## On Termination and Confluence Properties of Disjoint and Constructor-Sharing Conditional Rewrite Systems (1996)

Venue: | Theoretical Computer Science |

Citations: | 13 - 0 self |

### BibTeX

@ARTICLE{Gramlich96ontermination,

author = {Bernhard Gramlich},

title = {On Termination and Confluence Properties of Disjoint and Constructor-Sharing Conditional Rewrite Systems},

journal = {Theoretical Computer Science},

year = {1996},

volume = {165},

pages = {97--131}

}

### Years of Citing Articles

### OpenURL

### Abstract

We investigate the modularity behaviour of termination and confluence properties of (join) conditional term rewriting systems. We give counterexamples showing that the properties weak termination, weak innermost termination and (strong) innermost termination are not preserved in general under signature extensions. Then we develop sufficient conditions for the preservation of these properties under signature extensions, and more generally, for their modularity in the disjoint union case. This leads to new criteria for modularity of termination and completeness generalizing known results for unconditional systems. Finally, combining our analysis with recent related results on the preservation of semi-completeness, we show how to cover the (non-disjoint) case of combined conditional rewrite systems with shared constructors, too. 1 Introduction, Motivation and Overview Starting with the seminal work of Toyama [31] the investigation of preservation properties of term rewriting systems (TRSs...

### Citations

783 | Rewrite systems
- Dershowitz, Jouannaud
- 1990
(Show Context)
Citation Context ...ared constructors. 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. =-=[3]-=-, [12]). For brevity we shall make use of the following abbreviations (cf. e.g. [8], [12]) which apply to TRSs (and, if sensible, also to terms; in this case we shall also use notations like WN(s;R) i... |

581 | Term rewriting systems
- Klop
- 1992
(Show Context)
Citation Context ...constructors. 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. [3], =-=[12]-=-). For brevity we shall make use of the following abbreviations (cf. e.g. [8], [12]) which apply to TRSs (and, if sensible, also to terms; in this case we shall also use notations like WN(s;R) in orde... |

89 |
Conditional Rewrite Rules: Confluence and Termination
- Bergstra, Klop
- 1986
(Show Context)
Citation Context ...er 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 6 (=-=[1]-=-) Consider the CTRS R 1 = 8 ? ! ? : 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 n... |

89 |
Computing in Systems Described by Equations
- O’Donnell
- 1977
(Show Context)
Citation Context ...iting Systems Here we shall summarize some known and recently obtained new results on restricted termination and confluence properties of unconditional and conditional TRSs. Theorem 9 (cf. e.g. [29], =-=[24]-=-, [12]) 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(R) =) SIN(R... |

86 | Counterexamples to termination for the direct sum of term rewriting systems - Toyama - 1987 |

84 | Natural termination
- Dershowitz, Hoot
- 1995
(Show Context)
Citation Context ...n-disjoint unions of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], [27]. More general hierarchical combinations of TRSs have recently been dealt with in [17], =-=[2]-=-, [18], [8], [6]. Some preservation results for (disjoint and non-disjoint, but non-hierarchical) combinations of conditional TRSs (CTRSs for short) finally have been obtained in [20], [22]), [21], [9... |

80 |
Tree-manipulating systems and Church-Rosser theorems
- ROSEN
- 1973
(Show Context)
Citation Context ...m 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 9 (cf. e.g. =-=[29]-=-, [24], [12]) 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(R) =)... |

80 |
On the Church-Rosser property for the direct sum of term rewriting systems
- Toyama
- 1987
(Show Context)
Citation Context ...s, we show how to cover the (non-disjoint) case of combined conditional rewrite systems with shared constructors, too. 1 Introduction, Motivation and Overview Starting with the seminal work of Toyama =-=[31]-=- 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... |

65 |
Modular Properties of Term Rewriting Systems
- Middeldorp
- 1990
(Show Context)
Citation Context ...ient 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. [30], [19], [33], =-=[20]-=-, [14], [7], [8], [25]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], [27]. More general hierarchical combinations of TRSs have recentl... |

55 | Confluence of Conditional Rewrite Systems
- Dershowitz, Okada, et al.
- 1987
(Show Context)
Citation Context ...ed than unconditional rewriting. For instance, the rewrite relation may be undecidable even for complete CTRSs without extra variables in the conditions (cf. [11]). For some positive results see e.g. =-=[4]-=-. 3 Restricted Termination and Confluence Properties of Conditional Term Rewriting Systems Here we shall summarize some known and recently obtained new results on restricted termination and confluence... |

54 | Modular Properties of Composable Term Rewriting Systems
- Ohlebusch
- 1994
(Show Context)
Citation Context ...or the corresponding slightly more general case of composable CTRSs which may share constructors and must share all defining rules for some defined function symbol whenever that symbol is shared (cf. =-=[28]-=- and [16] for some recent results on composable (C)TRSs). This seems to be plausible but may be technically tedious to achieve. From a practical applicability point of view it seems to be more useful ... |

50 |
On termination of the direct sum of term rewriting systems
- Rusinowitch
- 1987
(Show Context)
Citation Context ...w 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. =-=[30]-=-, [19], [33], [20], [14], [7], [8], [25]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], [27]. More general hierarchical combinations of... |

49 | Generalized sufficient conditions for modular termination of rewriting
- Gramlich
- 1992
(Show Context)
Citation Context ...a for the preservation of termination, completeness (i.e., termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. [30], [19], [33], [20], [14], =-=[7]-=-, [8], [25]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], [27]. More general hierarchical combinations of TRSs have recently been deal... |

48 |
Modularity of simple termination of term rewriting systems
- Kurihara, Ohuchi
- 1990
(Show Context)
Citation Context ...riteria for the preservation of termination, completeness (i.e., termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. [30], [19], [33], [20], =-=[14]-=-, [7], [8], [25]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], [27]. More general hierarchical combinations of TRSs have recently been... |

37 |
A Sufficient Condition f r the Termination f the Direct Sum of Term Rewriting Systems
- Middeldorp
- 1989
(Show Context)
Citation Context ...btain 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. [30], =-=[19]-=-, [33], [20], [14], [7], [8], [25]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], [27]. More general hierarchical combinations of TRSs ... |

31 | Completeness of combinations of constructor systems
- Middeldorp, Toyama
- 1991
(Show Context)
Citation Context ...urihara & Ohuchi [15] and studied subsequently by many authors. Slightly less general are combinations of constructor systems (with disjoint sets of defined symbols) considered in Middeldorp & Toyama =-=[23]-=- (for unconditional TRSs) and Middeldorp [21] (for CTRSs). Since the proofs of many modularity results for the disjoint union case mainly rely on the layered structure of terms and on the rank decreas... |

31 |
Termination for the Direct Sum of Left. Linear Term Rewriting Systems
- Toyama, Klop, et al.
- 1989
(Show Context)
Citation Context ...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. [30], [19], =-=[33]-=-, [20], [14], [7], [8], [25]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], [27]. More general hierarchical combinations of TRSs have r... |

29 | On the modularity of termination of term rewriting systems
- Ohlebusch
- 1994
(Show Context)
Citation Context ...preservation of termination, completeness (i.e., termination plus confluence) and of other interesting properties of TRSs under disjoint combinations (cf. e.g. [30], [19], [33], [20], [14], [7], [8], =-=[25]-=-). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], [27]. More general hierarchical combinations of TRSs have recently been dealt with in [... |

28 | Relating innermost, weak, uniform and modular termination of term rewriting systems
- Gramlich
- 1992
(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. [30], [19], [33], [20], [14], [7], =-=[8]-=-, [25]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], [27]. More general hierarchical combinations of TRSs have recently been dealt wit... |

25 | Modular termination of term rewriting systems revisited, in: Recent Trends in Data Type Specification
- Fernández, Jouannaud
- 1995
(Show Context)
Citation Context ...s of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], [27]. More general hierarchical combinations of TRSs have recently been dealt with in [17], [2], [18], [8], =-=[6]-=-. Some preservation results for (disjoint and non-disjoint, but non-hierarchical) combinations of conditional TRSs (CTRSs for short) finally have been obtained in [20], [22]), [21], [9], [28]. As show... |

23 |
Modular properties of conditional term rewriting systems
- Middeldorp
- 1993
(Show Context)
Citation Context ...th in [17], [2], [18], [8], [6]. Some preservation results for (disjoint and non-disjoint, but non-hierarchical) combinations of conditional TRSs (CTRSs for short) finally have been obtained in [20], =-=[22]-=-), [21], [9], [28]. As shown in [33] (by a very involved proof), completeness is preserved under disjoint union of left-linear (unconditional) TRSs. Instead of left-linearity one may also require a st... |

16 |
Simple termination of hierarchical combinations of term rewriting systems
- Rao
- 1994
(Show Context)
Citation Context ...joint unions of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], [27]. More general hierarchical combinations of TRSs have recently been dealt with in [17], [2], =-=[18]-=-, [8], [6]. Some preservation results for (disjoint and non-disjoint, but non-hierarchical) combinations of conditional TRSs (CTRSs for short) finally have been obtained in [20], [22]), [21], [9], [28... |

15 |
Completeness of hierarchical combinations of term rewriting systems
- Rao
(Show Context)
Citation Context ...]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], [27]. More general hierarchical combinations of TRSs have recently been dealt with in =-=[17]-=-, [2], [18], [8], [6]. Some preservation results for (disjoint and non-disjoint, but non-hierarchical) combinations of conditional TRSs (CTRSs for short) finally have been obtained in [20], [22]), [21... |

13 | Sufficient conditions for modular termination of conditional term rewriting systems
- Gramlich
(Show Context)
Citation Context ...ction is trivial for CTRSs without extra variables but less trivial otherwise (see Lemma 43 below). For some sufficient conditions for modularity of termination of CTRSs we refer to [20], [22], [21], =-=[9]-=-, [27]. Here we shall concentrate on CTRSs satisfying some structural restrictions. In [8] we have proved that any (strongly) innermost terminating, locally confluent (unconditional) overlay system is... |

13 | Completeness of Combinations of Conditional Constructor Systems
- Middeldorp
- 1994
(Show Context)
Citation Context ...r direction is trivial for CTRSs without extra variables but less trivial otherwise (see Lemma 43 below). For some sufficient conditions for modularity of termination of CTRSs we refer to [20], [22], =-=[21]-=-, [9], [27]. Here we shall concentrate on CTRSs satisfying some structural restrictions. In [8] we have proved that any (strongly) innermost terminating, locally confluent (unconditional) overlay syst... |

12 | Vrijer. Modularity of confluence: A simplified proof
- Klop, Middeldorp, et al.
- 1994
(Show Context)
Citation Context ...m a term need not have a preserved reduct , i.e., a reduct with a stable layer structure. This property turned out to be crucial (and was easily verified) in the simplified proof of Toyama's theorem (=-=[13]-=-) stating that confluence is modular for disjoint unions of (unconditional) TRSs. By guaranteeing the existence of preserved reducts, the proof of [13] essentially carries over to constructor sharing ... |

12 | On the modularity of confluence of constructor-sharing term rewriting systems
- Ohlebusch
- 1994
(Show Context)
Citation Context ... unions of (unconditional) TRSs. By guaranteeing the existence of preserved reducts, the proof of [13] essentially carries over to constructor sharing combinations of TRSs as recognized by Ohlebusch (=-=[26]-=-). One straightforward criterion to ensure the crucial preservation property property is to require weak normalization of the involved TRSs which implies the modularity of semi-completeness for constr... |

8 | Conditional Rewrite Rules”, Theoretical Computer Science 33 - Kaplan - 1984 |

6 | On Termination and Confluence of Conditional Rewrite Systems
- Gramlich
- 1995
(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 (=-=[10]-=-). In the present paper we shall exploit this property and show how to obtain corresponding preservation results for disjoint and constructor-sharing unions of CTRSs. However, this generalization of [... |

3 |
Termination of combination of composable term rewriting systems
- Kurihara, Ohuchi
- 1994
(Show Context)
Citation Context ...rresponding slightly more general case of composable CTRSs which may share constructors and must share all defining rules for some defined function symbol whenever that symbol is shared (cf. [28] and =-=[16]-=- for some recent results on composable (C)TRSs). This seems to be plausible but may be technically tedious to achieve. From a practical applicability point of view it seems to be more useful to invest... |

3 | Modular Properties of Constructor-Sharing Conditional Term Rewriting Systems
- Ohlebusch
- 1995
(Show Context)
Citation Context ...TRSs under disjoint combinations (cf. e.g. [30], [19], [33], [20], [14], [7], [8], [25]). Non-disjoint unions of TRSs with common constructors have been considered e.g. in [23], [15], [7], [8], [25], =-=[27]-=-. More general hierarchical combinations of TRSs have recently been dealt with in [17], [2], [18], [8], [6]. Some preservation results for (disjoint and non-disjoint, but non-hierarchical) combination... |