## Decision Problems in Ordered Rewriting (1997)

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Venue: | In 13th IEEE Symposium on Logic in Computer Science (LICS |

Citations: | 16 - 7 self |

### BibTeX

@INPROCEEDINGS{Comon97decisionproblems,

author = {Hubert Comon and Paliath Narendran and Robert Nieuwenhuis and Michael Rusinowitch},

title = {Decision Problems in Ordered Rewriting},

booktitle = {In 13th IEEE Symposium on Logic in Computer Science (LICS},

year = {1997},

pages = {276--286}

}

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### Abstract

A term rewrite system (TRS) terminates iff its rules are contained in a reduction ordering ?. In order to deal with any set of equations, including inherently non-terminating ones (like commutativity), TRS have been generalised to ordered TRS (E; ?), where equations of E are applied in whatever direction agrees with ?. The confluence of terminating TRS is well-known to be decidable, but for ordered TRS the decidability of confluence has been open. Here we show that the confluence of ordered TRS is decidable if ordering constraints for ? can be solved in an adequate way, which holds in particular for the class of LPO orderings. For sets E of constrained equations, confluence is shown to be undecidable. Finally, ground reducibility is proved undecidable for ordered TRS. 1 Introduction Term rewrite systems (TRS) have been applied to many problems in symbolic computation, automated theorem proving, program synthesis and verification, and logic programming among others. Two fundamental pr...

### Citations

751 | Rewrite systems
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(Show Context)
Citation Context ...ally, in Section 6, we state some possible extensions and implications of our decidability result in other areas. 2 Ordered Rewriting and LPO Constraints We adopt the terminology and the notations of =-=[9]-=-. T (F ; X ) is the set of first-order terms built on alphabets F of function symbols and X of variable symbols. T (F) is the set of terms which do not contain any variables. An equation is a multiset... |

489 |
Simple word problems in universal algebras
- Knuth, Bendix
- 1970
(Show Context)
Citation Context ... it can be made total (on ground terms, which suffices as we will see). Confluence is undecidable in general, but for terminating TRS a decision procedure is given in Knuth and Bendix' landmark paper =-=[17]-=-: a TRS is confluent iff all its critical pairs are joinable. For ordered TRS, deciding confluence is more difficult. It has been a long standing open question, listed for instance as open problem #64... |

454 | Computation: finite and infinite machines - Minsky - 1967 |

453 | Termination of rewriting
- Dershowitz
- 1985
(Show Context)
Citation Context ... the (strict) tree embedding relation, the smallest transitive monotonic relation ? emb such that s ? emb t if t is a proper subterm of s. Moreover, it is total on T (F) if ?F is total on F (see e.g. =-=[8, 9]-=- for more details). LPO is a standard choice for ordered TRS, since it fulfils the necessary requirements and and it is easily and efficiently implementable (note that other general-purpose orderings ... |

116 |
Orderings for equational proofs
- Bachmair, Dershowitz, et al.
- 1986
(Show Context)
Citation Context ...f theoretical interest. Since Knuth and Bendix' paper in the late 60's, a lot of work has been devoted to the completion of standard TRS and to ordered (or unfailing) completion of ordered TRS (e.g., =-=[2, 12, 23, 18, 3]-=-). Ordered completion provably yields a confluent system after a (possibly infinite) number of iterations. It roughly amounts to a systematic closure of the TRS under inclusion of (simplifications of)... |

73 | Theorem proving with ordering and equality constrained clauses
- Nieuwenhuis, Rubio
- 1995
(Show Context)
Citation Context ...cidable (while ordered rewriting is still terminating). This relativizes the scope of our decidability result and also shows its significance. Unfortunately, in completion with constraint inheritance =-=[16, 22]-=-, constrained equations appear even if we start with only unconstrained ones. In such an equation e j c, the constraint 2 c records the conditions under which e is derivable. This restricts the number... |

66 |
Deduction with symbolic constraints. Revue Francaise d'Intelligence Artificielle, 4(3):9--52
- Kirchner, Kirchner, et al.
- 1990
(Show Context)
Citation Context ...cidable (while ordered rewriting is still terminating). This relativizes the scope of our decidability result and also shows its significance. Unfortunately, in completion with constraint inheritance =-=[16, 22]-=-, constrained equations appear even if we start with only unconstrained ones. In such an equation e j c, the constraint 2 c records the conditions under which e is derivable. This restricts the number... |

58 |
On word problems in equational theories
- Hsiang, Rusinowitch
- 1987
(Show Context)
Citation Context ...on-orientable axioms (like the commutativity axiom). Therefore they have been generalized to ordered TRS (E; ?), where rewriting is done by applying equations of E in whatever direction agrees with ? =-=[12]-=-. Hence ordered TRS can handle any set of equations E, being terminating by definition. For solving word problems by ordered rewriting, there must be one unique minimal element (w.r.t. ?) in each E-co... |

49 |
Proof by consistency in equational theories
- Bachmair
- 1988
(Show Context)
Citation Context ...nstances of s can be reduced. Going further, we investigate the use of ordered rewriting in a classical application of rewrite systems: the proof-by-consistency approach to proving inductive theorems =-=[14, 13, 1]-=-. Here again, the result is the opposite of what happens in the case of finite term rewriting systems: we show that ground reducibility is undecidable for ordered rewriting. The paper is organized as ... |

45 |
Proof by consistency
- Kapur, Musser
- 1987
(Show Context)
Citation Context ...nstances of s can be reduced. Going further, we investigate the use of ordered rewriting in a classical application of rewrite systems: the proof-by-consistency approach to proving inductive theorems =-=[14, 13, 1]-=-. Here again, the result is the opposite of what happens in the case of finite term rewriting systems: we show that ground reducibility is undecidable for ordered rewriting. The paper is organized as ... |

36 | Simple LPO constraint solving methods
- Nieuwenhuis
- 1993
(Show Context)
Citation Context ...iation. These steps precisely generate the right ordering relations that have to be analysed. The whole process of building the trees importantly relies on existing results for LPO constraint solving =-=[5, 20]-=-. This is also the case for our proof of finiteness of the trees. Our main result is finally that (E; ?) is confluent iff all leaves of the trees are tautologies. We emphasize that our decision proced... |

36 |
Semantic confluence tests and completion methods
- Plaisted
- 1985
(Show Context)
Citation Context .... This definition extends to ordered rewriting, replacing R with !E;? when E is a finite set of (unconstrained) equations. Ground reducibility is decidable for arbitrary finite term rewriting systems =-=[24]-=-. We show here that it is undecidable for finite sets of equations: Theorem 5.1 The problem: Input: A finite set of (unconstrained) equations E, a term t, a lexicographic path ordering. Question: Is t... |

34 |
Automatic proofs by induction in equational theories without constructors
- Jouannaud, Kounalis
- 1989
(Show Context)
Citation Context ...nstances of s can be reduced. Going further, we investigate the use of ordered rewriting in a classical application of rewrite systems: the proof-by-consistency approach to proving inductive theorems =-=[14, 13, 1]-=-. Here again, the result is the opposite of what happens in the case of finite term rewriting systems: we show that ground reducibility is undecidable for ordered rewriting. The paper is organized as ... |

30 | Equational inference, canonical proofs, and proof orderings
- Bachmair, Dershowitz
- 1994
(Show Context)
Citation Context ...f theoretical interest. Since Knuth and Bendix' paper in the late 60's, a lot of work has been devoted to the completion of standard TRS and to ordered (or unfailing) completion of ordered TRS (e.g., =-=[2, 12, 23, 18, 3]-=-). Ordered completion provably yields a confluent system after a (possibly infinite) number of iterations. It roughly amounts to a systematic closure of the TRS under inclusion of (simplifications of)... |

22 |
Simulation of Turing machines by a left-linear rewrite rule
- Dauchet
- 1989
(Show Context)
Citation Context ...ensure the existence and uniqueness of normal forms and a decidable word problem. Let us take a closer look at both properties. Termination of TRS is undecidable even for left-linear one-rule systems =-=[7]-=-. However, a TRS R terminates if there is some reduction ordering ? such that l ? r for each l ! r 2 R, and there are practical general-purpose methods for defining such ?. But standard TRS cannot dea... |

22 |
Ordered Rewriting and Confluence
- Martin, Nipkow
- 1990
(Show Context)
Citation Context ... + yoe, and, on the other, the further possible relations with uoe. In all cases, joinability follows. Note that it is not sufficient here to consider only relations between variables, as observed in =-=[18]-=-. In general the picture is not so simple as in the previous example. Joinability proofs may have more than one step, and the cases of ordering relations considered must be compatible with the ones of... |

20 | The first-order theory of lexicographic path orderings is undecidableâ€™, Theoretical Computer Science
- Comon, Treinen
(Show Context)
Citation Context ...our problem is to decide joinability of the critical pairs by rewriting with a set of constrained equations. This is not easy, because simplification with an arbitrary constrained rule is undecidable =-=[6]-=-: given a term s and a rule l ! r j c, it is undecidable whether or not all instances of s can be reduced. Going further, we investigate the use of ordered rewriting in a classical application of rewr... |

19 |
Complete sets of reductions with constraints
- Peterson
- 1990
(Show Context)
Citation Context ...f theoretical interest. Since Knuth and Bendix' paper in the late 60's, a lot of work has been devoted to the completion of standard TRS and to ordered (or unfailing) completion of ordered TRS (e.g., =-=[2, 12, 23, 18, 3]-=-). Ordered completion provably yields a confluent system after a (possibly infinite) number of iterations. It roughly amounts to a systematic closure of the TRS under inclusion of (simplifications of)... |

18 | The Saturate system
- Ganzinger, Nieuwenhuis, et al.
(Show Context)
Citation Context ...at our method is able to do (automatically), making it now also possible to decide whether the completion has finished. In fact, in [21] we reported on the use of similar trees in the Saturate system =-=[11]-=- as a successful method for proving the redundancy of critical pairs. Furthermore, in case of non-joinability, instead of adding the critical pair to E, one may choose to add the non-tautology leaves ... |

15 |
On ground-confluence of term rewriting systems
- Kapur, Narendran, et al.
- 1990
(Show Context)
Citation Context ...tems. We reduce the ground confluence problem for an lpo-terminating string rewriting system to the confluence of ordered rewriting of a finite set of constrained equations. The former is undecidable =-=[10]-=-. We consider the system R u;v of [10], Section 5. It only involves unary function symbols. No right hand side is reduced to a variable. This system is terminating; its termination can be proved using... |

14 | More Problems in Rewriting
- Dershowitz, Jouannaud, et al.
- 1993
(Show Context)
Citation Context ... iff all its critical pairs are joinable. For ordered TRS, deciding confluence is more difficult. It has been a long standing open question, listed for instance as open problem #64 in the RTA'93 list =-=[10]-=-. The main CNRS & LSV, ENS de Cachan, 61 Av. Pres. Wilson, F-94235 Cachan, France, comon@lsv.ens-cachan.fr. y SUNY at Albany, Dep. of Comp. Science, Albany, NY 12222 USA, dran@cs.albany.edu. z Technic... |

12 |
Practical results on the saturation of full first-order clauses: Experiments with the saturate system. (system description
- Nivela, Nieuwenhuis
- 1993
(Show Context)
Citation Context ...nce redundancy) of a critical pair are crucial. This is precisely what our method is able to do (automatically), making it now also possible to decide whether the completion has finished. In fact, in =-=[21]-=- we reported on the use of similar trees in the Saturate system [11] as a successful method for proving the redundancy of critical pairs. Furthermore, in case of non-joinability, instead of adding the... |

11 |
Solving inequations in term algebras
- Comon
- 1990
(Show Context)
Citation Context ...iation. These steps precisely generate the right ordering relations that have to be analysed. The whole process of building the trees importantly relies on existing results for LPO constraint solving =-=[5, 20]-=-. This is also the case for our proof of finiteness of the trees. Our main result is finally that (E; ?) is confluent iff all leaves of the trees are tautologies. We emphasize that our decision proced... |

11 |
On ground confluence of term rewriting systems
- Kapur, Narendan, et al.
- 1990
(Show Context)
Citation Context ...f the notion of confluence, since the weaker property of ground confluence, i.e. confluence when rewriting only ground terms over the given signature, is undecidable even for terminating standard TRS =-=[15]-=-. Example: Let E consist of the (clearly non-orientable) single equation (x+y)+z = z +(x+y). Then there is a critical pair cp which is (z + (x+y))+u = u+ ((x+y)+z). To show confluence, we have to prov... |

6 |
Automated Deduction with Constrained Clauses
- Rubio
- 1994
(Show Context)
Citation Context ... s) and :(s = t) is logically equivalent to (s ? t)s(t ? s). We will also consider the satisfaction of constraints over extended signatures. The relevance of such interpretations is explained in e.g. =-=[20]-=- and we will come back later to it. An assignment oe satisfies the constraint c over extended signatures (which we write oe j= e T (F) c) when there is an F 0 ' F and an orderingsF 0 extendingsF such ... |

2 |
A new ordering constraint solving method and its applications
- Nieuwenhuis
- 1993
(Show Context)
Citation Context ...blems in rewriting, 1993) is the decidability of confluence. Confluence is known to be undecidable for arbitrary finite term rewriting systems. What about ordered rewriting? The question is raised in =-=[15]-=- in general and for lexicographic path orderings in particular. In this [15], the problem is reduced to solving ordering constraints in the set of normal forms. However the question of confluence is l... |