## Paramodulation without Duplication (1995)

Venue: | Proceedings 10th IEEE Symposium on Logic in Computer Science, San Diego (Ca., USA |

Citations: | 11 - 6 self |

### BibTeX

@INPROCEEDINGS{Lynch95paramodulationwithout,

author = {Christopher Lynch},

title = {Paramodulation without Duplication},

booktitle = {Proceedings 10th IEEE Symposium on Logic in Computer Science, San Diego (Ca., USA},

year = {1995},

pages = {167--177},

publisher = {IEEE}

}

### OpenURL

### Abstract

The resolution (and paramodulation) inference systems are theorem proving procedures for first-order logic (with equality), but they can run exponentially long for subclasses which have polynomial time decision procedures, as in the case of SLD resolution and the Knuth-Bendix completion procedure, both in the ground case. Specialized methods run in polynomial time, but have not been extended to the full first-order case. We show a form of Paramodulation which does not copy literals, which runs in polynomial time for the ground case of the following four subclasses: Horn Clauses with any selection rule, any set of Unit Equalities (this includes Completion), Equational Horn Clauses with a certain selection rule, and Conditional Narrowing.

### Citations

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(Show Context)
Citation Context ...qualities. This shows that it is polynomial for completion. The search space is also polynomial for conditional narrowing, 2 and for 1 The existence of polynomialsized proofs has been studied by Haken=-=[6]-=- and others but Automated Deduction researchers are concerned with the eOEciency of searching for a proof. 2 See [11] for an explanation of this class. Equational Horn Clauses under certain selection ... |

224 |
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(Show Context)
Citation Context ...y are solved and avoid repeatedly solving them. These techniques generally only work for Horn clauses and not for rst order logic. Plaisted showed that only the MESON and Model Elimination strategies =-=[9]-=- with unit lemmas and caching [1], the simple and modied problem reduction formats [15, 16] and the Hyperlinking strategy [8] are polynomial. However, the rst two of these strategies (with caching) ar... |

198 |
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(Show Context)
Citation Context ...election rule that selects all maximal literals in each clause. Let Sel \Gamma be a selection rule that selects a negative literal in each clause containing 3 This is called Merging Paramodulation in =-=[3]-=-. We prefer it to Equational Factoring for this formalism. one, otherwise selects all maximal literals. Let Sel + be a selection rule which selects all positive literals in each clause containing one,... |

125 |
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(Show Context)
Citation Context ...ty. He also notes that none of the resolution strategies are polynomial, and suggests that maybe SLDresolution should be based on another strategy. We note that caching methods (often called memoing) =-=[22, 23]-=- have been applied to SLDresolution, but the technique is not complete for nonHorn clauses. Local Simplication [10] is a general strategy for rstorder equational theorem proving that always has polyno... |

77 | A Proof Procedure Using Connection Graphs
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(Show Context)
Citation Context ...se, it can be seen as an extension of the caching technique for logic programming and the congruence closure technique for completion to the general rst order case. The Clause Graph Resolution method =-=[7]-=- supercially resembles our method, but it copies literals, so it does not have a polynomial search space for the above cases. The Clause Graph Resolution method was developed to improve the efciency o... |

67 | Basic paramodulation
- Bachmair, Ganzinger, et al.
- 1995
(Show Context)
Citation Context ...with its multiset extension. We call a literal A maximal in C [[ ` ]] if A 2 C and there is a solution oe of ` such that AoesBoe for all B 2 C. We now present the Basic Paramodulation inference rules =-=[4, 13]-=-. They dioeer from the standard Paramodulation inference rules in that the most general unier is not applied to the conclusion of the inference. Instead, it is saved as an equational constraint in the... |

46 |
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(Show Context)
Citation Context ...with its multiset extension. We call a literal A maximal in C [[ ` ]] if A 2 C and there is a solution oe of ` such that AoesBoe for all B 2 C. We now present the Basic Paramodulation inference rules =-=[4, 13]-=-. They dioeer from the standard Paramodulation inference rules in that the most general unier is not applied to the conclusion of the inference. Instead, it is saved as an equational constraint in the... |

45 |
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(Show Context)
Citation Context ...ogic. Plaisted showed that only the MESON and Model Elimination strategies [9] with unit lemmas and caching [1], the simple and modied problem reduction formats [15, 16] and the Hyperlinking strategy =-=[8]-=- are polynomial. However, the rst two of these strategies (with caching) are not complete for rstorder logic; the second two are ineOEcient on nonHorn clauses, and have not been extended to equality. ... |

31 |
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(Show Context)
Citation Context ...or Horn clauses and not for rst order logic. Plaisted showed that only the MESON and Model Elimination strategies [9] with unit lemmas and caching [1], the simple and modied problem reduction formats =-=[15, 16]-=- and the Hyperlinking strategy [8] are polynomial. However, the rst two of these strategies (with caching) are not complete for rstorder logic; the second two are ineOEcient on nonHorn clauses, and ha... |

16 |
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(Show Context)
Citation Context ...or Horn clauses and not for rst order logic. Plaisted showed that only the MESON and Model Elimination strategies [9] with unit lemmas and caching [1], the simple and modied problem reduction formats =-=[15, 16]-=- and the Hyperlinking strategy [8] are polynomial. However, the rst two of these strategies (with caching) are not complete for rstorder logic; the second two are ineOEcient on nonHorn clauses, and ha... |

13 |
Canonical Equational Proofs. BirkhĂ¤user
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Citation Context ...ne of the premises of the inference is redundant. These cases of redundancy are easy to detect and quite useful to perform in practice. A theorem proving derivation models an automated theorem prover =-=[2]-=-. It is used to prove soundness and completeness results, but we have extended it to refer to complexity. If I is an inference system and S is a set of clauses, then an I-inference from S is an instan... |

13 |
A Fast Algorithm for Generating Reduced Ground Rewriting Systems from a set of Ground Equations
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(Show Context)
Citation Context ... positive unit equalities. Polynomial time algorithms, based on congruence closure, have been given to convert a set of equations without variables into an equivalent canonical set in polynomial time =-=[5, 21]-=-, but these algorithms are not Completion and do not work when the equations contain variables. Recently, it has been shown that ground completion with structure sharing is polynomial if a strategy is... |

12 |
Finding canonical rewriting systems equivalent to a finite set of ground equations in polynomial time
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(Show Context)
Citation Context ... positive unit equalities. Polynomial time algorithms, based on congruence closure, have been given to convert a set of equations without variables into an equivalent canonical set in polynomial time =-=[5, 21]-=-, but these algorithms are not Completion and do not work when the equations contain variables. Recently, it has been shown that ground completion with structure sharing is polynomial if a strategy is... |

9 |
Recursive query processing: the power of logic
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(Show Context)
Citation Context ...ty. He also notes that none of the resolution strategies are polynomial, and suggests that maybe SLDresolution should be based on another strategy. We note that caching methods (often called memoing) =-=[22, 23]-=- have been applied to SLDresolution, but the technique is not complete for nonHorn clauses. Local Simplication [10] is a general strategy for rstorder equational theorem proving that always has polyno... |

5 |
An e cient unication algorithm
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(Show Context)
Citation Context ...ds for only one equation. We have to be careful, because the graph may contain exponentially large clauses. However, by an extension of the Patterson-Wegman or Martelli-Montanari unication algorithms =-=[12, 14]-=-, the unication could still be performed in linear time. It is only necessary not to write out each clause. 2 Theorem 6 PWD \Gamma polynomially decides HCE . Proof. The left premise is a unit positive... |

5 |
Paramodulation and theorem proving in rst order theories with equality
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(Show Context)
Citation Context ...ng. 1 Introduction Since the early 1960's, automated deduction has been a popular topic of research. Resolution remains the most popular method, along with its extension to equality of Paramodulation =-=[20]-=-. Methods have been developed to restrict the search space of Resolution and Paramodulation proof procedures, and new theorem proving procedures have been developed. Much of this research has been to ... |

2 |
Caching and lemma use in model elimination theorem provers
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(Show Context)
Citation Context ... solving them. These techniques generally only work for Horn clauses and not for rst order logic. Plaisted showed that only the MESON and Model Elimination strategies [9] with unit lemmas and caching =-=[1]-=-, the simple and modied problem reduction formats [15, 16] and the Hyperlinking strategy [8] are polynomial. However, the rst two of these strategies (with caching) are not complete for rstorder logic... |

2 |
Linear uni#cation
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(Show Context)
Citation Context ...ds for only one equation. We have to be careful, because the graph may contain exponentially large clauses. However, by an extension of the Patterson-Wegman or Martelli-Montanari unication algorithms =-=[12, 14]-=-, the unication could still be performed in linear time. It is only necessary not to write out each clause. 2 Theorem 6 PWD \Gamma polynomially decides HCE . Proof. The left premise is a unit positive... |

1 |
Local Simplication
- Lynch
- 1994
(Show Context)
Citation Context ...e based on another strategy. We note that caching methods (often called memoing) [22, 23] have been applied to SLDresolution, but the technique is not complete for nonHorn clauses. Local Simplication =-=[10]-=- is a general strategy for rstorder equational theorem proving that always has polynomial size proofs in the case of SLDresolution of ground clauses, but the search space is exponential. 1 KnuthBendix... |

1 |
Theorem Proving with Equational Horn Clauses: Any Selection Rule is Complete
- Lynch
- 1994
(Show Context)
Citation Context ...owing, 2 and for 1 The existence of polynomialsized proofs has been studied by Haken[6] and others but Automated Deduction researchers are concerned with the eOEciency of searching for a proof. 2 See =-=[11]-=- for an explanation of this class. Equational Horn Clauses under certain selection rules. 2 Preliminaries Our setting is equational clauses. We have predicate symbols, function symbols and variables. ... |

1 |
The search eOEciency of theorem proving strategies
- Plaisted
- 1994
(Show Context)
Citation Context ...es. One class which can be decided in polynomial time is the class of propositional Horn Clause formulas. Goaldirected strategies are useful in theorem proving. In an important recent paper, Plaisted =-=[17, 18]-=- has examined the behavior of many general purpose goaldirected theorem proving methods on this class. His conclusion is that the only strategies which are polynomial are those strategies which use me... |

1 |
Polynomial Time Completion of Ground TermRewriting Systems. Unpublished note
- Plaisted, SattlerKlein
(Show Context)
Citation Context ...ork when the equations contain variables. Recently, it has been shown that ground completion with structure sharing is polynomial if a strategy is used which applies critical pairs in a certain order =-=[19]-=-. In this paper, we dene a method of performing paramodulation without copying literals. The terms are stored in a hypergraph. To perform resolution or paramodulation a hyperedge is added to the graph... |