## Using Rippling for Equational Reasoning (1996)

Venue: | In Proceedings 20th German Annual Conference on Artificial Intelligence KI-96 |

Citations: | 6 - 3 self |

### BibTeX

@INPROCEEDINGS{Hutter96usingrippling,

author = {Dieter Hutter},

title = {Using Rippling for Equational Reasoning},

booktitle = {In Proceedings 20th German Annual Conference on Artificial Intelligence KI-96},

year = {1996},

pages = {121--134},

publisher = {Springer}

}

### OpenURL

### Abstract

. This paper presents techniques to guide equational reasoning in a goal directed way. Suggested by rippling methods developed in the field of inductive theorem proving we use annotated terms to represent syntactical differences of formulas. Based on these annotations and on hierarchies of function symbols we define different abstractions of formulas which are used for planning of proofs. Rippling techniques are used to refine single planning steps, e.g. the application of a bridge lemma, on a next planning level. Fachbeitrag. Keywords: Automated reasoning, Theorem Proving, Rippling 1 Introduction Heuristics for judging similarities between formulas and subsequently reducing differences have been applied to automated deduction since the 1950s, when Newell, Shaw, and Simon built their first "logic machine" [NSS63]. Since the later 60s, a similar theme of difference identification and reduction appears in the field of resolution theorem proving [Mor69], [Dig85], [BS88]. Partial unifica...

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Citation Context ...ramodulation is used only when needed. In the field of inductive theorem proving syntactical differences between the induction hypothesis and induction conclusion are used in order to guide the proof =-=[Bun88]-=-, [BvHS91], [Hut90], [Hut]. This method to guide induction proofs is called rippling / coloring terms and a large amount of papers have been published which report its success on practical examples. F... |

93 | Experiments with proof plans for induction
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Citation Context ...ion is used only when needed. In the field of inductive theorem proving syntactical differences between the induction hypothesis and induction conclusion are used in order to guide the proof [Bun88], =-=[BvHS91]-=-, [Hut90], [Hut]. This method to guide induction proofs is called rippling / coloring terms and a large amount of papers have been published which report its success on practical examples. For equalit... |

61 |
Guiding inductive proofs
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(Show Context)
Citation Context ...d only when needed. In the field of inductive theorem proving syntactical differences between the induction hypothesis and induction conclusion are used in order to guide the proof [Bun88], [BvHS91], =-=[Hut90]-=-, [Hut]. This method to guide induction proofs is called rippling / coloring terms and a large amount of papers have been published which report its success on practical examples. For equality reasoni... |

44 |
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(Show Context)
Citation Context ...or judging similarities between formulas and subsequently reducing differences have been applied to automated deduction since the 1950s, when Newell, Shaw, and Simon built their first "logic mach=-=ine" [NSS63]-=-. Since the later 60s, a similar theme of difference identification and reduction appears in the field of resolution theorem proving [Mor69], [Dig85], [BS88]. Partial unification results in a special ... |

41 | A Calculus for and Termination of Rippling - BASIN, WALSH - 1996 |

32 | The Karlsruhe induction theorem proving system - Biundo, Hummel, et al. - 1986 |

32 |
E-resolution: extension of resolution to include the equality relation
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Citation Context ...n Newell, Shaw, and Simon built their first "logic machine" [NSS63]. Since the later 60s, a similar theme of difference identification and reduction appears in the field of resolution theore=-=m proving [Mor69]-=-, [Dig85], [BS88]. Partial unification results in a special kind of (E or RUE-) resolution step which introduces - in case of failure to unify - new inequalities that represents the differences of ter... |

27 | Difference unification - Basin, Walsh - 1992 |

25 | Colouring terms to control equational reasoning
- Hutter
- 1997
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Citation Context ...en needed. In the field of inductive theorem proving syntactical differences between the induction hypothesis and induction conclusion are used in order to guide the proof [Bun88], [BvHS91], [Hut90], =-=[Hut]-=-. This method to guide induction proofs is called rippling / coloring terms and a large amount of papers have been published which report its success on practical examples. For equality reasoning we u... |

16 | Some generalization heuristics in proofs by induction - Aubin - 1975 |

16 |
Semi-automated Mathematics
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Citation Context ...eta sum(x) = (x \Theta s(x) + s(x)) \Theta s(x)+sum(x) \Theta sum(x) Enlarging the skeleton again finishes the proof. Another Example We will illustrate our approach by another example of SAM's lemma =-=[GOBS69]-=- which is still a challenge equality problem [McC88] although it has also been solved by some resolution provers without paramodulation. Given two associative, commutative, and idempotent functions f ... |

14 | A methodology for equational reasoning - Cleve, Hutter - 1994 |

10 | A Guide to UNICOM, an Inductive Theorem Prover Based on Rewriting and Completion Techniques
- GRAMLICH, LINDNER
- 1991
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Citation Context ...an explicit calculation of ack(5; 5) which is expensive and unnecessary. Although this seems to be a rather artificial example, the same effects can be observed when proving theorems about series. In =-=[GL91]-=- huge terms denoting the coefficients of the polynomial are subject to time-consuming normalization without advancing the proof further. Also avoiding normalization preserves the structure of a theore... |

9 | The management of heuristic search in boolean experiments with RUE resolution
- Digricoli
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Citation Context ... Shaw, and Simon built their first "logic machine" [NSS63]. Since the later 60s, a similar theme of difference identification and reduction appears in the field of resolution theorem proving=-= [Mor69], [Dig85]-=-, [BS88]. Partial unification results in a special kind of (E or RUE-) resolution step which introduces - in case of failure to unify - new inequalities that represents the differences of terms. This ... |

8 |
Partial unification for graph based equational reasoning
- Blasius, Siekmann
- 1988
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Citation Context ...d Simon built their first "logic machine" [NSS63]. Since the later 60s, a similar theme of difference identification and reduction appears in the field of resolution theorem proving [Mor69],=-= [Dig85], [BS88]-=-. Partial unification results in a special kind of (E or RUE-) resolution step which introduces - in case of failure to unify - new inequalities that represents the differences of terms. This results ... |

6 |
Challenge equality problems in lattice theory
- McCune
- 1988
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Citation Context ...) \Theta sum(x) Enlarging the skeleton again finishes the proof. Another Example We will illustrate our approach by another example of SAM's lemma [GOBS69] which is still a challenge equality problem =-=[McC88]-=- although it has also been solved by some resolution provers without paramodulation. Given two associative, commutative, and idempotent functions f and g and additionally, the following set of axioms:... |

6 |
A.: Deduction in the verification support environment
- Hutter, Langenstein, et al.
- 1996
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Citation Context ...90% of these suggestions. About 50% of these candidates passing the difference matching test were used for the resulting proof. In general, the INKA-System is itself integrated into a tool called VSE =-=[VSE96]-=- for the formal development of software. Within this tool several large industrial case studies have been performed and up to now more than 20,000 lines of code have been verified which causes that th... |