## Approximating dependency graphs using tree automata techniques (2001)

Venue: | In Proc. IJCAR 2001, LNAI 2083 |

Citations: | 16 - 5 self |

### BibTeX

@INPROCEEDINGS{Middeldorp01approximatingdependency,

author = {Aart Middeldorp},

title = {Approximating dependency graphs using tree automata techniques},

booktitle = {In Proc. IJCAR 2001, LNAI 2083},

year = {2001},

pages = {593--610},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. The dependency pair method of Arts and Giesl is the most powerful technique for proving termination of term rewrite systems automatically. We show that the method can be improved by using tree automata techniques to obtain better approximations of the dependency graph. This graph determines the ordering constraints that need to be solved in order to conclude termination. We further show that by using our approximations the dependency pair method provides a decision procedure for termination of right-ground rewrite systems. 1

### Citations

1007 |
Term rewriting and all that
- Baader, Nipkow
- 1998
(Show Context)
Citation Context ...ursapproximations the dependency pair method provides a decision procedure for termination of right-ground rewrite systems. 2 Dependency Pairs We assume familiarity with the basics of term rewriting (=-=[5]-=-). A term rewrite system (TRS for short) consists of rewrite rules l → r that satisfy l /∈ V and Var(r) ⊆ Var(l). If these conditions are not imposed we find it useful to speak of extended TRSs (eTRSs... |

518 |
Simple word problems in universal algebras
- Knuth, Bendix
- 1970
(Show Context)
Citation Context ... constructing an explicit wellfounded order. These methods are fully automatable but have limited power. Well-known examples are the recursive path order of Dershowitz [10] and the Knuth-Bendix order =-=[18]-=-. 2. Semantic methods that compare terms by interpreting them in some wellfounded domain. These methods can be very powerful in theory but their implementations rely on heuristics that greatly reduce ... |

271 | Orderings for term-rewriting systems
- Dershowitz
- 1982
(Show Context)
Citation Context ...ic methods that compare terms by constructing an explicit wellfounded order. These methods are fully automatable but have limited power. Well-known examples are the recursive path order of Dershowitz =-=[10]-=- and the Knuth-Bendix order [18]. 2. Semantic methods that compare terms by interpreting them in some wellfounded domain. These methods can be very powerful in theory but their implementations rely on... |

229 | Termination of term rewriting using dependency pairs
- Arts, Giesl
- 2000
(Show Context)
Citation Context ...make clear that such eTRSs are very useful for automatically proving termination of TRSs. Below we recall the basic notions and results of the dependency pair technique of Arts and Giesl. We refer to =-=[2, 4, 12]-=- for motivations and further refinements. We adopt the notation of [13, 20]. Let R be a TRS over a signature F. As usual, root symbols of left-hand sides of rewrite rules are called defined. Let F ♯ d... |

157 | Tree automata techniques and applications
- Comon, Dauchet, et al.
- 1997
(Show Context)
Citation Context ...]). Let R be a TRS. 1. EDG(R) is computable. 2. DG(R) ⊆ EDG(R). ⊓⊔ 3 Tree Automata We briefly recall some basic definitions and results concerning tree automata. Much more information can be found in =-=[8]-=-. A (finite bottom-up) tree automaton is a quadruple A = (F, Q, Qf , ∆) consisting of a finite signature F, a finite set Q of states, disjoint from F, a subset Qf ⊆ Q of final states, and a set of tra... |

119 | On proving term rewriting systems are Noetherian - Lankford - 1979 |

91 | Termination of term rewriting by semantic labelling
- Zantema
- 1995
(Show Context)
Citation Context ...ier to prove and implies termination of the former system. Examples include the transformation order of Bellegarde and Lescanne [6], and Zantema’s distribution elimination [27] and semantic labelling =-=[28]-=-. Transformations differ in their degree of automation. Since termination is an undecidable property of rewrite systems, even for systems that consist of a single rewrite rule, no method will work in ... |

89 | Termination of term rewriting: interpretation and type elimination
- Zantema
- 1994
(Show Context)
Citation Context ... of the latter system is easier to prove and implies termination of the former system. Examples include the transformation order of Bellegarde and Lescanne [6], and Zantema’s distribution elimination =-=[27]-=- and semantic labelling [28]. Transformations differ in their degree of automation. Since termination is an undecidable property of rewrite systems, even for systems that consist of a single rewrite r... |

54 |
Decidable approximations of term rewriting systems
- Jacquemard
- 1996
(Show Context)
Citation Context ...pproximation; the tree automaton that recognizes (→∗ )[L] is defined as the limit of Rg a finite saturation process. This saturation process is similar to the ones defined in Comon [7] and Jacquemard =-=[16]-=-, but by working exclusively with deterministic tree automata, non-right-linear rewrite rules can be handled. For the strong and nv approximation simpler constructions using ground tree transducers ar... |

50 |
Two generalizations of the recursive path orderings, Unpublished note
- Kamin, Levy
- 1980
(Show Context)
Citation Context ...eory but their implementations rely on heuristics that greatly reduce this power. Well-known examples are the polynomial interpretations of Lankford [22] and the semantic path order of Kamin and Lévy =-=[17]-=-. 3. Transformation methods which do not attempt to prove termination directly but rather transform the given rewrite system into another rewrite system such that termination of the latter system is e... |

42 | Argument filtering transformation
- Kusakari, Nakamura, et al.
- 1999
(Show Context)
Citation Context ...ion of TRSs. Below we recall the basic notions and results of the dependency pair technique of Arts and Giesl. We refer to [2, 4, 12] for motivations and further refinements. We adopt the notation of =-=[13, 20]-=-. Let R be a TRS over a signature F. As usual, root symbols of left-hand sides of rewrite rules are called defined. Let F ♯ denote the union of F and {f ♯ | f is a defined symbol of R} where f ♯ has t... |

29 |
Decidability for left-linear growing term rewriting systems
- Nagaya, Toyama
(Show Context)
Citation Context ...loser to the original rule. The ambiguity in the definition of Rg causes no problems in the sequel.) Theorem 9. The approximation mappings s, nv, and g are regularity preserving. ⊓⊔ Nagaya and Toyama =-=[24]-=- proved the above result for the growing approximation; the tree automaton that recognizes (→∗ )[L] is defined as the limit of Rg a finite saturation process. This saturation process is similar to the... |

25 |
Termination by completion”, Applicable Algebra
- Bellegarde, Lescanne, et al.
- 1990
(Show Context)
Citation Context ...another rewrite system such that termination of the latter system is easier to prove and implies termination of the former system. Examples include the transformation order of Bellegarde and Lescanne =-=[6]-=-, and Zantema’s distribution elimination [27] and semantic labelling [28]. Transformations differ in their degree of automation. Since termination is an undecidable property of rewrite systems, even f... |

22 |
Right-linearfinite path overlapping term rewriting systems effectively preserve recognizability
- Takai, Kaji, et al.
- 2000
(Show Context)
Citation Context ...linear rewrite rules can be handled. For the strong and nv approximation simpler constructions using ground tree transducers are possible (see e.g. Durand and Middeldorp [11]). Recently, Takai et al. =-=[25]-=- introduced the class of left-linear inverse finite path overlapping rewrite systems and showed that the preceding theorem is true for the corresponding approximation mapping. Growing rewrite systems ... |

19 | System description: The dependency pair method
- Arts
- 2000
(Show Context)
Citation Context ... of Arts and Giesl. Consequently, we do not propose to eliminate the estimated dependency graph. Rather, our approximations should be tried only if tools based on the estimated dependency graph (like =-=[1]-=-) fail to prove termination or maybe in parallel to the search for suitable argument filterings and orderings to satisfy the resulting constraints. Clearly experimentation is needed to determine when ... |

18 | Modularity of termination using dependency pairs
- Giesl
- 1998
(Show Context)
Citation Context ...make clear that such eTRSs are very useful for automatically proving termination of TRSs. Below we recall the basic notions and results of the dependency pair technique of Arts and Giesl. We refer to =-=[2, 4, 12]-=- for motivations and further refinements. We adopt the notation of [13, 20]. Let R be a TRS over a signature F. As usual, root symbols of left-hand sides of rewrite rules are called defined. Let F ♯ d... |

14 | Termination of associative-commutative rewriting by dependency pairs
- MarchØ, Urbain
- 1998
(Show Context)
Citation Context ...o determine when to invoke our approximations. Currently we are working on an implementation of our algorithms. It is worthwhile to investigate whether our approach can be extended to AC termination (=-=[21, 23]-=-) and to innermost termination ([4]). For AC termination we do not expect any problems, but innermost termination seems more difficult. The reason is that the existence of an arrow from s → t to u → v... |

13 |
Computations in orthogonal rewriting systems, I and II
- Huet, Levy
- 1991
(Show Context)
Citation Context ...he approximation of the dependency graph defined by Kusakari and Toyama [19, 21]. Their approximation relies on the concepts of ω-reduction and Ω-reduction. The first concept stems from Huet and Lévy =-=[15]-=-. Let R be a TRS over a signature F. Let Ω be a fresh constant. The set of ground terms over the extended signature F ∪ {Ω} is denoted by TΩ(F). Given a term t ∈ T (F, V), the term in TΩ(F) obtained f... |

12 |
monadic second-order logic and tree automata
- Sequentiality
- 2000
(Show Context)
Citation Context ...t for the growing approximation; the tree automaton that recognizes (→∗ )[L] is defined as the limit of Rg a finite saturation process. This saturation process is similar to the ones defined in Comon =-=[7]-=- and Jacquemard [16], but by working exclusively with deterministic tree automata, non-right-linear rewrite rules can be handled. For the strong and nv approximation simpler constructions using ground... |

12 | Termination of linear rewriting systems (preliminary version
- Dershowitz
- 1981
(Show Context)
Citation Context ... whether these decidability results can be obtained with the dependency pair technique. The best known class of TRSs with a decidable termination problem is the class of right-ground TRSs (Dershowitz =-=[9]-=-). The following easy result states that in principle the dependency pair technique is very suitable for deciding termination of right-ground TRSs. Theorem 28. A right-ground TRS R is terminating if a... |

12 |
On Proving AC-Termination by AC-Dependency Pairs
- Kusakari
(Show Context)
Citation Context ...efine new approximations of the dependency graph. We compare our approximations with the one of Arts and Giesl in Section 5. We also include a comparison with the approximation of Kusakari and Toyama =-=[19, 21]-=-. In Section 6 we show that by using oursapproximations the dependency pair method provides a decision procedure for termination of right-ground rewrite systems. 2 Dependency Pairs We assume familiari... |

11 | Pushing the Frontiers of Combining Rewrite Systems Farther Outwards
- Giesl
- 2000
(Show Context)
Citation Context ... often better. Interestingly, we can automatically prove termination of rewrite systems outside the class of so-called DP quasi-simply terminating systems. This class, proposed by Giesl and Ohlebusch =-=[14]-=-, consists of all rewrite systems “where an automated termination proof using dependency pairs is potentially feasible”. The remainder of the paper is organized as follows. In the next section we brie... |

6 |
Veri of Erlang Processes by Dependency Pairs, Applicable Algebra
- Giesl
(Show Context)
Citation Context ...make clear that such eTRSs are very useful for automatically proving termination of TRSs. Below we recall the basic notions and results of the dependency pair technique of Arts and Giesl. We refer to =-=[2, 4, 12]-=- for motivations and further refinements. We adopt the notation of [13, 20]. Let R be a TRS over a signature F. As usual, root symbols of left-hand sides of rewrite rules are called defined. Let F ♯ d... |

5 | Eliminating Dummy Elimination
- Giesl
- 2000
(Show Context)
Citation Context ...ion of TRSs. Below we recall the basic notions and results of the dependency pair technique of Arts and Giesl. We refer to [2, 4, 12] for motivations and further refinements. We adopt the notation of =-=[13, 20]-=-. Let R be a TRS over a signature F. As usual, root symbols of left-hand sides of rewrite rules are called defined. Let F ♯ denote the union of F and {f ♯ | f is a defined symbol of R} where f ♯ has t... |

5 | AC – Termination and Dependency Pairs of Term Rewriting Systems
- Kusakari, Termination
- 2000
(Show Context)
Citation Context ...efine new approximations of the dependency graph. We compare our approximations with the one of Arts and Giesl in Section 5. We also include a comparison with the approximation of Kusakari and Toyama =-=[19, 21]-=-. In Section 6 we show that by using oursapproximations the dependency pair method provides a decision procedure for termination of right-ground rewrite systems. 2 Dependency Pairs We assume familiari... |

5 |
Tree automata and term rewrite systems
- Tison
- 2000
(Show Context)
Citation Context ... will turn out to be very important for automatically proving termination of TRSs that rely on non-linearity (i.e., by linearizing the rewrite rules the TRS becomes non-terminating). Theorem 6 (Tison =-=[26]-=-). The following problem is decidable: instance: tree automaton A, term t question: Σ(t) ∩ L(A) = ∅? Proof. First we transform A into an equivalent deterministic tree automaton B = (F, Q, Qf , ∆) with... |

3 | Applying rewriting techniques to the verification of erlang processes
- Arts, Giesl
- 1999
(Show Context)
Citation Context ...an arbitrary term is accepted by a given tree automaton. For a linear term t this is obvious since (1) Σ(t) is regular by Lemma 5, (2) regular languages are effectively closed under intersection, and =-=(3)-=- emptiness is decidable for regular languages. The point is that the problem remains decidable for non-linear terms. This extension will turn out to be very important for automatically proving termina... |

3 |
Decidable Call by Need Computations in Term Rewriting
- Durand, Middeldorp
- 1997
(Show Context)
Citation Context ...rform the expensive determinization of A. 4 Approximations In this section we define new approximations of the dependency graph. Our approximations are based on the framework of Durand and Middeldorp =-=[11]-=- for the study of decidable call-by-need computations in orthogonal term rewriting.sIf R is an eTRS over a signature F and L ⊆ T (F) then (→∗ R the set of all terms s ∈ T (F) such that s →∗ R )[L] den... |