### Abstract

A refinement of paramodulation, called hyperparamodulation, is the focus of attention in this paper. Clauses obtained by the use of this inference rule are, in effect, the result of a sequence of paramodulations into one coranon nucleus. Among the interesting properties of hyperparamodulation are: first, clauses are chosen from among the input and designated as nuclei or "into" clauses for paramodulation; second, terms in the nucleus are starred to restrict the domain of generalized equality substitution; third, total control is thus iteratively established over all possible targets for paramodulation during the entire run of the theorem-proving program; and fourth, application of demodulation is suspended until the hyperparamodulant is completed. In contrast to these four properties which are reminiscent of the spirit of hyper-resolution, the following differences exist: first, the nucleus and the starred terms therein, which are analogous to negative literals, are determined by the user rather than by syntax; second, nuclei are not restricted to being mixed clauses; and third, while hyper-resolution requires inferred clauses to be positive, no corresponding requirement exists for clauses inferred by hyperparamodulation. To illustrate the value of this refinement of paramodulation, we have chosen certain conjectures which arose during the study of Robbins algebra. A Robbins algebra is a set on which the functions o and n are defined such that o is both commutative and associative and such that for all x and y the following identity n(o(n(o(x,y)),n(o(x,n(y))))) =x holds. One may think of o as union and n as complement. Hie main interest in such algebras arises from the following open question: If S is a Robbins algebra, is S necessarily a Boolean algebra? The study of this open question entailed heavy use of an automated theorem-proving program to examine various conjectures. Certain computer proofs therein were obtained only after recourse to hyperparamodulation. (These lenmas were actually obtained prior to the work reported on here by Winker and Wos using a non-standard theorem-proving approach developed by Winker.).

Original language | English (US) |
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Title of host publication | 5th Conference on Automated Deduction |

Editors | Wolfgang Bibel, Robert Kowalski |

Publisher | Springer Verlag |

Pages | 208-219 |

Number of pages | 12 |

ISBN (Print) | 9783540100096 |

State | Published - Jan 1 1980 |

Event | 5th International Conference on Automated Deduction, CADE 1980 - Les Arcs, France Duration: Jul 8 1980 → Jul 11 1980 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 87 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 5th International Conference on Automated Deduction, CADE 1980 |
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Country | France |

City | Les Arcs |

Period | 7/8/80 → 7/11/80 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*5th Conference on Automated Deduction*(pp. 208-219). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 87 LNCS). Springer Verlag.

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*5th Conference on Automated Deduction.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 87 LNCS, Springer Verlag, pp. 208-219, 5th International Conference on Automated Deduction, CADE 1980, Les Arcs, France, 7/8/80.

**Hyperparamodulation : A refinement of paramodulation.** / Wos, L.; Overbeek, R.; Henschen, Lawrence Joseph.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Hyperparamodulation

T2 - A refinement of paramodulation

AU - Wos, L.

AU - Overbeek, R.

AU - Henschen, Lawrence Joseph

PY - 1980/1/1

Y1 - 1980/1/1

N2 - A refinement of paramodulation, called hyperparamodulation, is the focus of attention in this paper. Clauses obtained by the use of this inference rule are, in effect, the result of a sequence of paramodulations into one coranon nucleus. Among the interesting properties of hyperparamodulation are: first, clauses are chosen from among the input and designated as nuclei or "into" clauses for paramodulation; second, terms in the nucleus are starred to restrict the domain of generalized equality substitution; third, total control is thus iteratively established over all possible targets for paramodulation during the entire run of the theorem-proving program; and fourth, application of demodulation is suspended until the hyperparamodulant is completed. In contrast to these four properties which are reminiscent of the spirit of hyper-resolution, the following differences exist: first, the nucleus and the starred terms therein, which are analogous to negative literals, are determined by the user rather than by syntax; second, nuclei are not restricted to being mixed clauses; and third, while hyper-resolution requires inferred clauses to be positive, no corresponding requirement exists for clauses inferred by hyperparamodulation. To illustrate the value of this refinement of paramodulation, we have chosen certain conjectures which arose during the study of Robbins algebra. A Robbins algebra is a set on which the functions o and n are defined such that o is both commutative and associative and such that for all x and y the following identity n(o(n(o(x,y)),n(o(x,n(y))))) =x holds. One may think of o as union and n as complement. Hie main interest in such algebras arises from the following open question: If S is a Robbins algebra, is S necessarily a Boolean algebra? The study of this open question entailed heavy use of an automated theorem-proving program to examine various conjectures. Certain computer proofs therein were obtained only after recourse to hyperparamodulation. (These lenmas were actually obtained prior to the work reported on here by Winker and Wos using a non-standard theorem-proving approach developed by Winker.).

AB - A refinement of paramodulation, called hyperparamodulation, is the focus of attention in this paper. Clauses obtained by the use of this inference rule are, in effect, the result of a sequence of paramodulations into one coranon nucleus. Among the interesting properties of hyperparamodulation are: first, clauses are chosen from among the input and designated as nuclei or "into" clauses for paramodulation; second, terms in the nucleus are starred to restrict the domain of generalized equality substitution; third, total control is thus iteratively established over all possible targets for paramodulation during the entire run of the theorem-proving program; and fourth, application of demodulation is suspended until the hyperparamodulant is completed. In contrast to these four properties which are reminiscent of the spirit of hyper-resolution, the following differences exist: first, the nucleus and the starred terms therein, which are analogous to negative literals, are determined by the user rather than by syntax; second, nuclei are not restricted to being mixed clauses; and third, while hyper-resolution requires inferred clauses to be positive, no corresponding requirement exists for clauses inferred by hyperparamodulation. To illustrate the value of this refinement of paramodulation, we have chosen certain conjectures which arose during the study of Robbins algebra. A Robbins algebra is a set on which the functions o and n are defined such that o is both commutative and associative and such that for all x and y the following identity n(o(n(o(x,y)),n(o(x,n(y))))) =x holds. One may think of o as union and n as complement. Hie main interest in such algebras arises from the following open question: If S is a Robbins algebra, is S necessarily a Boolean algebra? The study of this open question entailed heavy use of an automated theorem-proving program to examine various conjectures. Certain computer proofs therein were obtained only after recourse to hyperparamodulation. (These lenmas were actually obtained prior to the work reported on here by Winker and Wos using a non-standard theorem-proving approach developed by Winker.).

UR - http://www.scopus.com/inward/record.url?scp=84936246670&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84936246670&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9783540100096

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 208

EP - 219

BT - 5th Conference on Automated Deduction

A2 - Bibel, Wolfgang

A2 - Kowalski, Robert

PB - Springer Verlag

ER -