## Combination Techniques for Non-Disjoint Equational Theories (1994)

Venue: | Proceedings 12th International Conference on Automated Deduction |

Citations: | 25 - 5 self |

### BibTeX

@INPROCEEDINGS{Domenjoud94combinationtechniques,

author = {Eric Domenjoud and Francis Klay and Christophe Ringeissen},

title = {Combination Techniques for Non-Disjoint Equational Theories},

booktitle = {Proceedings 12th International Conference on Automated Deduction},

year = {1994},

pages = {267--281},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

ion variables which are variables coming from an abstraction, either during preprocessing or during the algorithm itself. 3. Introduced variables which are variables introduced by the unification algorithms for each theory. We make the very natural assumption that the unification algorithm for each theory may recognize initial, abstraction and introduced variables and never assigns an introduced variable to a non-introduced one or an abstraction variable to an initial one. With this assumption, our combination algorithm will always make an introduced variable appear in at most one \Gamma i . We may thus also suppose that the domain of each solution does not contain an introduced variable. This does not compromise the soundness of our algorithm. The combination algorithm is described by the two rules given in figure 2. In the rule UnifSolve i , ae SF is obtained by abstracting aliens in the range of ae by fresh variables. ae F i is the substitution such that xae = xae SF ae F i for al...

### Citations

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Unification in the union of disjoint equational theories: Combining decision procedures
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(Show Context)
Citation Context ...which case, we speak of disjoint theories. Many authors studied the problem of combining unification algorithms for disjoint theories [6, 14, 13] and the best result is due to F. Baader and K. Schulz =-=[1]-=- who described a general technique for combining decision procedures for unifiability and, by the way, solved the problem of combining unification algorithms for non-finitary theories. This result was... |

80 |
Recursive unsolvability of a problem of Thue
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Citation Context ...lphabets, and ' and ' 0 be two morphisms from A to C . The Post correspondence problem for ' and ' 0 consists in finding a non-empty sequence ff 2 A + such that '(ff) = ' 0 (ff). Theorem23 (Post 1947 =-=[9]-=-). There exists no uniform algorithm for solving the Post correspondence problem. The problem remains undecidable when ' and ' 0 are injective. In the following, we shall consider families of theories... |

49 |
Complete sets of transformations for general Eunification
- Gallier, Snyder
- 1989
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Citation Context ...1 ; : : : ; t n ). Remarks: 1. Since matching modulo an arbitrary theory E is semi-decidable, t i 's may be effectively computed as soon as we know that they exist. Some general unification procedure =-=[5, 4]-=- may be used for this purpose. 2. the symbol matching problem reduces to a word problem if h is a constant. 3. If the symbol h does not occur at the top in E, then the symbol matching problem on h is ... |

48 |
Modularity of simple termination of term rewriting systems
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- 1990
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Citation Context ... we are able to extend such an ordering to a simplification ordering on T (F 0 ) such that any congruence class of (F 0 ; A) still contains a minimal element, then using a result of Kurihara & Ohuchi =-=[7]-=- we get a simplification ordering for the union of theories such that constructors are preserved. At last, we could imagine to share non-free constructors that might for instance be defined as constru... |

28 |
M'ethodes et outils de conception syst'ematique d'algorithmes d'unification dans les th'eories 'equationnelles. Th`ese de doctorat d"etat en informatique, Universit'e de Nancy 1
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Citation Context ... to the case where the signatures of E 1 and E 2 are disjoint, in which case, we speak of disjoint theories. Many authors studied the problem of combining unification algorithms for disjoint theories =-=[6, 14, 13]-=- and the best result is due to F. Baader and K. Schulz [1] who described a general technique for combining decision procedures for unifiability and, by the way, solved the problem of combining unifica... |

27 |
Combining matching algorithms: The regular case
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- 1991
(Show Context)
Citation Context ...last result was, up to now, the only modularity result for non-disjoint equational theories. Some authors also considered the problem of combining matching algorithms for disjoint equational theories =-=[8, 12]-=- which is again more complicated than just putting together two matching algorithms. We consider here the combination of non-disjoint theories, and give sufficient conditions under which algorithms fo... |

21 | Combination techniques and decision problems for disunification
- Baader, Schulz
- 1993
(Show Context)
Citation Context ... for unifiability and, by the way, solved the problem of combining unification algorithms for non-finitary theories. This result was extended in two different ways: F. Baader and K. Schulz themselves =-=[2]-=- extended it to take into account some restricted forms of disunification problems, and Ch. Ringeissen [10] described an extension where theories may share constants. This last result was, up to now, ... |

16 | Unification in a combination of equational theories with shared constants and its application to primal algebras
- Ringeissen
- 1992
(Show Context)
Citation Context ... theories. This result was extended in two different ways: F. Baader and K. Schulz themselves [2] extended it to take into account some restricted forms of disunification problems, and Ch. Ringeissen =-=[10]-=- described an extension where theories may share constants. This last result was, up to now, the only modularity result for non-disjoint equational theories. Some authors also considered the problem o... |

13 |
Combination of unification algorithms
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Citation Context ... to the case where the signatures of E 1 and E 2 are disjoint, in which case, we speak of disjoint theories. Many authors studied the problem of combining unification algorithms for disjoint theories =-=[6, 14, 13]-=- and the best result is due to F. Baader and K. Schulz [1] who described a general technique for combining decision procedures for unifiability and, by the way, solved the problem of combining unifica... |

9 |
Unification in combinations of collapse-free regular theories
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(Show Context)
Citation Context ... to the case where the signatures of E 1 and E 2 are disjoint, in which case, we speak of disjoint theories. Many authors studied the problem of combining unification algorithms for disjoint theories =-=[6, 14, 13]-=- and the best result is due to F. Baader and K. Schulz [1] who described a general technique for combining decision procedures for unifiability and, by the way, solved the problem of combining unifica... |

8 | P.: An improved general E-unification method
- Dougherty, Johann
- 1990
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Citation Context ...1 ; : : : ; t n ). Remarks: 1. Since matching modulo an arbitrary theory E is semi-decidable, t i 's may be effectively computed as soon as we know that they exist. Some general unification procedure =-=[5, 4]-=- may be used for this purpose. 2. the symbol matching problem reduces to a word problem if h is a constant. 3. If the symbol h does not occur at the top in E, then the symbol matching problem on h is ... |

8 |
Combinaison de resolution de contraintes. These de doctorat, universite de Nancy
- Ringeissen
- 1993
(Show Context)
Citation Context ...theories sharing constructors, provided that only constants are shared at the top. Even this restriction may be somehow relaxed if finitely many contexts built from constructors are shared at the top =-=[11]-=-. In this case, theory conflicts that occur in the combination algorithm may be solved by introducing explicitely a shared term taken from a finite set. Another extension, which in practice would be v... |

6 |
Unification dans les M'elanges de Th'eories 'equationnelles. Th`ese de doctorat, Universit'e Paris-Sud
- Boudet
- 1990
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Citation Context ...l unification algorithm in each theory: namely an algorithm for unification with free function symbols. The formalism we take for designing our combination algorithm is mostly borrowed from A. Boudet =-=[3]-=- and F. Baader & K. Schulz [1] with some modifications due to the special nature of the problem we address. 4.1 Preprocessing and data structure The algorithm we describe is devoted to the transformat... |

5 | Combination of matching algorithms
- Ringeissen
- 1994
(Show Context)
Citation Context ...last result was, up to now, the only modularity result for non-disjoint equational theories. Some authors also considered the problem of combining matching algorithms for disjoint equational theories =-=[8, 12]-=- which is again more complicated than just putting together two matching algorithms. We consider here the combination of non-disjoint theories, and give sufficient conditions under which algorithms fo... |