## Unravelings and Ultra-properties (1996)

Venue: | In Proceedings of the Fifth International Conference on Algebraic and Logic Programming (ALP'96), volume 1139 of LNCS |

Citations: | 26 - 3 self |

### BibTeX

@INPROCEEDINGS{Marchiori96unravelingsand,

author = {Massimo Marchiori},

title = {Unravelings and Ultra-properties},

booktitle = {In Proceedings of the Fifth International Conference on Algebraic and Logic Programming (ALP'96), volume 1139 of LNCS},

year = {1996},

pages = {107--121},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

Conditional rewriting is universally recognized as being much more complicated than unconditional rewriting. In this paper we study how much of conditional rewriting can be automatically inferred from the simpler theory of unconditional rewriting. We introduce a new tool, called unraveling, to automatically translate a conditional term rewriting system (CTRS) into a term rewriting system (TRS). An unraveling enables to infer properties of a CTRS by studying the corresponding ultra-property on the corresponding TRS. We show how to rediscover properties like decreasingness, and to give easy proofs of some existing results on CTRSs. Moreover, we show how unravelings provide a valuable tool to study modularity of CTRSs, automatically giving a multitude of new results.

### Citations

749 | Rewrite systems
- Dershowitz, Jouannaud
- 1990
(Show Context)
Citation Context ..., and it is shown this limit is just what obtained in this paper. 2 Preliminaries We assume knowledge of the basic notions regarding conditional term rewriting systems and term rewriting systems (cf. =-=[3, 15]-=-). In this paper we will mainly deal with join and normal CTRSs, that is in the first case rules are of the form l ! r ( s 1 #t 1 ; : : : ; s k #t k (with Var(r; s 1 ; t 1 ; : : : ; s k ; t k ) ` Var(... |

566 | Term rewriting systems
- Klop
- 1992
(Show Context)
Citation Context ..., and it is shown this limit is just what obtained in this paper. 2 Preliminaries We assume knowledge of the basic notions regarding conditional term rewriting systems and term rewriting systems (cf. =-=[3, 15]-=-). In this paper we will mainly deal with join and normal CTRSs, that is in the first case rules are of the form l ! r ( s 1 #t 1 ; : : : ; s k #t k (with Var(r; s 1 ; t 1 ; : : : ; s k ; t k ) ` Var(... |

88 |
Conditional rewrite rules: Confluence and termination
- Bergstra, Klop
- 1982
(Show Context)
Citation Context ...ry developed for term rewriting systems. This allows to automatically lift known results of TRSs to CTRSs. The idea of transforming CTRSs into TRSs dates back to the seminal work of Bergstra and Klop =-=[1]-=-, where they say that using such a transformation can be very useful to get better intuition on the behaviour of a CTRS. Later works include [6] and [11]. However, all these works only cover partial c... |

83 | Termination
- Dershowitz
- 1985
(Show Context)
Citation Context ...Ss ([25]).sTheorem 7.5 Ultra-termination is modular for non-overlapping composable CTRSs. Proof By Lemmata 5.1, 5.4, Theorem 7.1 and the modularity of termination for non-overlapping composable TRSs (=-=[2, 25]-=-).sAs far as non-duplication is concerned, it is known that termination is modular for non-duplicating TRSs ([32]). Middeldorp observed (cf. [22]) that this result does not carry over to CTRSs (even f... |

55 | Confluence of conditional rewrite systems
- Dershowitz, Okada, et al.
- 1987
(Show Context)
Citation Context ...TRSs have been chosen to be normal CTRSs). 6.1 Left-linear Normal CTRSs One might wonder why we bother about normal CTRSs, since they are encompassed by join CTRSs and, moreover, it has been shown in =-=[5, 4]-=- that every join CTRS can be simulated by a normal CTRS. The fact is that there is a fundamental difference w.r.t. join CTRSs as far as the properties that can be lifted from TRSs to CTRSs are concern... |

52 |
Simplifying conditional term rewriting systems: Unification, termination and confluence
- Kaplan
- 1987
(Show Context)
Citation Context ...in an effectively terminating finite CTRSs they should be decidable as well. To cope with such intuition, it has been proposed a set of properties like representative of such `effective termination' (=-=[14, 12, 5, 4]-=-): a CTRS R is `effectively terminating' if \Gamma! R is terminating and it is decidable whether s\Gamma! R t, s\Gamma! R t, s# R t, and if a term is in normal form. Three major criteria are known tha... |

51 | Modular properties of composable term rewriting systems
- Ohlebusch
- 1994
(Show Context)
Citation Context ...y of combinations w.r.t. some operator fi, it must be the case it is fi-compositional. The unravelings here developed are compositional w.r.t. all the modularity operators so far introduced (cf. e.g. =-=[25, 15]-=-), that is to say, in order of increasing power, the disjoint union \Phi (disjoint signatures), constructor-sharing union \Phi cs (sharing only of constructor symbols), and composable union \Phi comp ... |

50 |
Conditional rewrite rules
- Kaplan
- 1984
(Show Context)
Citation Context ... is that the rewrite relation is defined recursively. A bad side-effect of this greater flexibility is that termination of a CTRS does not imply any more the decidability of the rewrite relation (cf. =-=[13]-=-). So, while for TRSs termination implies effective computability, for CTRSs basic questions like `is a term a normal form' or `does s reduce to t', `are s and t joinable' and so on can be undecidable... |

50 |
On termination of the direct sum of term rewriting systems
- Rusinowitch
- 1987
(Show Context)
Citation Context ... Theorem 7.1 and the modularity of termination for non-overlapping composable TRSs ([2, 25]).sAs far as non-duplication is concerned, it is known that termination is modular for non-duplicating TRSs (=-=[32]-=-). Middeldorp observed (cf. [22]) that this result does not carry over to CTRSs (even for decreasing CTRSs). So, he tried to give stronger conditions restoring modularity, and managed to prove modular... |

49 | Generalized sufficient conditions for modular termination of rewriting. Applicable Algebra
- Gramlich
(Show Context)
Citation Context ... get: Theorem 7.6 Ultra-termination is modular for non-ultraduplicating composable CTRSs. Proof By Lemmata 5.1, 5.4, Theorem 7.1 and the modularity of termination for non-duplicating composable TRSs (=-=[9, 25]-=-).s9 One of the most powerful results is the modularity of C E -termination (cf. [28, 9]), that has been extended to finitely branching disjoint CTRSs ([8]). It is an open problem whether this result ... |

47 |
Modularity of simple termination of term rewriting systems
- Kurihara, Ohuchi
- 1990
(Show Context)
Citation Context ...t is of particular practical importance: Theorem 7.9 Ultra-simplifyingness is modular for composable CTRSs. Proof By Theorem 7.1, Lemma 5.4, and the modularity of simplifyingness for composable TRSs (=-=[16]-=-).sWhen comparing the above result with the aforementioned result [24], by Lemma 5.6 we obtain that it is strictly more powerful . Other similar results can be obtained lifting the modularity results ... |

35 |
A sufficient condition for the termination of the direct sum of term rewriting systems
- Middeldorp
- 1989
(Show Context)
Citation Context ...Proof By Theorems 7.1, Lemma 5.4, and the modularity of simple termination for composable TRSs ([16]).sAnother important result that can be lifted from TRSs to CTRSs is that obtained by Middeldorp in =-=[21]-=-: he proved that if one of two terminating TRSs is both non-collapsing and non-duplicating, then their disjoint sum is terminating as well. Ohlebusch ([25]) extended this result to composable union of... |

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 |

29 | On the modularity of termination of term rewriting systems
- Ohlebusch
- 1994
(Show Context)
Citation Context .... Proof By Lemmata 5.1, 5.4, Theorem 7.1 and the modularity of termination for non-duplicating composable TRSs ([9, 25]).s9 One of the most powerful results is the modularity of C E -termination (cf. =-=[28, 9]-=-), that has been extended to finitely branching disjoint CTRSs ([8]). It is an open problem whether this result holds for disjoint CTRSs as well. Recall that a TRS T is C E -terminating if T \Phi for(... |

28 | Relating innermost, weak, uniform and modular termination of term rewriting systems
- Gramlich
- 1992
(Show Context)
Citation Context ... are known for composable unions. Theorem 7.12 Ultra innermost termination is modular for composable CTRSs. Proof By Theorems 7.1, 4.5 and the modularity of innermost termination for composable TRSs (=-=[7, 25]-=-). Consistency Properties So far, there were no results on the modularity of CON for CTRSs. Theorem 7.13 Ultra-CON is modular for CTRSs. Proof By Theorems 7.1, 7.3 and 4.3, since CON is modular for TR... |

24 |
Modular aspects of properties of term rewriting systems related to normal forms
- Middeldorp
- 1989
(Show Context)
Citation Context ..., since UN ! is modular for left-linear TRSs ([19]).sTheorem 7.25 Ultra-UN is modular for left-linear SP normal CTRSs. Proof By Lemma 6.6 and Theorems 7.1, 7.3 and 6.16, since UN is modular for TRSs (=-=[20]-=-).sTheorem 7.26 Ultra-NF is modular for left-linear SP normal CTRSs. Proof By Lemma 6.6 and Theorems 7.1, 7.3 and 6.17, since NF is modular for left-linear TRSs ([22]).sNote that in the above three re... |

23 |
Modular properties of conditional term rewriting systems
- Middeldorp
- 1993
(Show Context)
Citation Context ...of termination for non-overlapping composable TRSs ([2, 25]).sAs far as non-duplication is concerned, it is known that termination is modular for non-duplicating TRSs ([32]). Middeldorp observed (cf. =-=[22]-=-) that this result does not carry over to CTRSs (even for decreasing CTRSs). So, he tried to give stronger conditions restoring modularity, and managed to prove modularity imposing confluence. This ca... |

23 |
Uni in a Combination of Arbitrary Disjoint Equational Theories
- Schmidt-Schau
- 1989
(Show Context)
Citation Context ...onsistency Properties So far, there were no results on the modularity of CON for CTRSs. Theorem 7.13 Ultra-CON is modular for CTRSs. Proof By Theorems 7.1, 7.3 and 4.3, since CON is modular for TRSs (=-=[33]-=-).s10 7.2 Normal CTRSs So far, there is not a single result specific for the modularity of normal CTRSs. This class has somehow been neglected in view of the fact, as said earlier, that every join CTR... |

22 | Termination for direct sums of left-linear complete term rewriting systems - Toyama, Klop, et al. - 1995 |

21 |
Reductive conditional term rewriting systems
- Jouannaud, Waldmann
- 1985
(Show Context)
Citation Context ...in an effectively terminating finite CTRSs they should be decidable as well. To cope with such intuition, it has been proposed a set of properties like representative of such `effective termination' (=-=[14, 12, 5, 4]-=-): a CTRS R is `effectively terminating' if \Gamma! R is terminating and it is decidable whether s\Gamma! R t, s\Gamma! R t, s# R t, and if a term is in normal form. Three major criteria are known tha... |

16 |
Simple termination of hierarchical combinations of term rewriting systems
- Rao
- 1994
(Show Context)
Citation Context ...lects the fact their analysis is extremely hard. It is not surprising, therefore, that so far there are no results on the modularity of hierarchical combinations (but for a slight extension stated in =-=[30]-=-). Even in this setting, however, unravelings allow to lift existing results from TRSs to CTRSs. Formally, two TRSs R 1 and R 2 are said to form a hierarchical combination (cf. [25]) if R 1 does not h... |

15 |
Completeness of hierarchical combinations of term rewriting systems
- Rao
(Show Context)
Citation Context ...pecific important results. By Corollary 6.11, we can lift by unravelings also all the other results obtained for the completeness hierarchical combinations of TRSs: those of Dershowitz ([2]) and Rao (=-=[29]-=-) on completeness. Another result is obtained by lifting the recent result of Verma ([37]) on the modularity of confluence: he treats a kind of combination, called lr-combination, that is even more ge... |

14 | Modularity of completeness revisited - Marchiori - 1995 |

13 | Sufficient conditions for modular termination of conditional term rewriting systems
- Gramlich
(Show Context)
Citation Context ...ion for non-duplicating composable TRSs ([9, 25]).s9 One of the most powerful results is the modularity of C E -termination (cf. [28, 9]), that has been extended to finitely branching disjoint CTRSs (=-=[8]-=-). It is an open problem whether this result holds for disjoint CTRSs as well. Recall that a TRS T is C E -terminating if T \Phi for(X; Y ) ! X; or(X; Y ) ! Y g is terminating. Interestingly, the lift... |

12 |
Operational and Semantic Equivalence Between Recursive Programs
- Raoult, Vuillemin
- 1978
(Show Context)
Citation Context ...22 Ultra-confluence is modular for left-linear constructor-sharing normal CTRSs. Proof By Lemma 6.6 and Theorems 7.1, 7.3 and 6.8, since confluence is modular for left-linear constructorsharing TRSs (=-=[31]-=-).sNote that if confluence is proven to be modular for left-linear composable TRSs (a conjecture which is widely believed to hold), then we will automatically be able to say that ultra-confluence is m... |

11 | On the modularity of confluence of constructor-sharing term rewriting systems
- Ohlebusch
- 1994
(Show Context)
Citation Context ...composable CTRSs. Theorem 7.29 Ultra strict semicompleteness is modular for composable CTRSs. Proof By Lemma 6.6 and Theorems 7.1, 7.3 and 6.21, since semicompleteness is modular for composable TRSs (=-=[27, 25]-=-).s12 8 Hierarchical Combinations In Section 7 we treated all the so far known composition operators, but for the most recent (and involved) case: hierarchical combinations. Hierarchical combinations ... |

11 |
E.: \Modular termination of r-consistent and left-linear term rewriting systems"; Theoretical Computer Science 149
- Marchiori, M, et al.
- 1995
(Show Context)
Citation Context ...implies decreasingness.sTheorem 7.19 Decreasingness is modular for left-linear CON ! normal CTRSs. Proof By Theorems 6.12, 7.1, Lemma 6.6 and the modularity of termination for left-linear CON ! TRSs (=-=[19, 34]-=-).sTheorem 7.20 If one of two decreasing left-linear normal CTRSs is both CON ! and ultra-C E -terminating, then their disjoint union is decreasing. Proof By Theorems 6.12, 7.1, Lemma 6.6 and the resu... |

11 |
Conditional rewrite rules: con uence and termination
- Bergstra, Klop
- 1986
(Show Context)
Citation Context ...ry developed for term rewriting systems. This allows to automatically lift known results of TRSs to CTRSs. The idea of transforming CTRSs into TRSs dates back to the seminal work of Bergstra and Klop =-=[1]-=-, where they say that using such a transformation can be very useful to get better intuition on the behaviour of a CTRS. Later works include [6] and [11]. However, all these works only cover partial c... |

9 | How to transform canonical decreasing CTRSs into equivalent canonical TRSs
- Hintermeier
- 1994
(Show Context)
Citation Context ...s back to the seminal work of Bergstra and Klop [1], where they say that using such a transformation can be very useful to get better intuition on the behaviour of a CTRS. Later works include [6] and =-=[11]-=-. However, all these works only cover partial cases, since their aims are different: in [6] Giovannetti and Moiso seek for a transformation that completely preserves the operational behaviour of a TRS... |

9 | On the modularity of normal forms in rewriting
- Marchiori
- 1996
(Show Context)
Citation Context ...implies decreasingness.sTheorem 7.19 Decreasingness is modular for left-linear CON ! normal CTRSs. Proof By Theorems 6.12, 7.1, Lemma 6.6 and the modularity of termination for left-linear CON ! TRSs (=-=[19, 34]-=-).sTheorem 7.20 If one of two decreasing left-linear normal CTRSs is both CON ! and ultra-C E -terminating, then their disjoint union is decreasing. Proof By Theorems 6.12, 7.1, Lemma 6.6 and the resu... |

9 | Notes on the elimination of conditions - Giovannetti, Moiso - 1987 |

9 |
Generalized su cient conditions for modular termination of rewriting
- Gramlich
- 1994
(Show Context)
Citation Context ... get: Theorem 7.6 Ultra-termination is modular for non-ultraduplicating composable CTRSs. Proof By Lemmata 5.1, 5.4, Theorem 7.1 and the modularity of termination for non-duplicating composable TRSs (=-=[9, 25]-=-). 9sOne of the most powerful results is the modularityofCE-termination (cf. [28, 9]), that has been extended to nitely branching disjoint CTRSs ([8]). It is an open problem whether this result holds ... |

7 |
A su cient condition for the termination of the direct sum of term rewriting systems
- Middeldorp
- 1989
(Show Context)
Citation Context ...Proof By Theorems 7.1, Lemma 5.4, and the modularity of simple termination for composable TRSs ([16]). Another important result that can be lifted from TRSs to CTRSs is that obtained by Middeldorp in =-=[21]-=-: he proved that if one of two terminating TRSs is both non-collapsing and non-duplicating, then their disjoint sum is terminating as well. Ohlebusch ([25]) extended this result to composable union of... |

6 | Notes on the eliminations of conditions - Giovanetti, Moiso |

6 | On modularity of termination and confluence properties of conditional rewrite systems
- Gramlich
- 1994
(Show Context)
Citation Context ...terminating. Proof By Theorem 7.1, Lemmata 5.4, 5.1, and the aforementioned result of [25].sFinally, we tackle innermost termination: this property has been proved to be modular for disjoint CTRSs in =-=[10]-=-. No results are known for composable unions. Theorem 7.12 Ultra innermost termination is modular for composable CTRSs. Proof By Theorems 7.1, 4.5 and the modularity of innermost termination for compo... |

4 | Combinations of Simplifying Conditional Term Rewriting Systems
- Ohlebusch
- 1992
(Show Context)
Citation Context ...l the other properties of left-linear normal CTRSs, and Uotherwise. 7.1 Join CTRSs Termination No results on the modularity of termination for composable CTRSs are so far known, but for the result of =-=[24]-=- showing the modularity of simplifyingness for composable CTRSs. We will now show what we can automatically obtain using unravelings. Theorem 7.4 Ultra-termination is modular for non-collapsing compos... |

3 | Bubbles in modularity
- Marchiori
- 1998
(Show Context)
Citation Context ...20 If one of two decreasing left-linear normal CTRSs is both CON ! and ultra-C E -terminating, then their disjoint union is decreasing. Proof By Theorems 6.12, 7.1, Lemma 6.6 and the result proved in =-=[17]-=- for TRSs.sConfluence Confluence is modular for non-collapsing CTRSs ([22]), but no results were so far known for the modularity of confluence for non-collapsing CTRSs even in the constructor-sharing ... |

3 | Modular Properties of Constructor-Sharing Conditional Term Rewriting Systems
- Ohlebusch
- 1995
(Show Context)
Citation Context ...CTRSs. Proof By Lemma 6.6 and Theorems 7.1, 7.3 and 6.21, since innermost normalization is modular for composable TRSs ([25]).sFinally, semi-completeness: it is modular for constructor-sharing CTRSs (=-=[26]-=-), but no results are known for composable CTRSs. Theorem 7.29 Ultra strict semicompleteness is modular for composable CTRSs. Proof By Lemma 6.6 and Theorems 7.1, 7.3 and 6.21, since semicompleteness ... |

3 | Unique normal forms and confluence of rewrite systems: Persistence
- Verma
- 1995
(Show Context)
Citation Context ...er results obtained for the completeness hierarchical combinations of TRSs: those of Dershowitz ([2]) and Rao ([29]) on completeness. Another result is obtained by lifting the recent result of Verma (=-=[37]-=-) on the modularity of confluence: he treats a kind of combination, called lr-combination, that is even more general than those considered in the aforementioned papers. We can lift his result obtainin... |

2 |
Su cient conditions for modular termination of conditional term rewriting systems
- Gramlich
- 1993
(Show Context)
Citation Context ...rmination for non-duplicating composable TRSs ([9, 25]). 9sOne of the most powerful results is the modularityofCE-termination (cf. [28, 9]), that has been extended to nitely branching disjoint CTRSs (=-=[8]-=-). It is an open problem whether this result holds for disjoint CTRSs as well. Recall that a TRS T is CE-terminating if T for(X� Y ) ! X� or(X� Y ) ! Y g is terminating. Interestingly, the lifted ultr... |

2 |
On the modularity of con uence of constructorsharing term rewriting systems
- Ohlebusch
- 1994
(Show Context)
Citation Context ...composable CTRSs. Theorem 7.29 Ultra strict semicompleteness is modular for composable CTRSs. Proof By Lemma 6.6 and Theorems 7.1, 7.3 and 6.21, since semicompleteness is modular for composable TRSs (=-=[27, 25]-=-). 12s8 Hierarchical Combinations In Section 7 we treated all the so far known composition operators, but for the most recent (and involved) case: hierarchical combinations. Hierarchical combinations ... |

2 |
Uni cation in a combination of arbitrary disjoint equational theories
- Schmidt-Schau
- 1989
(Show Context)
Citation Context ...onsistency Properties So far, there were no results on the modularity of CON for CTRSs. Theorem 7.13 Ultra-CON is modular for CTRSs. Proof By Theorems 7.1, 7.3 and 4.3, since CON is modular for TRSs (=-=[33]-=-). 10s7.2 Normal CTRSs So far, there is not a single result speci c for the modularity of normal CTRSs. This class has somehow been neglected in view of the fact, as said earlier, that every join CTRS... |

2 |
Unique normal forms and con uence of rewrite systems: Persistence
- Verma
- 1995
(Show Context)
Citation Context ...er results obtained for the completeness hierarchical combinations of TRSs: those of Dershowitz ([2]) and Rao ([29]) on completeness. Another result is obtained by lifting the recent result of Verma (=-=[37]-=-) on the modularity of con uence: he treats a kind of combination, called lr-combination, that is even more general than those considered in the aforementioned papers. We can lift his result obtaining... |

1 |
A rationale for conditional equational programming. TCS
- Dershowitz, Okada
- 1990
(Show Context)
Citation Context ...rmally shown that left-linearity does play a relevant role for normal CTRSs, and that this class of CTRSs has a behaviour very similar to TRSs: For instance, the fundamental notion of decreasingness (=-=[5, 4]-=-) is provided with a much more meaningful justification, since it just corresponds to `ultra-termination' (termination of the unraveled CTRS). Moreover, results specifically obtained for normal CTRSs ... |

1 |
On modularity of termination and con uence properties of conditional rewrite systems
- Gramlich
- 1994
(Show Context)
Citation Context ...a-terminating. Proof By Theorem 7.1, Lemmata 5.4, 5.1, and the aforementioned result of [25]. Finally,wetackle innermost termination: this property has been proved to be modular for disjoint CTRSs in =-=[10]-=-. No results are known for composable unions. Theorem 7.12 Ultra innermost termination is modular for composable CTRSs. Proof By Theorems 7.1, 4.5 and the modularity of innermost termination for compo... |