## Termination Modulo Combinations of Equational Theories

Citations: | 6 - 5 self |

### BibTeX

@MISC{Durán_terminationmodulo,

author = {Francisco Durán and Salvador Lucas and José Meseguer},

title = {Termination Modulo Combinations of Equational Theories},

year = {}

}

### OpenURL

### Abstract

Abstract. Rewriting with rules R modulo axioms E is a widely used technique in both rule-based programming languages and in automated deduction. Termination methods for rewriting systems modulo specific axioms E (e.g., associativity-commutativity) are known. However, much less seems to be known about termination methods that can be modular in the set E of axioms. In fact, current termination tools and proof methods cannot be applied to commonly occurring combinations of axioms that fall outside their scope. This work proposes a modular termination proof method based on semantics- and termination-preserving transformations that can reduce the proof of termination of rules R modulo E to an equivalent proof of termination of the transformed rules modulo a typically much simpler set B of axioms. Our method is based on the notion of variants of a term recently proposed by Comon and Delaune. We illustrate its practical usefulness by considering the very common case in which E is an arbitrary combination of associativity, commutativity, left- and right-identity axioms for various function symbols. 1

### Citations

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(Show Context)
Citation Context ...eal with the termination of → R∪R ′ /E∪E ′. There is also a rich body of related work on rewriting modulo E and coherence issues, including the just-discussed [13]. Early papers include those by Huet =-=[16]-=-, and particularly by Peterson and Stickel [26], who first studied the coherence of rewriting modulo A and AC axioms. Their ideas were later extended to general sets E of axioms in, e.g., [17, 2]. Per... |

214 | Unification theory
- Baader, Schulz
- 1998
(Show Context)
Citation Context ...nal theories, where different function symbols may satisfy different axioms. Although well-known modularity results exist for matching and unification modulo combinations of equational theories, e.g. =-=[1, 23, 27]-=-, the modularity aspects of termination modulo combinations of such theories do not seem to have been systematically studied. Indeed, at present there is a practical impossibility of proving many rewr... |

208 | Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations
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(Show Context)
Citation Context ...current tool support, and § 6 covers related work and conclusions. All proofs of technical results can be found in [7]. 2 Preliminaries 2.1 Order-sorted term rewriting We summarize here material from =-=[15, 19]-=- on order-sorted algebra and ordersorted rewriting. We start with a partially ordered set (S, ≤) of sorts, where s ≤ s ′ is interpreted as subsort inclusion. The connected components of (S, ≤) are the... |

145 |
Completion of a set of rules modulo a set of equations
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(Show Context)
Citation Context ...can try to make it so by completing the set of rules R to a set of rules ˜ R by a Knuth-Bendix-like completion procedure that computes critical pairs between equations in E and rules in R (see, e.g., =-=[17, 29]-=- for the strong coherence completion that we use here, and [13] for the equivalent notion of extension completion). As we will further discuss in § 4, for theories E that are combinations of A, C, LU,... |

135 |
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(Show Context)
Citation Context ...r any terms u, v with sorts in the same connected component, the relation u →R,E v holds if there is a position p in u, a rule l → r in R, and a substitution σ such that u|p =E lσ and v = u[rσ]p (see =-=[26]-=-). Of course, →R,E⊆→ R/E. The important question is the completeness question: can any → R/E-step be simulated by a →R,E-step? We say that R satisfies the E-completeness property if for any u, v with ... |

88 | AProVE 1.2: Automatic termination proofs in the dependency pair framework
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(Show Context)
Citation Context ...e running example have been obtained with these commands, and then MTT has been used to obtain a version of the specification in TPDB notation, which was used to prove its AC-termination using AProVE =-=[14]-=-. Specifically, the specification proved was the one obtained using the C;Uk;B transformation in MTT (see [8] for details on this transformation). Notice that sort information is key to prove the term... |

80 | Termination of rewriting systems by polynomial interpretations and its implementation - Cherifa, Lescanne - 1987 |

75 |
Advanced topics in term rewriting
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- 2002
(Show Context)
Citation Context ...how our running example is proved. Our work bears also some relationship to modularity methods for termination of TRSs. A very good survey of the literature on such methods up to 2002 can be found in =-=[24]-=-. They are very much orthogonal to ours. They consider the problem of when two TRSs, R1 and R2, both terminating, are such that R1 ∪ R2 is also terminating. Most of the literature seems to focus on th... |

39 | The finite variant property: How to get rid of some algebraic properties
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- 2005
(Show Context)
Citation Context ...second key idea is to generate the transformed rules ̂ R by computing the ∆, B-variants of the left-hand sides l for the rules l → r in R. The notion of variant has been proposed by Comon and Delaune =-=[5]-=- and has been further developed in [9, 10]. Intuitively, given a term t, a ∆, B-variant of t is a ∆, Bcanonical form u of an instance of t by some substitution θ; more precisely, it is a pair (u, θ). ... |

31 | Completion for rewriting modulo a congruence
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(Show Context)
Citation Context ...by Huet [16], and particularly by Peterson and Stickel [26], who first studied the coherence of rewriting modulo A and AC axioms. Their ideas were later extended to general sets E of axioms in, e.g., =-=[17, 2]-=-. Perhaps the two papers closest in spirit to ours are those by Marché [21] and Viry [29], since in both of them the idea of decomposing a set of axioms E as a union ∆ ∪ B with ∆ convergent and cohere... |

29 |
Operational semantics of OBJ3
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(Show Context)
Citation Context ...current tool support, and § 6 covers related work and conclusions. All proofs of technical results can be found in [7]. 2 Preliminaries 2.1 Order-sorted term rewriting We summarize here material from =-=[15, 19]-=- on order-sorted algebra and ordersorted rewriting. We start with a partially ordered set (S, ≤) of sorts, where s ≤ s ′ is interpreted as subsort inclusion. The connected components of (S, ≤) are the... |

27 |
Combining matching algorithms: The regular case
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- 1991
(Show Context)
Citation Context ...nal theories, where different function symbols may satisfy different axioms. Although well-known modularity results exist for matching and unification modulo combinations of equational theories, e.g. =-=[1, 23, 27]-=-, the modularity aspects of termination modulo combinations of such theories do not seem to have been systematically studied. Indeed, at present there is a practical impossibility of proving many rewr... |

21 | Proving operational termination of membership equational programs
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(Show Context)
Citation Context ...e specification in TPDB notation, which was used to prove its AC-termination using AProVE [14]. Specifically, the specification proved was the one obtained using the C;Uk;B transformation in MTT (see =-=[8]-=- for details on this transformation). Notice that sort information is key to prove the termination of the resulting specification. 6 Related Work and Conclusions This work is related to a wide body of... |

19 | Normalised rewriting and normalised completion
- Marché
- 1994
(Show Context)
Citation Context ... the coherence of rewriting modulo A and AC axioms. Their ideas were later extended to general sets E of axioms in, e.g., [17, 2]. Perhaps the two papers closest in spirit to ours are those by Marché =-=[21]-=- and Viry [29], since in both of them the idea of decomposing a set of axioms E as a union ∆ ∪ B with ∆ convergent and coherent modulo B is used. In particular, [29] developed in detail the treatment ... |

17 |
Equational rules for rewriting logic
- Viry
- 2002
(Show Context)
Citation Context ...can try to make it so by completing the set of rules R to a set of rules ˜ R by a Knuth-Bendix-like completion procedure that computes critical pairs between equations in E and rules in R (see, e.g., =-=[17, 29]-=- for the strong coherence completion that we use here, and [13] for the equivalent notion of extension completion). As we will further discuss in § 4, for theories E that are combinations of A, C, LU,... |

14 | D.: Dependency pairs for equational rewriting
- Giesl, Kapur
- 2001
(Show Context)
Citation Context ...Unfortunately, it applies also to termination methods themselves. For example, perhaps the most general termination modulo proof method known, namely, the Giesl-Kapur dependency pairs modulo E method =-=[13]-=- assumes that E has non-collapse equational axioms (thus excluding the identity axiom for __) and a finitary E-unification algorithm (which also excludes the associativity axiom for the list concatena... |

13 | Termination of Associative-Commutative Rewriting by Dependency Pairs - Marche, Urbain - 1998 |

12 | A total AC-compatible ordering based on RPO - Rubio, Nieuwenhuis - 1995 |

11 | Termination and completion modulo associativity, commutativity and identity - Jouannaud, Marché - 1992 |

7 | R.: Variant narrowing and equational unification - Escobar, Meseguer, et al. - 2009 |

7 | Dummy elimination in equational rewriting - Ferreira - 1996 |

6 |
Effectively checking the finite variant property
- Escobar, Meseguer, et al.
(Show Context)
Citation Context ...nsformed rules ̂ R by computing the ∆, B-variants of the left-hand sides l for the rules l → r in R. The notion of variant has been proposed by Comon and Delaune [5] and has been further developed in =-=[9, 10]-=-. Intuitively, given a term t, a ∆, B-variant of t is a ∆, Bcanonical form u of an instance of t by some substitution θ; more precisely, it is a pair (u, θ). Some variants are more general than others... |

5 | Combination of matching algorithms
- Ringeissen
- 1994
(Show Context)
Citation Context ...nal theories, where different function symbols may satisfy different axioms. Although well-known modularity results exist for matching and unification modulo combinations of equational theories, e.g. =-=[1, 23, 27]-=-, the modularity aspects of termination modulo combinations of such theories do not seem to have been systematically studied. Indeed, at present there is a practical impossibility of proving many rewr... |

4 | Equational termination by semantic labelling - Ohsaki, Middeldorp, et al. - 2000 |

1 | Variant Narrowing and Extreme Termination
- Escobar, Meseguer, et al.
- 2009
(Show Context)
Citation Context ...ies, for deduction, confluence, and termination purposes. The transformation can be defined in general. Furthermore, if B has a finitary unification algorithm it can be implementd by variantnarrowing =-=[11]-=-, which when EE has the finite variant property makes ̂ R finite if R is finite. Definition 3 (R ↦→ ̂ R transformation). Let R = (Σ, E, R) be an order-sorted rewrite theory where E satisfies the requi... |