## Discrete Normalization and Standardization in Deterministic Residual Structures (1996)

Venue: | In ALP '96 [ALP96 |

Citations: | 10 - 3 self |

### BibTeX

@INPROCEEDINGS{Khasidashvili96discretenormalization,

author = {Zurab Khasidashvili and John Glauert},

title = {Discrete Normalization and Standardization in Deterministic Residual Structures},

booktitle = {In ALP '96 [ALP96},

year = {1996},

pages = {135--149}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We prove a version of the Standardization Theorem and the Discrete Normalization (DN) Theorem in stable Deterministic Residual Structures, which are Abstract Reduction Systems with axiomatized residual relation, and model orthogonal rewrite systems. The latter theorem gives a strategy for construction of reductions L'evy-equivalent (or permutation-equivalent) to a given, finite or infinite, regular (or semi-linear) reduction, based on the neededness concept of Huet and L'evy. This and other results of this paper add to the understanding of L'evy-equivalence of reductions, and consequently, L'evy's reduction space. We demonstrate how elements of this space can be used to give denotational semantics to known functional languages in an abstract manner. 1 Introduction Long ago, Curry and Feys [CuFe58] proved that repeated contraction of leftmostoutermost redexes in any normalizable -term eventually yields its normal form, even if the term possesses infinite reductions as well. The reaso...

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Citation Context ...us) [Kha92]. Sekar and Ramakrishnan [SeRa93] study a normalizing strategy which in each multi-step contracts a necessary set of redexes. A different approach to normalization is developed in Kennaway =-=[Ken89]-=- and in Antoy and Middeldorp [AnMi94]. Antoy et al. [AEH94] designed a needed narrowing strategy. Gardner [Gar94] described a complete way of encoding neededness information using a type assignment sy... |

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Citation Context ...normal forms as well as w.r.t. head-normal forms, in the -calculus, proving correctness of the two needed strategies for computing normal forms and head-normal forms, respectively. Kennaway and Sleep =-=[KeSl89]-=- generalized the needed strategy to orthogonal Combinatory Reduction systems (CRSs) of Klop [Klo80]. Maranget [Mar92] studied a strategy that computes a weak head-normal form of a term in an OTRS. Nor... |

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Citation Context ...cessary set of redexes. A different approach to normalization is developed in Kennaway [Ken89] and in Antoy and Middeldorp [AnMi94]. Antoy et al. [AEH94] designed a needed narrowing strategy. Gardner =-=[Gar94]-=- described a complete way of encoding neededness information using a type assignment system, in the -calculus. Kennaway et al. [KKSV95] studied needed strategy for infinitary OTRSs. In [GlKh94], the p... |

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Citation Context ...or computing normal forms and head-normal forms, respectively. Kennaway and Sleep [KeSl89] generalized the needed strategy to orthogonal Combinatory Reduction systems (CRSs) of Klop [Klo80]. Maranget =-=[Mar92]-=- studied a strategy that computes a weak head-normal form of a term in an OTRS. Normalization w.r.t. another interesting set of `normal forms', that of constructor head-normal forms in constructor OTR... |

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Citation Context ...ntoy et al. [AEH94] designed a needed narrowing strategy. Gardner [Gar94] described a complete way of encoding neededness information using a type assignment system, in the -calculus. Kennaway et al. =-=[KKSV95]-=- studied needed strategy for infinitary OTRSs. In [GlKh94], the present authors found natural conditions on a set S of terms, in an OERS, called stability, that are necessary and sufficient for the fo... |

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Citation Context ...Nivat [Niv75], and was later extended to the -calculus by L'evy [L'ev76] (the so called L'evy-Longo trees or Lazy-trees [Lon83, AbOn93]) and Barendregt [Bar84] (the Bohm trees); and to TRSs by Boudol =-=[Bou85]-=-. In Nivat's and Boudol's semantics, undefined are terms which cannot be reduced to a root stable form -- a term that cannot be reduced to a redex. In Bohm semantics, -terms not having a head-normal f... |

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Citation Context ...Gardner [Gar94] described a complete way of encoding neededness information using a type assignment system, in the -calculus. Kennaway et al. [KKSV95] studied needed strategy for infinitary OTRSs. In =-=[GlKh94]-=-, the present authors found natural conditions on a set S of terms, in an OERS, called stability, that are necessary and sufficient for the following Relative Normalization (RN for short) theorem to h... |

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Citation Context ...ormal forms', that of constructor head-normal forms in constructor OTRSs, is studied by Nocker [Nok94]. Khasidashvili defined a similar normalizing strategy, the essential strategy, for the -calculus =-=[Kha88]-=-, OTRSs [Kha93], and Orthogonal Expression Reduction Systems (OERS) [Kha94], a form of higher-order rewriting similar to Klop's CRSs ? This work was supported by the Engineering and Physical Sciences ... |

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Citation Context ... work was supported by the Engineering and Physical Sciences Research Council of Great Britain under grant GR/H 41300 (which subsumes Term Rewriting and the -calculus) [Kha92]. Sekar and Ramakrishnan =-=[SeRa93]-=- study a normalizing strategy which in each multi-step contracts a necessary set of redexes. A different approach to normalization is developed in Kennaway [Ken89] and in Antoy and Middeldorp [AnMi94]... |

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Citation Context ...to which s is reducible) is stable and regular, then the task can be reduced to construction of a minimal , or rather a least (w.r.t. L'evy's ordering \Theta on reductions) G s -normalizing reduction =-=[GlKh94a]-=-, since it must end at s. However, as we already know [GlKh94], G s need not be stable, mainly because of syntactic accidents [L'ev80]. For example, the graph fI(x); xg of s = I(x), in the TRS with th... |

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Citation Context ...meaningless to be -terms that do not have neither hnfs nor weak hnfs (which are simply abstractions xt). Reasonable concepts of meaningless terms in (infinitary) TRSs and the -calculus are studied in =-=[KOV96]-=-. Lisper's definition of abstract semantics allows one to describe all the above tree semantics, but is not suitable for all known interpretations, and we generalize it slightly by taking f to be a fu... |

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Citation Context ...shortest in hP 0 i L . Then jQ=ujsjQj \Gamma 1 since exactly one residual of u is contracted in Q, and jSE(P )jsjQj follows from the induction assumption jP 0 jsjQ=uj. 6 Denotational Semantics Lisper =-=[Lis96]-=- defined semantics for an ARS A = (T er; Red; !) 4 as follows: let C = (C; v; ?) be a cpo and let f : T er ! C be a monotone function, meaning that t ! s implies f(t) v f(s). Then semantics of a term ... |

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Citation Context ... used in Section 5 to prove the Discrete Normalization Theorem, the main result of this paper. Section 6 is devoted to semantics of DRSs. Conclusions appear in Section 7. More details can be found in =-=[KhGl96]-=-. 2 Deterministic Residual Structures In this section we recall Deterministic Residual Structures (DRSs), introduced in [GlKh96] but largely based on [L'ev80]. DRSs are Abstract Reduction Systems (ARS... |

3 |
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Citation Context ...essential. In fact, the notion of self-essential reduction is the best approximation to the notion of standard reduction for DRSs, and it captures the essence of standardization in many respects (see =-=[KhGl96a]-=- for the use of self-essential standard reductions in an abstract optimality theory). Therefore: 3 Here we also use the fact that Q \Theta P , so that every finite initial part of Q can be considered ... |

2 |
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Citation Context ...e concatenation of P and Q. PsQ denotes that P is an initial part of Q. We use U; V; W to denote sets of redexes of a term. DRSs descend from Stark's Determinate Concurrent Transition Systems (DCTSs) =-=[Sta89]-=- and Abstract Reduction Systems of Gonthier et al. [GLM92]. Unlike DCTSs, the residual relation in DRSs may be duplicating, and unlike ARSs of [GLM92], we do not have a nesting relation on redexes. Se... |