## Proving Ground Completeness of Resolution by Proof Planning (1997)

Citations: | 2 - 1 self |

### BibTeX

@MISC{Kerber97provingground,

author = {Manfred Kerber and Arthur Christian Sehn},

title = {Proving Ground Completeness of Resolution by Proof Planning},

year = {1997}

}

### OpenURL

### Abstract

A lot of the human ability to prove hard mathematical theorems can be ascribed to a problem-specific problem solving know-how. Such knowledge is intrinsically incomplete. In order to prove related problems human mathematicians, however, can go beyond the acquired knowledge by adapting their know-how to new related problems. These two aspects, having rich experience and extending it by need, can be simulated in a proof planning framework: the problem-specific reasoning knowledge is represented in form of declarative planning operators, called methods; since these are declarative, they can be mechanically adapted to new situations by so-called metamethods. In this contribution we apply this framework to two prominent proofs in theorem proving, first, we present methods for proving the ground completeness of binary resolution, which essentially correspond to key lemmata, and then, we show how these methods can be reused for the proof of the ground completeness of lock resolution.

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Citation Context ... which a human mathematician would intuitively consider as an instance of homomorphy. A well-known example is the rippling method developed in Bundy's group, which has been extended from rippling-out =-=[5]-=- to a method covering a wide range of related problems [6]. The intention of our work can be compared to Ireland's approach of proof critics [17]. While proof critics are specific and attached to sing... |

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Citation Context ...thods essentially consist of pre- and postconditions for the method to allow connecting them to proof plans. These proof plans are the basic elements of a more general planning framework (e.g. CL A M =-=[7, 8, 6]-=-). The reasoning power of such plan-based systems is not derived from a (theoretically) complete underlying reasoning calculus, but relies on domain-specific problem solving knowledge. Indeed the meth... |

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Citation Context ...thods essentially consist of pre- and postconditions for the method to allow connecting them to proof plans. These proof plans are the basic elements of a more general planning framework (e.g. CL A M =-=[7, 8, 6]-=-). The reasoning power of such plan-based systems is not derived from a (theoretically) complete underlying reasoning calculus, but relies on domain-specific problem solving knowledge. Indeed the meth... |

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Citation Context ...thods essentially consist of pre- and postconditions for the method to allow connecting them to proof plans. These proof plans are the basic elements of a more general planning framework (e.g. CL A M =-=[7, 8, 6]-=-). The reasoning power of such plan-based systems is not derived from a (theoretically) complete underlying reasoning calculus, but relies on domain-specific problem solving knowledge. Indeed the meth... |

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Citation Context ...undy's group, which has been extended from rippling-out [5] to a method covering a wide range of related problems [6]. The intention of our work can be compared to Ireland's approach of proof critics =-=[17]-=-. While proof critics are specific and attached to single methods, meta-methods embody general problem independent procedures for adapting arbitrary methods which meet some applicability conditions. T... |

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Citation Context ...hed to single methods, meta-methods embody general problem independent procedures for adapting arbitrary methods which meet some applicability conditions. The related work of Giunchiglia and Traverso =-=[10]-=- to represent tactics in a logical meta-language has a similar motivation as well, namely to mechanically adapt existing tactics. Their formalism is more expressive since they can represent procedural... |

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Citation Context ... problem of mechanically modifying procedures. In previous work, we have developed a declarative approach for representing methods that enables a mechanical modification and adaptation to novel tasks =-=[15, 16]-=-. In the following section we shortly recall the general framework (the description of the formal framework as well as parts of this introduction are based on [13]) and then present the user-written m... |

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Citation Context ...[3], in particular we use the lemmata proposed there for working out the main theorem. This proof is the most advanced case study that has been done with the\Omega -MKRP proof development environment =-=[14]-=- up to now and is first described in [20]. In this work we show that our general framework can not only be applied to specify and automatically solve this problem, but that this proof can even be tran... |

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Citation Context ...f this introduction are based on [13]) and then present the user-written methods for proving the ground completeness of binary resolution. In doing so we follow Bledsoe's formalisation of the problem =-=[3]-=-, in particular we use the lemmata proposed there for working out the main theorem. This proof is the most advanced case study that has been done with the\Omega -MKRP proof development environment [14... |

8 | Adapting methods to novel tasks in proof planning
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(Show Context)
Citation Context ... problem of mechanically modifying procedures. In previous work, we have developed a declarative approach for representing methods that enables a mechanical modification and adaptation to novel tasks =-=[15, 16]-=-. In the following section we shortly recall the general framework (the description of the formal framework as well as parts of this introduction are based on [13]) and then present the user-written m... |

7 | Omega\Gamma mkrp -- a proof development environment - Huang, Kerber, et al. - 1994 |

5 |
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(Show Context)
Citation Context ...oof plans for them); the other hypotheses are standard definitions and properties of set theory. In Bledsoe's approach these hypotheses are not necessary, since they are basic to the Str : +ve-Prover =-=[12]-=- for set theory. These properties are easy properties of the subset relation and can straightforwardly be proved either by calling an automated theorem prover or by further planning processes. The sup... |

5 |
DECLAME { eine deklarative Sprache zur Reprasentation von Methoden
- Sehn
- 1995
(Show Context)
Citation Context ...posed there for working out the main theorem. This proof is the most advanced case study that has been done with the\Omega -MKRP proof development environment [14] up to now and is first described in =-=[20]-=-. In this work we show that our general framework can not only be applied to specify and automatically solve this problem, but that this proof can even be transferred to a similar problem, namely the ... |

4 | Adaptation of declaratively represented methods in proof planning
- Huang, Kerber, et al.
- 1995
(Show Context)
Citation Context ...on and adaptation to novel tasks [15, 16]. In the following section we shortly recall the general framework (the description of the formal framework as well as parts of this introduction are based on =-=[13]-=-) and then present the user-written methods for proving the ground completeness of binary resolution. In doing so we follow Bledsoe's formalisation of the problem [3], in particular we use the lemmata... |

1 |
A Linear Proof Format for Resolution with Merging and a New Technique for Establishing Completeness
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- 1970
(Show Context)
Citation Context ...as: exl(S) = [ P C2S jCj] \Gamma jSj, where jCj is the number of literals in the clause C and jSj is the number of clauses in the clause set S (this so-called k--parameter technique was introduced in =-=[1]-=-). We now turn to the proof of the completeness of binary ground resolution. The basic idea is to use an inductive argument on the excess literal number of the clause set S. We do not describe the bas... |