## Complete Monotonic Semantic Path Orderings (2000)

Venue: | In Proc. 17th CADE, LNAI 1831 |

Citations: | 31 - 8 self |

### BibTeX

@INPROCEEDINGS{Borralleras00completemonotonic,

author = {Cristina Borralleras and Maria Ferreira and Albert Rubio},

title = {Complete Monotonic Semantic Path Orderings},

booktitle = {In Proc. 17th CADE, LNAI 1831},

year = {2000},

pages = {346--364},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS). In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination, i.e., a TRS is terminating if and only if its rules are included in some MSPO. Hence MSPO is a complete termination method. On the practical side, it can be easily automated using as ingredients standard interpretations and general-purpose orderings like RPO. This is shown to be a sufficiently powerful way to handle several non-trivial examples and to obtain methods like dummy elimination or dependency pairs as particular cases. Finally, we obtain some positive modularity results for termination based on MSPO. 1 Introduction Rewrite systems are sets of rules (directed equations) used to compute by repeatedly replacing parts of a given formula with equal ones until the simplest po...

### Citations

779 | Rewrite Systems - Dershowitz, Jouannaud - 1990 |

578 | Term Rewriting Systems - Klop - 1992 |

469 | Termination of rewriting - Dershowitz - 1987 |

272 | Orderings for term-rewriting systems
- Dershowitz
- 1982
(Show Context)
Citation Context ...relation are called simplification orderings, and its well-foundedness follows from Kruskal's theorem [Kru60]. Inside this class, path orderings, and in particular the recursive path ordering 1 (RPO) =-=[Der82]-=-, have received a special attention (see [Der87, DJ90]). Unfortunately, although these orderings are simple and easy to use, they turn out, in many cases, to be a weak termination proving tool, as the... |

229 | Termination of term rewriting using dependency pairs - Arts, Giesl - 2000 |

126 |
the tree theorem, and Vazsonyi's conjecture
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- 1960
(Show Context)
Citation Context ...s to check that lsr for every rule l ! r in the system. Monotonic orderings including the subterm relation are called simplification orderings, and its well-foundedness follows from Kruskal's theorem =-=[Kru60]-=-. Inside this class, path orderings, and in particular the recursive path ordering 1 (RPO) [Der82], have received a special attention (see [Der87, DJ90]). Unfortunately, although these orderings are s... |

89 | Termination of term rewriting: interpretation and type elimination - Zantema - 1994 |

60 | Commutation, transformation, and termination - Bachmair, Dershowitz - 1986 |

51 |
Attempts for generalizing the recursive path ordering. Unpublished manuscript
- Kamin, Levy
- 1980
(Show Context)
Citation Context ...or proving termination of (non-simply terminating) TRS. As an alternative to transformation methods, more powerful term orderings can be used. Due to its simplicity, the Semantic Path Ordering (SPO) (=-=[KL80]-=-) becomes a potential well-known candidate: in SPO the scheme of RPO is generalized by replacing the precedence on function symbols by any (well-founded) underlying (quasi-)ordering involving the whol... |

51 | The higher-order recursive path ordering - Jouannaud, Rubio - 1999 |

49 | Generalized sufficient conditions for modular termination of rewriting
- Gramlich
- 1992
(Show Context)
Citation Context ...R [ fG(x; y) ! x; G(x; y) ! yg terminates for some new symbol G. For C " -termination we have the following results: ffl [Ohl94] C " -termination is a modular property for disjoint unions of=-= TRS. ffl [Gra94] C " -=--termination is a modular property for constructor-sharing unions of finite TRS. Lemma 6.1. If \Xi ` B and R is included insmspo then R is C " -terminating. Note that since B includes the subterm... |

49 | Solving symbolic ordering constraints - Comon |

42 | Argument filtering transformation - Kusakari, Nakamura, et al. - 1999 |

34 | Automatically Proving Termination Where Simplification Orderings Fail
- Arts, Giesl
- 1997
(Show Context)
Citation Context ...his problem, different transformation methods have been developed, e.g. [BD86, BL90, Zan94, FZ93, Ste95, Xi98]. By transforming the TRS into a set of ordering constraints, the dependency pairs method =-=[AG97]-=- has become a successful general technique for proving termination of (non-simply terminating) TRS. As an alternative to transformation methods, more powerful term orderings can be used. Due to its si... |

29 | On the modularity of termination of term rewriting systems
- Ohlebusch
- 1994
(Show Context)
Citation Context ...ed terminating under non-deterministic collapses, denoted C " -terminating, if R [ fG(x; y) ! x; G(x; y) ! yg terminates for some new symbol G. For C " -termination we have the following res=-=ults: ffl [Ohl94] C " -=--termination is a modular property for disjoint unions of TRS. ffl [Gra94] C " -termination is a modular property for constructor-sharing unions of finite TRS. Lemma 6.1. If \Xi ` B and R is incl... |

27 | Dummy Elimination: Making Termination Easier
- Ferreira, Zantema
- 1995
(Show Context)
Citation Context ...cular classes of underlying quasi-orderings, which can be automatically generated. As a hint of the power of the resulting family of MSPO's, it is shown that some known methods like dummy elimination =-=[FZ93]-=- or dependency pairs [AG97] (without the dependency graph refinement) are particular instances. 2 Using these classes of underlying quasi-orderings the termination proofs of several non-simply termina... |

25 | Termination by completion”, Applicable Algebra - Bellegarde, Lescanne, et al. - 1990 |

22 | Automatic termination proofs with transformation orderings - Steinbach - 1995 |

22 | A fully syntactic AC-RPO - Rubio - 1631 |

19 | Transforming termination by selflabelling - Middeldorp, Ohsaki, et al. |

18 | Modularity of termination using dependency pairs - Giesl - 1998 |

18 | TALP: a tool for the termination analysis of logic programs - Ohlebusch, Claves, et al. |

11 | Pushing the Frontiers of Combining Rewrite Systems Farther Outwards - Giesl - 2000 |

6 | Bachmair and Nachum Dershowitz. Commutation, transformation, and termination - Leo - 1986 |

4 | Towards automated termination proofs through “freezing - Xi - 1998 |

3 |
On a monotonic semantic path ordering
- Geser
- 1992
(Show Context)
Citation Context ...automatically checked once the ingredients, i.e. the underlying (base) quasi-orderings, of the MSPO are provided. The first and only other, as far as we know, monotonic version of SPO is due to Geser =-=[Ges92]-=-. On the one hand, this proposal is not as general as ours (in fact, as we will show, it does not characterize termination), and on the other hand, it is less suitable for practical implementations. I... |

1 | Term Riwriting and all that - Baader, Nipkow - 1998 |