## Integrating computer algebra into proof planning (1998)

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Venue: | Journal of Automated Reasoning |

Citations: | 42 - 27 self |

### BibTeX

@ARTICLE{Kerber98integratingcomputer,

author = {Manfred Kerber and Michael Kohlhase and Volker Sorge},

title = {Integrating computer algebra into proof planning},

journal = {Journal of Automated Reasoning},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Mechanised reasoning systems and computer algebra systems have different objectives. Their integration is highly desirable, since formal proofs often involve both of the two di erent tasks, proving and calculating. Even more importantly, proof and computation are often interwoven and not easily separable. In this contribution we advocate an integration of computer algebra into mechanised reasoning systems at the proof plan level. This approach allows to view the computer algebra algorithms as methods, that is, declarative representations of the problem solving knowledge speci c to a certain mathematical domain. Automation can be achieved in many cases bysearching for a hierarchic proof plan at the methodlevel using suitable domain-speci c control knowledge about the mathematical algorithms. In other words, the uniform framework of proof planning allows to solve a large class of problems that are not automatically solvable by separate systems. Our approach also gives an answer to the correctness problems inherent insuch an integration. We advocate an approach where the computer algebra system produces high-level protocol information that can be processed by aninterface to derive proof plans. Such a proof plan in turn can be expanded to proofs at di erent levels of abstraction, so the approach iswell-suited for producing a high-level verbalised explication as well as for a low-level machine checkable calculus-level proof. We present an implementation of our ideas and exemplify them using an automatically solved example. Changes in the criterion of `rigour of the proof ' engender major revolutions in mathematics.

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Citation Context ...s left to be proved by RSYS. In any case, the proof found by RSYS must eventually be transformed into a PDS, since this is the proof-theoretic basis of �MEGA. For automated theorem provers like OTTER =-=[32]-=-, we described the integration in [25] and the necessary proof transformation to PDS in [27], so we will not pursue this matter here. The integration of CAS follows the same paradigm and is the main t... |

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Citation Context ...INTEGRATING COMPUTER ALGEBRA INTO PROOF PLANNING 335 The goal of proof planning is to fill gaps in a given PDS by forward and backward reasoning [26] (proof plans were first introduced by Bundy; see =-=[10, 11]-=-). Thus, from an abstract point of view the planning process is the process of exploring the search space of planning states that is generated by the plan operators in order to find a complete plan fr... |

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Citation Context ...INTEGRATING COMPUTER ALGEBRA INTO PROOF PLANNING 335 The goal of proof planning is to fill gaps in a given PDS by forward and backward reasoning [26] (proof plans were first introduced by Bundy; see =-=[10, 11]-=-). Thus, from an abstract point of view the planning process is the process of exploring the search space of planning states that is generated by the plan operators in order to find a complete plan fr... |

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Citation Context ...ter algebra systems. munication. Its rôle is to automate the broadcasting of messages by transforming output of one system into data that can be processed by the other. ⋆ Unlike other approaches (see =-=[23, 19]-=-, for example), we do not want to change the logic inside our MRS. In the same line, we do not want to change the computational behavior of the computer algebra algorithms. In order to achieve this go... |

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Citation Context ...1. Proof Explanation Proof presentation is one important feature of a mathematical assistant that has been neglected by traditional deduction systems. �MEGA employs an extension of the PROVERB system =-=[27]-=- developed by our group that allows for presenting proofs and proof plans in natural language. In order to produce coherent texts that resemble those found in mathematical textbooks, PROVERB employs s... |

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Citation Context ... explanation or proof of their solution to the problem at hand, even though some mathematical theories like that of Gröbner bases can be successfully applied to theorem proving in elementary geometry =-=[13, 29, 14, 35]-=-.330 M. KERBER, M. KOHLHASE AND V. SORGE 1.3. CONTRIBUTIONS OF THIS PAPER Not only can a mutual simulation of the tasks of an MRS and a CAS be quite inefficient, but more important, the daily work of... |

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Citation Context ...rated from this information. The proof plan in turn can be expanded by the MRS into a proof verifying the concrete computation. ⋆ This is an adaptation of the general approach on combining systems in =-=[16]-=-.346 M. KERBER, M. KOHLHASE AND V. SORGE The interface itself can be roughly divided into two parts: the translation part and the plan generator. The first performs syntax translations between �MEGA ... |

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Citation Context ...nguages of CAS differ from system to system, thereby complicating the integration of such systems as well as the cooperation between them. This deficiency has been attacked in the OpenMath initiative =-=[1]-=-, which strives for a standard CAS communication protocol. Currently the main emphasis is on standardizing the syntax and the computational behavior of the mathematical objects, while their properties... |

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Citation Context ...se, the proof found by RSYS must eventually be transformed into a PDS, since this is the proof-theoretic basis of �MEGA. For automated theorem provers like OTTER [32], we described the integration in =-=[25]-=- and the necessary proof transformation to PDS in [27], so we will not pursue this matter here. The integration of CAS follows the same paradigm and is the main topic of this paper, so we will develop... |

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Citation Context ...the same right, one can of course follow the converse direction, namely, to approach the integration from the point of the CAS; and indeed such approaches are also successfully undertaken (see, e.g., =-=[15, 9]-=-). The question about the granularity of integration is treated uniformly by all these experiments. The application of the CAS is treated as another (derived) rule of inference at the level of the (ta... |

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Citation Context ...riant of Gentzen’s calculus of natural deduction (ND).INTEGRATING COMPUTER ALGEBRA INTO PROOF PLANNING 335 The goal of proof planning is to fill gaps in a given PDS by forward and backward reasoning =-=[26]-=- (proof plans were first introduced by Bundy; see [10, 11]). Thus, from an abstract point of view the planning process is the process of exploring the search space of planning states that is generated... |

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Citation Context ...the same right, one can of course follow the converse direction, namely, to approach the integration from the point of the CAS; and indeed such approaches are also successfully undertaken (see, e.g., =-=[15, 9]-=-). The question about the granularity of integration is treated uniformly by all these experiments. The application of the CAS is treated as another (derived) rule of inference at the level of the (ta... |

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Citation Context ...oth of their main activities. Indeed, two independent systems can hardly cover their needs, since in many cases the tasks of proving and calculating are hardly separable. As pointed out by Buchberger =-=[7]-=- the integration problem is still unsolved, but it can be expected that a successful combination of these systems will lead to “a drastic improvement of the intelligence level” of such support systems... |

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Citation Context ...ter algebra systems. munication. Its rôle is to automate the broadcasting of messages by transforming output of one system into data that can be processed by the other. ⋆ Unlike other approaches (see =-=[23, 19]-=-, for example), we do not want to change the logic inside our MRS. In the same line, we do not want to change the computational behavior of the computer algebra algorithms. In order to achieve this go... |

3 |
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Citation Context ... explanation or proof of their solution to the problem at hand, even though some mathematical theories like that of Gröbner bases can be successfully applied to theorem proving in elementary geometry =-=[13, 29, 14, 35]-=-.330 M. KERBER, M. KOHLHASE AND V. SORGE 1.3. CONTRIBUTIONS OF THIS PAPER Not only can a mutual simulation of the tasks of an MRS and a CAS be quite inefficient, but more important, the daily work of... |

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Citation Context ...of inference at the level of the (tactic) calculus, so the granularity of integration depends on the granularity of the calculus or the tactics involved. In the first category of attempts (see, e.g., =-=[21, 3]-=-) one essentially trusts that the CAS properly work; hence, their results are directly incorporated into the proof. All these experiments are at least partly motivated by achieving a broader applicabi... |

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Citation Context ...ch reasoner’s results can (on demand) be transformed into a sub-PDS. The level to which the proofs have to be expanded depends on the sophistication of the proof checker. As pointed out by Barendregt =-=[4]-=-, a more complex proof-checker that accepts proofs in a more expressive formalism may drastically reduce the length of the communicated proofs. If the high-level justifications are not expanded but ac... |

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Citation Context ...d approach of integrating computer algebra systems into a particular kind of mechanized reasoning system consists in the meta-theoretic extension of the reasoning system as proposed, for instance, in =-=[6, 24]-=- and been realized in NUPRL [17]. In this approach a constructive mechanized reasoning system is basically used as its own meta-system. The constructive features are exploited to synthesize a correct ... |

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Citation Context ...rd reflects the fact that the study of symbolic calculation has long been an established and fruitful subfield of mathematics that has developed the mathematical theory and tools. Most modern systems =-=[34, 12, 28]-=- have in common that the algebraic algorithms are integrated in a very comfortable graphical user interface that includes formula editing, visualization of mathematical objects, and even an interface ... |