times and prod are defined by the axioms (times-1) times(0; y)... |