## On Modularity of Termination and Confluence Properties of Conditional Rewrite Systems (1994)

Venue: | 4 th ALP 1994, LNCS 850 |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Gramlich94onmodularity,

author = {Bernhard Gramlich},

title = {On Modularity of Termination and Confluence Properties of Conditional Rewrite Systems},

booktitle = {4 th ALP 1994, LNCS 850},

year = {1994},

pages = {186--203},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

. We investigate the modularity behaviour of termination and confluence properties of conditional term rewriting systems. In particular, we show how to obtain sufficient conditions for the modularity of weak termination, weak innermost termination, (strong) innermost termination, (strong) termination, confluence and completeness of conditional rewrite systems. 1 Introduction, Motivation and Overview Starting with the seminal work of Toyama [Toy87b] the investigation of preservation properties of term rewriting systems (TRSs for short) under various forms of combinations has become a very interesting and active area of research. From a practical point of view this field has a great potential in applications of rewriting techniques since it provides the theoretical basis for a systematic construction of large systems of rewrite rules with some desired properties from smaller ones with corresponding properties. Vice versa, it is also crucial for analyzing properties of large systems by d...

### Citations

751 | Rewrite systems
- Dershowitz, Jouannaud
- 1992
(Show Context)
Citation Context ...operties of CTRSs. Finally we summarize known and new results and discuss some open problems and related work. 2 Preliminaries We assume familiarity with the basic theory for term rewriting (cf. e.g. =-=[DJ90]-=-, [Klo92]). For brevity we shall make use of the following abbreviations which apply to TRSs (and, if sensible, also to terms): SN (strongly) terminating (strongly normalizing), WN weakly terminating ... |

565 | Term Rewriting Systems
- Klop
- 1992
(Show Context)
Citation Context ... of CTRSs. Finally we summarize known and new results and discuss some open problems and related work. 2 Preliminaries We assume familiarity with the basic theory for term rewriting (cf. e.g. [DJ90], =-=[Klo92]-=-). For brevity we shall make use of the following abbreviations which apply to TRSs (and, if sensible, also to terms): SN (strongly) terminating (strongly normalizing), WN weakly terminating (weakly n... |

88 |
Conditional rewrite rules: Confluence and termination
- Bergstra, Klop
- 1982
(Show Context)
Citation Context ...r all substitutions which satisfy the correspondingly instantiated conditions. Moreover, the critical pair lemma does not hold for CTRSs in general as shown e.g. by the following example. Example 1. (=-=[BK86]-=-) Consider the join CTRS R = ae x # f(x) =)f(x) ! a b ! f(b) : Here we get f(b) ! a due to b # f(b) and hence f(f(b)) ! f(a). We also have f(f(b)) ! a because of f(b) # f(f(b)). But a and f(a) do not ... |

85 |
Counterexamples to termination for the direct sum of term rewriting systems
- Toyama
- 1987
(Show Context)
Citation Context ...re corresponding properties are often easier to verify. In [Toy87b] it was shown that confluence is indeed preserved under disjoint unions of (unconditional) TRSs whereas termination is not (cf. also =-=[Toy87a]-=-). This phenomenon was the starting point of a couple of investigations about how to obtain sufficient criteria for the preservation of termination, completeness (i.e. termination plus confluence) and... |

83 | Termination
- Dershowitz
- 1985
(Show Context)
Citation Context ... with common constructors have been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], [Ohl93a]. More general hierarchical combinations of TRSs have recently been dealt with in [Kri93], =-=[Der93]-=-, [Kri94], [Gra93b], [FJ93]. Some preservation results for (disjoint and non-disjoint, but non-hierarchical) combinations of conditional TRSs (CTRSs for short) finally have been obtained in [Mid90] (c... |

73 |
On the Church-Rosser property for the direct sum of term rewriting systems
- Toyama
- 1987
(Show Context)
Citation Context ...ion, (strong) innermost termination, (strong) termination, confluence and completeness of conditional rewrite systems. 1 Introduction, Motivation and Overview Starting with the seminal work of Toyama =-=[Toy87b]-=- the investigation of preservation properties of term rewriting systems (TRSs for short) under various forms of combinations has become a very interesting and active area of research. From a practical... |

71 |
Tree manipulation systems and Church-Rosser theorems
- Rosen
- 1973
(Show Context)
Citation Context ... Rewriting Systems Here we shall summarize some known and recently obtained new results on restricted termination and confluence properties of unconditional and conditional TRSs. Theorem 5. (cf. e.g. =-=[Ros73]-=-, [O'D77], [Klo92]) Let R be an orthogonal, i.e. non-overlapping and left-linear (unconditional) TRS. Then R is confluent, and the following properties hold: (1a) 8t : [ WIN(t) =) SIN(t) ] . (1b) WIN(... |

61 |
Modular Properties of Term Rewriting Systems
- Middeldorp
- 1990
(Show Context)
Citation Context ...teria for the preservation of termination, completeness (i.e. termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. [Rus87], [Mid89], [TKB89], =-=[Mid90]-=-, [KO90], [Gra92a], [Gra92b], [Ohl93b]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], [Ohl93a]. More general hierarch... |

56 | Confluence of conditional rewrite systems - Dershowitz, Okada, et al. - 1987 |

50 | Conditional rewrite rules - Kaplan - 1984 |

50 |
On termination of the direct sum of term rewriting systems
- Rusinowitch
- 1987
(Show Context)
Citation Context ...ow to obtain sufficient criteria for the preservation of termination, completeness (i.e. termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. =-=[Rus87]-=-, [Mid89], [TKB89], [Mid90], [KO90], [Gra92a], [Gra92b], [Ohl93b]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], [Ohl... |

49 | Generalized sufficient conditions for modular termination of rewriting. Applicable Algebra
- Gramlich
- 1994
(Show Context)
Citation Context ...servation of termination, completeness (i.e. termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. [Rus87], [Mid89], [TKB89], [Mid90], [KO90], =-=[Gra92a]-=-, [Gra92b], [Ohl93b]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], [Ohl93a]. More general hierarchical combinations ... |

47 |
Modularity of simple termination of term rewriting systems
- Kurihara, Ohuchi
- 1990
(Show Context)
Citation Context ... the preservation of termination, completeness (i.e. termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. [Rus87], [Mid89], [TKB89], [Mid90], =-=[KO90]-=-, [Gra92a], [Gra92b], [Ohl93b]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], [Ohl93a]. More general hierarchical com... |

35 |
A sufficient condition for the termination of the direct sum of term rewriting systems
- Middeldorp
- 1989
(Show Context)
Citation Context ...ain sufficient criteria for the preservation of termination, completeness (i.e. termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. [Rus87], =-=[Mid89]-=-, [TKB89], [Mid90], [KO90], [Gra92a], [Gra92b], [Ohl93b]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], [Ohl93a]. Mor... |

31 | Completeness of combinations of constructor systems - Middeldorp, Toyama - 1991 |

30 |
Termination for the direct sum of left-linear term rewriting systems
- Toyama, Klop, et al.
- 1989
(Show Context)
Citation Context ...cient criteria for the preservation of termination, completeness (i.e. termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. [Rus87], [Mid89], =-=[TKB89]-=-, [Mid90], [KO90], [Gra92a], [Gra92b], [Ohl93b]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], [Ohl93a]. More general... |

29 | On the modularity of termination of term rewriting systems
- Ohlebusch
- 1994
(Show Context)
Citation Context ...tion, completeness (i.e. termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. [Rus87], [Mid89], [TKB89], [Mid90], [KO90], [Gra92a], [Gra92b], =-=[Ohl93b]-=-). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], [Ohl93a]. More general hierarchical combinations of TRSs have recentl... |

27 | Relating innermost, weak, uniform and modular termination of term rewriting systems
- Gramlich
- 1992
(Show Context)
Citation Context ...of termination, completeness (i.e. termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. [Rus87], [Mid89], [TKB89], [Mid90], [KO90], [Gra92a], =-=[Gra92b]-=-, [Ohl93b]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], [Ohl93a]. More general hierarchical combinations of TRSs ha... |

23 |
Modular properties of conditional term rewriting systems
- Middeldorp
- 1993
(Show Context)
Citation Context ...], [Gra93b], [FJ93]. Some preservation results for (disjoint and non-disjoint, but non-hierarchical) combinations of conditional TRSs (CTRSs for short) finally have been obtained in [Mid90] (cf. also =-=[Mid93b]-=-), [Mid93a], [Gra93c], [Ohl93b]. As shown in [TKB89] (by a very involved proof), completeness is preserved under disjoint union of left-linear (unconditional) TRSs. Instead of left-linearity ? This re... |

16 |
Simple termination of hierarchical combinations of term rewriting systems
- Rao
- 1994
(Show Context)
Citation Context ...mon constructors have been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], [Ohl93a]. More general hierarchical combinations of TRSs have recently been dealt with in [Kri93], [Der93], =-=[Kri94]-=-, [Gra93b], [FJ93]. Some preservation results for (disjoint and non-disjoint, but non-hierarchical) combinations of conditional TRSs (CTRSs for short) finally have been obtained in [Mid90] (cf. also [... |

15 |
Completeness of hierarchical combinations of term rewriting systems
- Rao
(Show Context)
Citation Context ...s of TRSs with common constructors have been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], [Ohl93a]. More general hierarchical combinations of TRSs have recently been dealt with in =-=[Kri93]-=-, [Der93], [Kri94], [Gra93b], [FJ93]. Some preservation results for (disjoint and non-disjoint, but non-hierarchical) combinations of conditional TRSs (CTRSs for short) finally have been obtained in [... |

13 | Sufficient conditions for modular termination of conditional term rewriting systems
- Gramlich
(Show Context)
Citation Context ...Some preservation results for (disjoint and non-disjoint, but non-hierarchical) combinations of conditional TRSs (CTRSs for short) finally have been obtained in [Mid90] (cf. also [Mid93b]), [Mid93a], =-=[Gra93c]-=-, [Ohl93b]. As shown in [TKB89] (by a very involved proof), completeness is preserved under disjoint union of left-linear (unconditional) TRSs. Instead of left-linearity ? This research was supported ... |

12 | Completeness of Combinations of Conditional Constructor Systems
- Middeldorp
- 1994
(Show Context)
Citation Context ...by Middeldorp. Theorem12. ([Mid90], [Mid93b]) Confluence is a modular property of CTRSs. For some sufficient conditions for modularity of termination of CTRSs we refer to [Mid90] (cf. also [Mid93b]), =-=[Mid93a]-=-, [Gra93c]). Here we shall concentrate on CTRSs satisfying some structural restrictions. In [Gra92b], [Gra93b] we have proved that any (strongly) innermost terminating, locally confluent (unconditiona... |

11 | On the modularity of confluence of constructor-sharing term rewriting systems
- Ohlebusch
- 1994
(Show Context)
Citation Context ...s87], [Mid89], [TKB89], [Mid90], [KO90], [Gra92a], [Gra92b], [Ohl93b]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], =-=[Ohl93a]-=-. More general hierarchical combinations of TRSs have recently been dealt with in [Kri93], [Der93], [Kri94], [Gra93b], [FJ93]. Some preservation results for (disjoint and non-disjoint, but non-hierarc... |

5 |
Modularity properties of term rewriting systems revisited. Rapport de Recherche 875
- Fernandez, Jouannaud
- 1993
(Show Context)
Citation Context ...ve been considered e.g. in [MT91], [KO92], [Gra92a], [Gra92b], [Ohl93b], [Ohl93a]. More general hierarchical combinations of TRSs have recently been dealt with in [Kri93], [Der93], [Kri94], [Gra93b], =-=[FJ93]-=-. Some preservation results for (disjoint and non-disjoint, but non-hierarchical) combinations of conditional TRSs (CTRSs for short) finally have been obtained in [Mid90] (cf. also [Mid93b]), [Mid93a]... |

2 |
New abstract criteria for termination and confluence of conditional rewrite systems. SEKI-Report SR-93-17
- Gramlich
- 1993
(Show Context)
Citation Context ... crucial point is that for such TRSs (strong) innermost termination implies already (strong) termination. Recently we have been able to show that this latter property does indeed also hold for CTRSs (=-=[Gra93a]-=-). In the present paper we shall exploit this property and show how to obtain corresponding preservation results for disjoint unions of CTRSs. However, this generalization of [Gra92b] to the condition... |

2 | Computing in Systems Described by Equations, LNCS 58 - O'Donnell - 1977 |