## Higher-Order Equational Unification via Explicit Substitutions (1996)

Venue: | in Proceedings of the tenth UNIF Workshop |

Citations: | 5 - 3 self |

### BibTeX

@INPROCEEDINGS{Kirchner96higher-orderequational,

author = {Claude Kirchner and Christophe Ringeissen},

title = {Higher-Order Equational Unification via Explicit Substitutions},

booktitle = {in Proceedings of the tenth UNIF Workshop},

year = {1996},

pages = {61--75},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We show how to reduce the unification problem modulo fij- conversion and a first-order equational theory E, into a first-order unification problem in a union of two non-disjoint equational theories including E and a calculus of explicit substitutions. A rule-based unification procedure in this combined theory is described and may be viewed as an extension of the one initially designed by G. Dowek, T. Hardin and C. Kirchner for performing unification of simply typed -terms in a first-order setting via the oe-calculus of explicit substitutions. Additional rules are used to deal with the interaction between E and oe. 1 Introduction Unification modulo an equational theory plays an important role in automated deduction and in logic programming systems. For example, Prolog[NM88] is based on higher-order unification, ie. unification modulo the fij-conversion. In order to design more expressive higher-order logic programming systems enhanced with a first-order equational theory E,...

### Citations

390 | Explicit substitutions
- Abadi, Cardelli, et al.
- 1991
(Show Context)
Citation Context ...ubstitution-calculus was originally introduced for describing the implementation of -calculi and led to the design of several calculi among which the oe-calculus based on a first-order rewrite system =-=[ACCL91]-=-. Recently, G. Dowek, T. Hardin and C. Kirchner [DHK95] have shown how to reduce the higher-order unification problem into a first-order unification problem modulo a first-order theory of explicit sub... |

177 |
An overview of >'Prolog
- Nadathur, Miller
- 1988
(Show Context)
Citation Context ... deal with the interaction between E and oe. 1 Introduction Unification modulo an equational theory plays an important role in automated deduction and in logic programming systems. For example, Prolog=-=[NM88]-=- is based on higher-order unification, ie. unification modulo the fij-conversion. In order to design more expressive higher-order logic programming systems enhanced with a first-order equational theor... |

162 |
Solving equations in abstract algebras: a rule-based survey of unification
- Jouannaud, Kirchner
- 1991
(Show Context)
Citation Context ... right-hand side of axioms). But this work could be generalized to arbitrary theories E at the cost of more complicated rules (and proofs). Following the rule-based approach developed for unification =-=[JK91]-=-, our unification procedure may be viewed as a set of transformation rules together with a given strategy. The interest of this unification procedure lies in the result that a higher-order Eunificatio... |

104 |
Unification in the union of disjoint equational theories: Combining decision procedures
- Baader, Schulz
- 1996
(Show Context)
Citation Context ...isjoint combination of first-order equational theories. Therefore, we cannot reuse the well-known techniques developed for combining unification algorithms in the union of signature-disjoint theories =-=[BS96]-=-, and those developed in [DKR94] for some specific non-disjoint unions of theories cannot be applied to this particular case. Thus, we are designing in this work a complete oeE-unification procedure s... |

102 | Higher-order unification via explicit substitutions
- Dowek, Hardin, et al.
(Show Context)
Citation Context ...ibing the implementation of -calculi and led to the design of several calculi among which the oe-calculus based on a first-order rewrite system [ACCL91]. Recently, G. Dowek, T. Hardin and C. Kirchner =-=[DHK95]-=- have shown how to reduce the higher-order unification problem into a first-order unification problem modulo a first-order theory of explicit substitutions. A natural extension is now to express highe... |

56 |
Combining algebra and higher-order types
- Breazu-Tannen
- 1988
(Show Context)
Citation Context ...der equational theory E, one should consider higher-order E-unification, ie. unification modulo = fij and =E . The problem of combining - calculi with first-order equational theories was initiated in =-=[BT88]-=- and the higherordersE-unification problem has been already successfully studied [QW96, NQ91, Sny90] by extending the techniques developed [Hue75] for unification of simply typed -terms. On the side o... |

41 |
A Unification Algorithm for Typed Lambda-Calculus
- Huet
- 1975
(Show Context)
Citation Context ... with first-order equational theories was initiated in [BT88] and the higherordersE-unification problem has been already successfully studied [QW96, NQ91, Sny90] by extending the techniques developed =-=[Hue75]-=- for unification of simply typed -terms. On the side of functional programming languages implementations, the operation of substitution (issued from fi-reduction) is expensive and explicit substitutio... |

25 | Combination techniques for non-disjoint equational theories - Domenjoud, Klay, et al. - 1994 |

22 | Contributions `a l"etude de -calculs avec des substitutions explicites - R'ios - 1993 |

9 | Modular Higher-order E-Unification - Nipkow, Qian - 1991 |

5 | Higher-order E-unification - Snyder - 1990 |

3 | Modular higher-order equational preunification - Qian, Wang - 1996 |

2 | Extensions of lambda-calculi with explicit substitutions preserving the church-rosser's property. Manuscript - Pagano |

1 | A left linear variant of oe. Manuscript - Munoz |