## Normalization of Typable Terms by Superdevelopments (1999)

Venue: | Computer Science Logic'98, Springer LNCS 1584 |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Khasidashvili99normalizationof,

author = {Zurab Khasidashvili and Adolfo Piperno},

title = {Normalization of Typable Terms by Superdevelopments},

booktitle = {Computer Science Logic'98, Springer LNCS 1584},

year = {1999},

pages = {260--282},

publisher = {Springer}

}

### OpenURL

### Abstract

. We define a class of hyperbalanced l-terms by imposing syntactic constraints on the construction of l-terms, and show that such terms are strongly normalizing. Furthermore, we show that for any hyperbalanced term, the total number of superdevelopments needed to compute its normal form can be statically determined at the beginning of reduction. To obtain the latter result, we develop an algorithm that, in a hyperbalanced term M, statically detects all inessential (or unneeded)subterms which can be replaced by fresh variables without effecting the normal form of M; that is, full garbage collection can be performed before starting the reduction. Finally, we show that, modulo a restricted h-expansion, all simply typable l-terms are hyperbalanced, implying importance of the class of hyperbalanced terms. 1 Introduction The termination of b-reduction for typed terms is one of the most studied topics in l- calculus. After classical proofs of Tait [21] and Girard [8], many interesting proo...

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Citation Context ... more. Furthermore, we use our static garbage-collection algorithm also in our normalization theorem for hyperbalanced terms, to compute (exact) lengths of normalizing reductions by superdevelopments =-=[20, 14]-=-. That is, we show that the amount of superdevelopments needed to reduce a hyperbalanced term M to its normal form can be determined at the beginning of reduction just by analyzing the structure of M.... |

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Citation Context ...e normalization of terms typable in the intersection discipline, but, due to lack of space, this result cannot be presented here. More precisely, it has been applied to strict intersection types (see =-=[7, 3]-=-) using the assignment system described in [19], in which a notion of extensional derivation can be introduced. In such a scenario, in a way similar to Curry types, the number of superdevelopments nee... |

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Citation Context ...n non-creating OERSs (where no redexcreation is possible). It is relevant to remark here that, because redex-creation is restricted in persistent OERSs, weak and strong normalization become decidable =-=[12]-=-. Note that, whenever in an extensional derivation for a term M a syntactical componentsN of it has been assigned an arrow type of functionality k, then N must have exactly k initial abstractions. Usi... |

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Citation Context ... occurrences of x in L are the substituted occurrences of N. The descendants of other subterms of M are intuitively clear. This definition immediately extends to (many-step) b-reductions. We refer to =-=[13]-=- for a more precise definition of descendants for Expression Reduction Systems (ERSs) in general. In the next section, we will use descendants to develop our static garbage-collection algorithm for hy... |

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Citation Context ...e normalization of terms typable in the intersection discipline, but, due to lack of space, this result cannot be presented here. More precisely, it has been applied to strict intersection types (see =-=[7, 3]-=-) using the assignment system described in [19], in which a notion of extensional derivation can be introduced. In such a scenario, in a way similar to Curry types, the number of superdevelopments nee... |

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Citation Context ...dexes. (In Section 7 we will briefly remark on the relationship between substitution paths and other concepts of paths in the l-calculus [1].) By using the so called `decreasing redex labelling lemma =-=[16, 15]-=-, strong normalization of hyperbalanced terms can be deduced from the fact that any b-step decreases the lengths of paths relative to the contracted redex and does not increase the lengths of other pa... |

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Citation Context ...pe s such that A `! M:s. The set of terms typable in TA! will be denoted by LTA! . Moreover, D:A `! M:s denotes a derivation in TA! proving the typing A `! M:s. Definition 6.2 (EXTENSIONAL DERIVATION =-=[18, 19]-=-) A derivation D:A `! M:s is extensional if and only if every head-variable or application term appearing in D as the subject of a rule is not assigned an arrow type. We write A ` E ! M:s if there exi... |

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Citation Context ...pe s such that A `! M:s. The set of terms typable in TA! will be denoted by LTA! . Moreover, D:A `! M:s denotes a derivation in TA! proving the typing A `! M:s. Definition 6.2 (EXTENSIONAL DERIVATION =-=[18, 19]-=-) A derivation D:A `! M:s is extensional if and only if every head-variable or application term appearing in D as the subject of a rule is not assigned an arrow type. We write A ` E ! M:s if there exi... |

1 |
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Citation Context ... the contracted redex and does not increase the lengths of other paths. A strong normalization proof for simply typed l-calculus using a similar idea (but completely different technically) appears in =-=[2]-=-. Arrow-decoration of hyperbalanced terms is useful for other purposes too. For example, it allows us to develop a static garbage-collection algorithm; that is, we can detect all inessential [11] subt... |

1 |
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Citation Context ...rs in [2]. Arrow-decoration of hyperbalanced terms is useful for other purposes too. For example, it allows us to develop a static garbage-collection algorithm; that is, we can detect all inessential =-=[11]-=- subterms in a hyperbalanced term M without any transformation of M, and inessential subterms can be replaced by fresh variables without effecting the normal form of M. Once garbage collection is done... |

1 |
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Citation Context ... and it is worth investigating the precise relationship. In particular, we expect that every arrow of the form i : lx 1 : : :x m :s 7! x, in a hyperbalanced term M, determines exactly m redexfamilies =-=[17]-=-, where m is the weight of i. Thus the length of the normalizing leftmostoutermost complete-family reduction starting from M coincides with the sum of weights of arrows in M ffi augmented by the numbe... |