## E-Unification by Means of Tree Tuple Synchronized Grammars (1996)

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Citations: | 9 - 3 self |

### BibTeX

@MISC{Limet96e-unificationby,

author = {Sébastien Limet and Pierre Réty},

title = {E-Unification by Means of Tree Tuple Synchronized Grammars},

year = {1996}

}

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### Abstract

: The goal of this paper is both to give a E-unification procedure that always terminates, and to decide unifiability. For this, we assume that the equational theory is specified by a confluent and constructor-based rewrite system, and that four additional restrictions are satisfied. We give a procedure that represents the (possibly infinite) set of solutions thanks to a tree tuple synchronized grammar, and that can decide unifiability thanks to an emptiness test. Moreover we show that if only three of the four additional restrictions are satisfied then unifiability is undecidable. 1 Introduction First order E-unification [29] is a tool that plays an important role in automated deduction, in particular in functional logic programming and for solving symbolic constraints (see [4] for an extensive survey of the area). It consists in finding instances to variables that make two terms equal modulo an equational theory given by a set of equalities, i.e. it amounts to solve an equation (ca...

### Citations

779 | Rewrite Systems
- Dershowitz, Jouannaud
- 1990
(Show Context)
Citation Context ...ect. 9 concludes the paper, and discusses future extensions to this work. 2 Preliminaries We recall some basic notions that concern rewriting techniques. For more details see Dershowitz and Jouannaud =-=[14]. Let � -=-be a finite set of symbols and V be an infinite set of variables, T��V is the first-order term algebra over � and V . � is partitioned in two parts: the set F of function symbols, andthesetC o... |

296 | Uniform narrowing strategies
- Echahed
- 1992
(Show Context)
Citation Context ...ion is undecidable and may have infinitely many solutions. This is why E-unification procedures, like narrowing, often loop, enumerating an infinite set of unifiers or computing unproductive branches =-=[3, 4, 5, 6, 7, 8, 9, 10]-=-. From the programming point of view, it is very important to avoid infinite loops when possible. When solving equations in a computation (of a functional logic program, for instance), most of the tim... |

238 | Unification theory
- Baader, Snyder
- 2001
(Show Context)
Citation Context ...First order E-unification [1] is a tool that plays an important role in automated deduction, in particular in functional logic programming and for solving symbolic constraints (see Baader and Siekmann=-=[2]-=- for an extensive survey of the area). It consists of finding instances to variables that make two terms equal modulo to an equational theory given by a set of equalities, i.e. it amounts to solving a... |

169 |
Canonical forms and unification
- Hullot
- 1980
(Show Context)
Citation Context ...ion is undecidable and may have infinitely many solutions. This is why E-unification procedures, like narrowing, often loop, enumerating an infinite set of unifiers or computing unproductive branches =-=[3, 4, 5, 6, 7, 8, 9, 10]-=-. From the programming point of view, it is very important to avoid infinite loops when possible. When solving equations in a computation (of a functional logic program, for instance), most of the tim... |

154 |
Logic programming with functions and predicates: The language BABEL
- Moreno-Navarro, Rodr'iguez-Artalejo
- 1992
(Show Context)
Citation Context ...ion is undecidable and may have infinitely many solutions. This is why E-unification procedures, like narrowing, often loop, enumerating an infinite set of unifiers or computing unproductive branches =-=[3, 4, 5, 6, 7, 8, 9, 10]-=-. From the programming point of view, it is very important to avoid infinite loops when possible. When solving equations in a computation (of a functional logic program, for instance), most of the tim... |

114 |
Building-in equational theories
- Plotkin
- 1972
(Show Context)
Citation Context ...fiability thanks to an emptiness test. Moreover we show that if only three of the four additional restrictions are satisfied then unifiability is undecidable. 1 Introduction First order E-unification =-=[28]-=- is a tool that plays an important role in automated deduction, in particular in functional logic programming and for solving symbolic constraints (see [4] for an extensive survey of the area). It con... |

29 | Basic narrowing revisited - Nutt, R'ety, et al. - 1989 |

25 | Combination techniques for nondisjoint equational theories
- Domenjoud, Klay, et al.
(Show Context)
Citation Context ... of h��� � � h 0 ��� forbidding � � � and � � �. This is therefore impossible. 2 Lemma 4.4 Forbidding �in is necessary to decide unifiability. Proof. The following rewrit=-=e system comes from Domenjoud [24]. Let R2 � f f�ai�x��y� � �i�f�x� ai�y��� j i �1�����n f 0 �ai�x��y� � �-=-�� 0 i �f0 �x� ai�y��� j i �1�����n f���y� � h�y� f 0 ���y��h�y� g where �i and � 0 i respectively denote ��ai� and � 0 �ai�. The ... |

21 |
Basic Paramodulation and Decidable Theories
- Nieuwenhuis
- 1996
(Show Context)
Citation Context ...y a well founded ordering. Comon et al. [18] show that decidability also holds for shallow rewrite systems (the sides of rewrite rules have variables occurring at a depth of at most one). Nieuwenhuis =-=[19]-=- extends the shallow theories to standard theories that allow non-shallow variables.sE-unification by means of tree tuple synchronized grammars 73 The restriction of Kapur and Narendran [20], extended... |

16 |
Narrowing directed by a graph of terms
- Chabin, Rety
- 1991
(Show Context)
Citation Context |

14 |
Enumerating outer narrowing derivations for constructor-based term rewriting systems
- You
- 1989
(Show Context)
Citation Context |

12 |
Narrowing based procedures for equational disunification
- Fern'andez
- 1992
(Show Context)
Citation Context ... a decision procedure for unifiability. Moreover, we have shown that each restriction is needed to decide unifiability. In future, it would be nice to use TTSGs to deal with equational disunification =-=[25, 26]-=-, i.e. finding substitutions that are not the solution of a given equation. This may be achieved if it is possible to compute the set minus between two languages recognized by TTSGs. TTSG productions ... |

11 | Syntacticness, cycle-syntacticness, and shallow theories
- Comon, Haberstrau, et al.
- 1994
(Show Context)
Citation Context ...terion: every rewrite rule’s lhs is flat (f�s1�����sn�is flat if 8i 2 �1�n�, siis either a variable or a ground data term), and the rewrite rules are oriented by a well founded o=-=rdering. Comon et al. [18]-=- show that decidability also holds for shallow rewrite systems (the sides of rewrite rules have variables occurring at a depth of at most one). Nieuwenhuis [19] extends the shallow theories to standar... |

10 | Complete Axiomatizations of Some Quotient Term Algebras, Theoretical Computer Science 118
- Comon
- 1993
(Show Context)
Citation Context ... a decision procedure for unifiability. Moreover, we have shown that each restriction is needed to decide unifiability. In future, it would be nice to use TTSGs to deal with equational disunification =-=[25, 26]-=-, i.e. finding substitutions that are not the solution of a given equation. This may be achieved if it is possible to compute the set minus between two languages recognized by TTSGs. TTSG productions ... |

8 | Solving a Unification Problem under Constrained Substitutions Using Tree Automata
- Kaji, Fujiwara, et al.
- 1997
(Show Context)
Citation Context ...ction is flat in the sense of Christian [17]. Thanks to the notion of iterated substitution, Mitra is able to represent finitely the infinite set of unifiers and decide upon unifiability. Kaji et al. =-=[22]-=- give a procedure that, when it terminates, decides upon unifiability by means of tree automata. They assume linearity for the goal, right linearity and (nearly) left linearity for the TRS. Unfortunat... |

6 |
Narrowing strategies for arbitrary canonical systems
- Bockmayr, Krischer, et al.
- 1995
(Show Context)
Citation Context |

6 |
Some new decidability results on positive and negative set constraints
- Gilleron, Tison, et al.
- 1994
(Show Context)
Citation Context ...que that can express the symmetric binary trees, and whose emptiness is decidable. Some authors have already used tree languages to represent infinite sets of solutions. For example inGilleron et al. =-=[11]-=-, they are used to solve set constraints, but without synchronization. The notion of synchronization has already appeared in string grammars, for example, as in parallel communicating grammar systems ... |

5 |
Positive first order logic is NP-complete
- Kozen
- 1981
(Show Context)
Citation Context ...Results In term rewriting, some authors have already established decidability results for unifiability, assuming some restrictions on the TRS. The first result assumed the rewrite system to be ground =-=[15]. Hu-=-llot [3] extended it to rewrite systems whose rhs’s are either variables or ground terms (Mitra [16] allows rhs’s to be data terms). Actually, these results are very restrictive because they forbi... |

5 |
Matching, unification and complexity
- Kapur, Narendran
- 1987
(Show Context)
Citation Context ...euwenhuis [19] extends the shallow theories to standard theories that allow non-shallow variables.sE-unification by means of tree tuple synchronized grammars 73 The restriction of Kapur and Narendran =-=[20]-=-, extended by Mitra [16] imposes that for every rule, every subterm of the rhs having a function symbol on top is a strict subterm of the lhs. For all these restrictions the theory is finitary, i.e. t... |

4 |
Tree Grammars with Multilinear Interpretation
- Guan, Hotz, et al.
- 1992
(Show Context)
Citation Context ...et constraints, but without synchronization. The notion of synchronization has already appeared in string grammars, for example, as in parallel communicating grammar systems [12] and in tree grammars =-=[13]-=-. However the TTSGs are not identical to the coupled context-free grammars of Guan et al. [13] because we need a finer control of synchronizations which is achieved thanks to a tuple of integers. The ... |

4 |
Semantic Unification for Convergent Rewrite Systems
- Mitra
- 1994
(Show Context)
Citation Context ...assuming some restrictions on the TRS. The first result assumed the rewrite system to be ground [15]. Hullot [3] extended it to rewrite systems whose rhs’s are either variables or ground terms (Mitr=-=a [16] allows rhs’s to-=- be data terms). Actually, these results are very restrictive because they forbid recursivity. Christian [17] defines a new criterion: every rewrite rule’s lhs is flat (f�s1�����sn�is ... |

4 | Semantic Unification for Convergent Systems
- Mitra
- 1994
(Show Context)
Citation Context ... proofs are thus based on the fact that there exists a complete narrowing strategy whose search space is always finite. As regards non-finitary theories, a decidability result is established by Mitra =-=[16, 21]-=- for constructorbased rewrite systems, assuming that, for every function symbol f, there is at most one rewrite rule among the rules defining f that does not have a data term as the rhs. Moreover, thi... |

4 | Some Independence Results for Equational Unification - Otto, Narendran, et al. - 1995 |

3 | A strict border for the decidability of E-unification for recursive functions
- Faßbender, Maneth
(Show Context)
Citation Context ...position of s�x� with s�x � y�. Note that our algorithm works (and terminates) for this example when solving linear equations, since our restrictions are satisfied (see Sect. 4). Faßbender =-=and Maneth [23]-=- give a decision procedure for unifiability without representing the set of solutions. However, they need very strong restrictions. Only one function can be defined, and every constructor and every fu... |

3 |
M'ethodes de sch'ematisation pour la d'emonstration automatique
- Amaniss
- 1996
(Show Context)
Citation Context ...f (tuples of) ground terms. In automated deduction, several authors have studied recurrent schematizations of infinite sets of terms, like �-terms, I-terms, R-terms and primal grammars (see elsewher=-=e [27, 28, 29] for an overview). A languag-=-e like f n �a� � ����f n �a� � �z � p �n�p2IN can be defined both by a primal grammar and by a TTSG that does not come from a unification problem. On the other hand, the sy... |

2 |
Building-in equational theories. Machine Intelligence
- Plotkin
- 1972
(Show Context)
Citation Context ...nly three of the four additional restrictions are satisfied then unifiability is undecidable. Keywords: E-unification, narrowing, tree languages, decidability 1 Introduction First order E-unification =-=[1]-=- is a tool that plays an important role in automated deduction, in particular in functional logic programming and for solving symbolic constraints (see Baader and Siekmann[2] for an extensive survey o... |

2 | Regular Substitution Sets: a Means of Controlling E-Unification
- Burghardt
- 1995
(Show Context)
Citation Context |

2 | Pattern Languages versus Parallel Communicating Grammar Systems
- Dumitrescu, Paun, et al.
- 1996
(Show Context)
Citation Context ..., they are used to solve set constraints, but without synchronization. The notion of synchronization has already appeared in string grammars, for example, as in parallel communicating grammar systems =-=[12]-=- and in tree grammars [13]. However the TTSGs are not identical to the coupled context-free grammars of Guan et al. [13] because we need a finer control of synchronizations which is achieved thanks to... |

2 |
Some Termination Criteria for Narrowing and E-Unification
- Christian
- 1992
(Show Context)
Citation Context ...to rewrite systems whose rhs’s are either variables or ground terms (Mitra [16] allows rhs’s to be data terms). Actually, these results are very restrictive because they forbid recursivity. Christ=-=ian [17] defines a new criteri-=-on: every rewrite rule’s lhs is flat (f�s1�����sn�is flat if 8i 2 �1�n�, siis either a variable or a ground data term), and the rewrite rules are oriented by a well founded order... |

1 | Overview of Existing Recurrent Schematizations
- Hermann
- 1996
(Show Context)
Citation Context ...f (tuples of) ground terms. In automated deduction, several authors have studied recurrent schematizations of infinite sets of terms, like �-terms, I-terms, R-terms and primal grammars (see elsewher=-=e [27, 28, 29] for an overview). A languag-=-e like f n �a� � ����f n �a� � �z � p �n�p2IN can be defined both by a primal grammar and by a TTSG that does not come from a unification problem. On the other hand, the sy... |

1 | Classification, Extension and Applications of some Schematization Methods of Infinite Sets of Terms
- Amaniss
- 1996
(Show Context)
Citation Context ...f (tuples of) ground terms. In automated deduction, several authors have studied recurrent schematizations of infinite sets of terms, like �-terms, I-terms, R-terms and primal grammars (see elsewher=-=e [27, 28, 29] for an overview). A languag-=-e like f n �a� � ����f n �a� � �z � p �n�p2IN can be defined both by a primal grammar and by a TTSG that does not come from a unification problem. On the other hand, the sy... |

1 |
Some Termination Criteria for Narrowingand E-Unification
- Christian
- 1992
(Show Context)
Citation Context ...] to rewrite systems whose rhs's are either variables or ground terms (S. Mitra in [23] allows rhs's to be data-terms). Actually these results are very restrictive because they forbid recursivity. In =-=[8]-=-, J. Christian defines a new criterion : every rewrite rule lhs is flat (f(s 1 ; : : : ; s n ) is flat if 8i 2 [1; n], s i is either a variable or a ground data-term) and the rewrite rules are oriente... |