## Disunification: a Survey (1991)

Venue: | Computational Logic: Essays in Honor of Alan |

Citations: | 56 - 9 self |

### BibTeX

@INPROCEEDINGS{Comon91disunification:a,

author = {Hubert Comon},

title = {Disunification: a Survey},

booktitle = {Computational Logic: Essays in Honor of Alan},

year = {1991},

pages = {322--359},

publisher = {MIT Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

Solving an equation in an algebra of terms is known as unification. Solving more complex formulas combining equations and involving in particular negation is called disunification. With such a broad definition, many works fall into the scope of disunification. The goal of this paper is to survey these works and bring them together in a same framework. R'esum'e On appelle habituellement (algorithme d') unification un algorithme de r'esolution d'une 'equation dans une alg`ebre de termes. La r'esolution de formules plus complexes, comportant en particulier des n'egations, est appel'ee ici disunification. Avec une d'efinition aussi 'etendue, de nombreux travaux peuvent etre consid'er'es comme portant sur la disunification. L'objet de cet article de synth`ese est de rassembler tous ces travaux dans un meme formalisme. Laboratoire de Recherche en Informatique, Bat. 490, Universit'e de Paris-Sud, 91405 ORSAY cedex, France. E-mail: comon@lri.lri.fr i Contents 1 Syntax 5 1.1 Basic Defini...

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137 |
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114 |
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104 | Equational problems and disunification
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101 |
Equations and inequations on finite and infinite trees
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81 | Equational Logic as a Programming Language - O’Donnell - 1985 |

79 |
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- 1981
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Citation Context ... in this volume). However, E-unification is decidable for some special theories such as AC-theories where E consists of a set of commutativity and associativity axioms for some functions symbols in F =-=[Sti81]-=- (see also the survey on unification in this volume). Since such cases are of practical interest, the problem arises to study equational problems in the algebra T (F )= =E (or T (F; X)==E ). Indeed, f... |

59 |
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51 | Solving symbolic ordering constraints - Comon - 1990 |

50 | Order-sorted equational computation - Smolka, Nutt, et al. - 1989 |

45 |
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Citation Context ...e symbol is equality (let's call it an equational formula) in an algebra A shows the decidability of the theory of A and provides a complete axiomatization of A. The work of logicians such as Mal'cev =-=[Mal71]-=- thus falls into the scope of disunification. There are also completely different approaches: solving certain kinds of equational formulas comes up when compiling functional programs, when dealing wit... |

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31 |
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Citation Context ...Mah88a, Com88b] [Mah88a] [Mah88a] [Com89, Com88b] Equational theory free quasi-free compact AC with decidable unification Equational Formulas: F finite [CL89, Mah88a] [Com88b] undecidable undecidable =-=[Tre90]-=- undecidable F infinite [Mah88a, Kun87b] [Com88b] \Delta \Delta undecidable Existential formulas: F finite \Delta \Delta [Com88b] [Com88b] undecidable [Fer91] F infinite [Col84] \Delta \Delta \Delta [... |

27 |
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Citation Context ...pleteness results when A is an arbitrary quotient of T (F ). This would indeed imply in particular the decidability of unification modulo any set of equational axioms which is known to be undecidable =-=[Dav77]-=-. It is not even possible to find a semi-decision procedure for the satisfiability of equational formulas in any quotient of T (F ) since the first order theory of T (F )= =E is complete. Hence, such ... |

25 |
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Citation Context ... / ODD(x) As shown, for example, in [BMPT90, Com88b] this automatic computation relies on the complement of the left hand sides defining EVEN. Another use of complement problems is developed by Kunen =-=[Kun87a]-=-. He uses them to represent infinite sets of answers to negative queries when dealing with negation as failure. For these reasons, we want to be able to compute a suitable representation of all the gr... |

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22 |
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21 |
A constraint sequent calculus
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Citation Context ...n some further papers. Among them, let me cite [LMM86], [Com88b] where the lemma is extended to the case of some equational theories, [LM89] where the lemma is extended to systems of inequalities and =-=[LM90]-=- where this lemma is set up in a general framework. The independence of disequations lemma plays a central role not only for showing the existence of a solution but also in the transformation rules of... |

19 |
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Citation Context ...ations and a simple operational semantics. "Logic program" should be understood here in a broad sense: a logic program may be a (constrained) set of clauses [JL87] or a (constrained) set of =-=equations [KK89]-=-, or any other set of formulas in some logic. Enhancement of expressiveness is obvious, but some particular applications are shown in [KK89]. For example, it is possible to deal with AC function symbo... |

16 |
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Citation Context ...iable in U is mapped to an element of A. 2.3 Examples 2.3.1 A = T (F ) or A = T (F; X) This is the interpretation used for complement problems (for example [Sch87, Sch88b], [Thi84], [BMPT90], [LM87], =-=[Com86]-=- and others). This interpretation is also considered in e.g [Ven87], [CL89], [Mah88a], [Bur88], [Kun87b, Kun87a, Nic87]. It appears that the main point for such interpretations is to know whether F is... |

16 | Inductive proofs by specification transformation - Comon - 1989 |

15 |
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- 1990
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13 |
Unification et disunification. Théories et applications. Thèse de Doctorat d’Université, Institut Polytechnique de Grenoble
- Comon
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(Show Context)
Citation Context ... [[t]] (i.e. without negation). In other words, the class of finite unions [[t 1 ]] [ : : : [ [[t n ]] is not closed under complement. That is the reason why constrained terms have been introduced in =-=[Com88b]-=-. A constrained term is a pair t & d, where t is a term and d is a conjunction of non-trivial disequations whose variables are those of t. Then [[t &d]] is the set of all ground instances of t which s... |

13 | Extending Equation Solving and Constraint Handling in Logic Programming - Dincbas, Simonis, et al. - 1987 |

11 |
Solving disequations in equational theories
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- 1988
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11 |
Solving inequations in term algebras
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- 1990
(Show Context)
Citation Context ... sentence is undecidable. By contrast, he shows that the existential fragment is decidable. Interpreting ? as a recursive path ordering leads to similar results: the existential fragment is decidable =-=[Com90c]-=- and the first order theory of embedding 10 is undecidable [Tre90]. It is still open whether the first order theory of a recursive path ordering extending a total precedence on F is decidable (althoug... |

11 |
Refined compilation of pattern-matching for functional languages
- Schnoebelen
- 1988
(Show Context)
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10 | Complete Axiomatizations of Some Quotient Term Algebras, Theoretical Computer Science 118
- Comon
- 1993
(Show Context)
Citation Context ... axioms fulfill this condition but not associativity. Also, equational theories which are defined by ground equations do have a conservative extension which is generated by a quasi-free set of axioms =-=[Com91a]. There ar-=-e however "few" equational theories satisfying these requirements. Theorem 8 EST is a complete axiomatization of T (F )= =E when F is infinite and E is quasi-free. EST [ fDCAg is a complete ... |

9 |
Decidability of the existential theory of infinite terms with subterm relation
- Tulipani
- 1994
(Show Context)
Citation Context ...his result has been extended to other ordinal notations (in particular, recursive path orderings) [JO91]. Venkataraman's results on the theory of subterm ordering have been extended to rational trees =-=[Tul91]. Th-=-e set of transformation rules we gave in this survey for disunification in finite trees (with slight modifications) has been shown confluent [Del90], showing that �� T (F ) can be decided by rewri... |

8 |
A general complete E-unification procedure
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- 1987
(Show Context)
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7 | On solving equations and disequations - Buntine, Bürckert - 1994 |

7 | Order-sorted algebra I: Partial and overloaded operators, errors and inheritance - Goguen, Meseguer - 1987 |

7 | Lazy pattern matching in the ML language - Laville - 1987 |

6 | An effective method for handling initial algebras - Comon - 1988 |

6 |
Independence of negative constraints
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- 1989
(Show Context)
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6 |
A Deduction Procedure for First Order Programs
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- 1989
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5 |
Solving disequations
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- 1987
(Show Context)
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4 |
préordres sur les arbres associés à des ensembles inévitables et preuves de terminaison de systèmes de réécriture. Thèse d’Etat
- Puel
- 1987
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Citation Context .... Simplifications of such complement problems as well as properties of the corresponding constrained terms are studied in [PS90]. Classical complement problems are also studied and used by L. Puel in =-=[Pue87]-=-. Here, the purpose is to decompose terms into patterns of a given (unavoidable) set in order to compare them using a pattern path ordering. 1.6 Related Work Using a Similar Formalism Solving equation... |