## Persistency of Confluence (1997)

Citations: | 23 - 6 self |

### BibTeX

@MISC{Aoto97persistencyof,

author = {Takahito Aoto and Yoshihito Toyama},

title = {Persistency of Confluence },

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

A property P of term rewriting systems (TRSs, for short) is said to be persistent if for any many-sorted TRS R, R has the property P if and only if its underlying unsorted TRS (R) has the property P. This notion was introduced by H. Zantema (1994). In this paper, it is shown that confluence is persistent.

### Citations

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(Show Context)
Citation Context ...htforward modification of the proof of modularity of confluence, whose details are given in this report. 1 Introduction Confluence is a property that is widely studied in the theory of term rewriting =-=[5]-=-,[2],[9]. One of the approaches to guarantee confluence of a given TRS is to infer it from those of its subsystems. Y. Toyama showed that confluence of the disjoint sum of TRSs can be inferred from th... |

364 |
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(Show Context)
Citation Context ...rward modification of the proof of modularity of confluence, whose details are given in this report. 1 Introduction Confluence is a property that is widely studied in the theory of term rewriting [5],=-=[2]-=-,[9]. One of the approaches to guarantee confluence of a given TRS is to infer it from those of its subsystems. Y. Toyama showed that confluence of the disjoint sum of TRSs can be inferred from those ... |

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(Show Context)
Citation Context ... 0 ]. Then rank(s 0 ) ? rank(t 0 ) by 10 Proposition 3.8, hence ks 0 k AE kt 0 k. From this follows ksk AE ktk, since this s 0 is a special subterm of s. But the relation AE is well-founded (e.g. see =-=[1]-=-) and hence there can not be infinite collapsing reduction sequences. 4 Persistency of confluence Let s 1 ; : : : ; s n and t 1 ; : : : ; t n be terms. We write hs 1 ; : : : ; s n i / ht 1 ; : : : ; t... |

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(Show Context)
Citation Context ... the property P iff its underlying unsorted TRS h\Theta(F ); \Theta(R)i has the property P. Here \Theta denotes the sort elimination function. The notion of persistency is formulated by H. Zantema in =-=[10]-=-. He showed some of persistent properties including confluence are also modular w.r.t. disjoint union. Thus the notion of persistency is thought to be an extension of the notion of modularity. Also he... |

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(Show Context)
Citation Context ...pproaches to guarantee confluence of a given TRS is to infer it from those of its subsystems. Y. Toyama showed that confluence of the disjoint sum of TRSs can be inferred from those of its components =-=[8]. Followin-=-g standard terminology, we say "a property P is modular" if P is inferred from those of its components. However, unlike other important properties in the theory of term rewriting, only few r... |

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(Show Context)
Citation Context ...n of Toyama's proof in [8] seems applicable to show persistency of confluence. In this report, we rather show it by a straightforward modification of a simplified proof of modularity of confluence in =-=[4]-=-. 2 Preliminaries 2.1 Sorted term rewriting systems In this subsection, we review some basic notions of sorted term rewriting and fix the notations that will be used in this report. Let S be a set of ... |

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(Show Context)
Citation Context ...ed from those of its components. However, unlike other important properties in the theory of term rewriting, only few results which relax the disjoint limitation for modularity of confluence is known =-=[6]-=-, [3]. A property P of TRSs is said to be persistent if for any many-sorted TRS hF ; Ri, hF ; Ri has the property P iff its underlying unsorted TRS h\Theta(F ); \Theta(R)i has the property P. Here \Th... |

11 |
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Citation Context ...d modification of the proof of modularity of confluence, whose details are given in this report. 1 Introduction Confluence is a property that is widely studied in the theory of term rewriting [5],[2],=-=[9]-=-. One of the approaches to guarantee confluence of a given TRS is to infer it from those of its subsystems. Y. Toyama showed that confluence of the disjoint sum of TRSs can be inferred from those of i... |

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2 |
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(Show Context)
Citation Context ...om those of its components. However, unlike other important properties in the theory of term rewriting, only few results which relax the disjoint limitation for modularity of confluence is known [6], =-=[3]-=-. A property P of TRSs is said to be persistent if for any many-sorted TRS hF ; Ri, hF ; Ri has the property P iff its underlying unsorted TRS h\Theta(F ); \Theta(R)i has the property P. Here \Theta d... |

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