## A Total, Ground Path Ordering for Proving Termination of AC-Rewrite Systems (1997)

Venue: | Proc. 8th RTA, LNCS 1232 |

Citations: | 16 - 3 self |

### BibTeX

@INPROCEEDINGS{Kapur97atotal,,

author = {Deepak Kapur and G. Sivakumar},

title = {A Total, Ground Path Ordering for Proving Termination of AC-Rewrite Systems},

booktitle = {Proc. 8th RTA, LNCS 1232},

year = {1997},

pages = {142--156},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

. A new path ordering for showing termination of associativecommutative (AC) rewrite systems is defined. If the precedence relation on function symbols is total, the ordering is total on ground terms, but unlike the ordering proposed by Rubio and Nieuwenhuis, this ordering can orient the distributivity property in the proper direction. The ordering is defined in a natural way using recursive path ordering with status as the underlying basis. This settles a longstanding problem in termination orderings for AC rewrite systems. The ordering can be used to define an ordering on nonground terms. 1 Introduction Rewriting techniques reduce the search space for finding proofs substantially because of the ability to orient equality, which is symmetric, into a terminating directed rewrite relation (!), which is anti-symmetric, using well founded orderings. Rules are used for "simplifying" expressions by repeatedly replacing instances of left-hand sides by the corresponding right-hand s...

### Citations

456 | Termination of Rewriting
- Dershowitz
- 1987
(Show Context)
Citation Context ...in proofs and computations. Syntactic "path orderings" based on a precedence relationson function symbols have been developed to prove termination of a set of rewrite rules. A comprehensive =-=survey is [5]-=-. As in the example above, many interesting and useful rewrite systems have operators (like +,,,,\Phi) which are associative and commutative. Also, distributivity axioms like as(b + c) ! (asb) + (asc)... |

116 |
On Proving Term Rewriting Systems are Noetherian
- Lankford
- 1979
(Show Context)
Citation Context ...ntly used. Developing path orderings for associative-commutative (AC) rewrite systems has been quite a challenge at least for a decade. A number of attempts have been reported in [1, 15, 4]; see also =-=[11, 2]-=- for polynomial orderings as well as [6] for other approaches. In 1990, Kapur, Sivakumar and Zhang proposed a general ordering scheme based on recursive path ordering with status (rpos) without any re... |

80 |
Termination of rewriting systems by polynomial interpretations and its implementation
- Cherifa, Lescanne
- 1987
(Show Context)
Citation Context ...ntly used. Developing path orderings for associative-commutative (AC) rewrite systems has been quite a challenge at least for a decade. A number of attempts have been reported in [1, 15, 4]; see also =-=[11, 2]-=- for polynomial orderings as well as [6] for other approaches. In 1990, Kapur, Sivakumar and Zhang proposed a general ordering scheme based on recursive path ordering with status (rpos) without any re... |

64 | An overview of Rewrite Rule Laboratory (RRL - Kapur, Zhang - 1995 |

49 |
Termination ordering for Associative-Commutative Rewriting Systems
- Bachmair, Plaisted
- 1985
(Show Context)
Citation Context ...b) + (asc) are frequently used. Developing path orderings for associative-commutative (AC) rewrite systems has been quite a challenge at least for a decade. A number of attempts have been reported in =-=[1, 15, 4]-=-; see also [11, 2] for polynomial orderings as well as [6] for other approaches. In 1990, Kapur, Sivakumar and Zhang proposed a general ordering scheme based on recursive path ordering with status (rp... |

31 |
A.: A precedence-based total AC-compatible ordering
- Nieuwenhuis, Rubio
- 1993
(Show Context)
Citation Context ...bio and Nieuwenhuis introduced a total ordering on ground terms for a total precedence relation on function symbols by elevating arguments of an AC operator whose root is smaller than the AC operator =-=[14]-=-, an idea proposed in [9]. They also demonstrated how their ordering (henceforth called RN's ordering) can be lifted to nonground terms. The main weakness of their ordering is that it does not orient ... |

23 |
Extension of the associative path ordering to a chain of associative commutative symbols
- Delor, Puel
- 1993
(Show Context)
Citation Context ...b) + (asc) are frequently used. Developing path orderings for associative-commutative (AC) rewrite systems has been quite a challenge at least for a decade. A number of attempts have been reported in =-=[1, 15, 4]-=-; see also [11, 2] for polynomial orderings as well as [6] for other approaches. In 1990, Kapur, Sivakumar and Zhang proposed a general ordering scheme based on recursive path ordering with status (rp... |

12 |
Any ground associative commutative theory has a finite canonical system
- Narendran, Rusinowitch
- 1991
(Show Context)
Citation Context ...not total on equivalence classes of AC ground terms even when the precedence relation on function symbols is total. A total ordering on AC-ground terms was first proposed by Narendran and Rusinowitch =-=[13]-=- based on polynomial interpretations, which was used to show that every ground AC theory has a canonical rewrite system. Unfortunately, that ordering does not orient distributivity appropriately. It i... |

7 | Orderings, ACTheories and Symbolic Constraint Solving
- Comon, Nieuwenhuis, et al.
(Show Context)
Citation Context ...cedence relation on function symbols is given, thus highlighting the new ideas and constructions. The ordering can be extended to nonground terms using methods based on constraint solving proposed in =-=[3]-=-, or using an approximation when dealing with nonground AC-terms. We plan to develop an extension of the ordering on ground terms to nonground terms without having to resort to constraint solving or a... |

6 | Proving termination of associative-commutative rewriting systems by rewriting - Gnaedig, Lescanne - 1986 |

5 |
A Path Ordering for Proving Termination of AC Rewrite Systems
- Kapur, Sivakumar, et al.
- 1995
(Show Context)
Citation Context ...In 1990, Kapur, Sivakumar and Zhang proposed a general ordering scheme based on recursive path ordering with status (rpos) without any restrictions on the precedence relation between function symbols =-=[9]-=-. Their ordering scheme however has a weakness: it is not total on equivalence classes of AC ground terms even when the precedence relation on function symbols is total. A total ordering on AC-ground ... |

5 |
Normalized rewriting and normalized completion
- Marche
- 1994
(Show Context)
Citation Context ...ticular, even ifs? +, as(b + c) ! (asb) + (asc). Because of this anomaly, the use of their ordering for orienting axioms of familiar algebraic structures with AC operators becomes problematic. Marche =-=[12]-=- proposed doing a lexicographic combination of associative path orderings extended by Delor and Puel [4] and RN's ordering to address this problem. In this paper, we propose a path ordering based on r... |

3 |
Maximal extensions of simplification orderings
- Kapur, Sivakumar
- 1995
(Show Context)
Citation Context ...nd terms without having to resort to constraint solving or approximations. The proposed ordering can also be used as a basis to define a maximal ordering on nonground AC terms using ideas proposed in =-=[7]-=-. 2 Rewrite systems and Simplification Orderings Let F be a set of function symbols, and T (F; X) be the set of terms built using function symbols in F and variables in X: Each function symbol f 2 F h... |

2 |
Path and decomposition orderings for proving ACtermination. Seki-Report, SR-89-18, University of Kaiserslautern. See also "Improving associative path orderings
- Steinbach
- 1989
(Show Context)
Citation Context ...b) + (asc) are frequently used. Developing path orderings for associative-commutative (AC) rewrite systems has been quite a challenge at least for a decade. A number of attempts have been reported in =-=[1, 15, 4]-=-; see also [11, 2] for polynomial orderings as well as [6] for other approaches. In 1990, Kapur, Sivakumar and Zhang proposed a general ordering scheme based on recursive path ordering with status (rp... |