## Perpetuality and Uniform Normalization (1997)

Venue: | In Proc. of the 6 th International Conference on Algebraic and Logic Programming, ALP'97 |

Citations: | 4 - 2 self |

### BibTeX

@INPROCEEDINGS{Khasidashvili97perpetualityand,

author = {Zurab Khasidashvili and Mizuhito Ogawa},

title = {Perpetuality and Uniform Normalization},

booktitle = {In Proc. of the 6 th International Conference on Algebraic and Logic Programming, ALP'97},

year = {1997},

pages = {240--255},

publisher = {Submitted}

}

### OpenURL

### Abstract

. We define a perpetual one-step reduction strategy which enables one to construct minimal (w.r.t. L'evy's ordering \Theta on reductions) infinite reductions in Conditional Orthogonal Expression Reduction Systems. We use this strategy to derive two characterizations of perpetual redexes, i.e., redexes whose contractions retain the existence of infinite reductions. These characterizations generalize existing related criteria for perpetuality of redexes. We give a number of applications of our results, demonstrating their usefulness. In particular, we prove equivalence of weak and strong normalization (the uniform normalization property) for various restricted -calculi, which cannot be derived from previously known perpetuality criteria. 1 Introduction The objective of this paper is to study sufficient conditions for uniform normalization, UN, of a term in an orthogonal (first or higher-order) rewrite system, and for the UN property of the rewrite system itself. Here a term is UN if ei...

### Citations

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Citation Context ... A` = a. The descendant f(b) of the redexsf(a) after contraction of a is not a redex since the assignment A` = b is not admissible, hence the system is not orthogonal. As in the case of the -calculus =-=[Bar84]-=-, for any co-initial (i.e., with the same initial term) reductions P and Q, one can define in OCERSs the notion of residual of P under Q, written P=Q, due to L'evy [L'ev80]. We write P \Theta Q if P=Q... |

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Citation Context ...ms (ERSs) [Kha92], by proving that all non-erasing redexes are perpetual in orthogonal ERSs. For orthogonal Term Rewriting Systems (OTRSs), a very powerful perpetuality criterion was obtained by Klop =-=[Klo92]-=- in terms of critical redexes. These are redexes that are not perpetual, i.e., reduce 1-terms to strongly normalizable terms (SN-terms). Klop showed that any critical redex u must erase an argument po... |

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Citation Context ...llowing restrictions both on arguments of redexes and on the contexts in which the redexes can be contracted. Various interesting typed -calculi, including the simply typed -calculus and the system F =-=[Bar92]-=-, can directly be encoded as OCERSs (see also [KOR93]); --calculi with specific reduction strategies (such as the call-byvalues-calculus [Plo75]) can also be naturally encoded as OCERSs. ERSs are very... |

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Citation Context ...rms aresI -terms with the constant ffi K . The authors show that the -ffi k -calculus has the UN property. This result follows from Corollary 4.6 only if the j-rule is dropped. However, Klop shows in =-=[Klo80]-=- that j-redexes are perpetual, and we hope that our results can be generalized to weakly-orthogonal CERSs (and thus cover the j-rule since j-redexes are non-erasing) using van Oostrom and van Raamsdon... |

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Citation Context ...ying the UN property reduces to studying perpetuality of redexes. The latter has already been studied quite extensively in the literature. The classical results in this direction are Church's Theorem =-=[CR36]-=-, stating that thesI -calculus is uniformly normalizing, and the Conservation Theoremsof Barendregt et al [BBKV76, Bar84], stating that fi I -redexes are perpetual in the -calculus. Bergstra and Klop ... |

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Citation Context ...ey of the relationship between various formats of higher-order rewriting, such as [Klo80, Kha92, Wol93, Nip93, OR94]. Restricted rewriting systems with substitutions were first studied in [Pkh77] and =-=[Acz78]-=-. We refer to [Klo92] for a survey of results concerning conditional TRSs. Terms in CERSs are built from the alphabet like in the first order case. The symbols having binding power (likesin -calculus ... |

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Citation Context ... hence is a LV -normal form. This is not surprising, however, as LV -redexes are disjoint, 4 and there is no duplication or erasure of (admissible) redexes. 5.3 De Groote's fi IS -reduction De Groote =-=[dGr93]-=- introduced fi S -reduction on -terms by the following rule: fi S : (((x:M)N)O) ! ((x:(MO)N ), where x 62 FV (M; O). He proved that the fi IS - calculus is uniformly normalizing. Clearly, this is an i... |

38 |
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Citation Context ...tions. We claim that our results are also valid for Klop's orthogonal substructure CRSs [KOR93]. However, they cannot be generalized (at least, directly) to orthogonal Pattern Rewrite Systems (OPRSs) =-=[Nip93]-=-, as witnessed by the following example due to van Oostrom [Oos97]. It shows that already the Conservation Theorem fails for OPRSs (i.e., non-erasing steps need not be perpetual): Let R = fg(M:N:X(x:M... |

33 |
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Citation Context ...[Plo75]) can also be naturally encoded as OCERSs. ERSs are very close to the more familiar format of CRSs of Klop [Klo80], and we claim that all our results are valid for orthogonal CRSs as well (see =-=[Raa96]-=- for a detailed comparison of various forms of higher-order rewriting). However, using an example due to van Oostrom [Oos97], we will demonstrate that our results cannot be extended to higher-order re... |

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Citation Context ...ained two criteria for perpetuality of redexes in orthogonal CERSs, and demonstrated their usefulness in applications. We claim that our results are also valid for Klop's orthogonal substructure CRSs =-=[KOR93]-=-. However, they cannot be generalized (at least, directly) to orthogonal Pattern Rewrite Systems (OPRSs) [Nip93], as witnessed by the following example due to van Oostrom [Oos97]. It shows that alread... |

31 |
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Citation Context ... to prove our perpetuality criteria, we first generalize the constrictings(or zoom-in) perpetual strategy, independently discovered by Plaisted [Pla93], Srensen [Sr95], Gramlich [Gra96], and Melli`es =-=[Mel96]-=- (with small differences), from term rewriting and the -calculus to OCERSs. These strategies specify a construction of infinite reductions (whenever possible) such that all steps are performed in some... |

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Citation Context ...the -calculi. In order to prove our perpetuality criteria, we first generalize the constrictings(or zoom-in) perpetual strategy, independently discovered by Plaisted [Pla93], Srensen [Sr95], Gramlich =-=[Gra96]-=-, and Melli`es [Mel96] (with small differences), from term rewriting and the -calculus to OCERSs. These strategies specify a construction of infinite reductions (whenever possible) such that all steps... |

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Citation Context ...y redex (if any) otherwise [Bar84]. A redex (not an occurrence) is called perpetual iff its occurrence in any (admissible) context is perpetual. Finally, let us recall the concept of external redexes =-=[HL91]-=-. These are redexes whose residuals or descendants can never occur in an argument of another redex. Any external redex is outermost, but not vice versa. (For example, consider the OTRS R = ff(x; g(y))... |

24 |
Weak orthogonality implies confluence: the higher-order case
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Citation Context ... be generalized to weakly-orthogonal CERSs (and thus cover the j-rule since j-redexes are non-erasing) using van Oostrom and van Raamsdonk's technique for simulating fij reductions with fi-reductions =-=[OR94]-=-. 5.5 Honsell &Lenisa's fi N o-calculus Honsell and Lenisa [HL93] define a similar reduction, fi N o -reduction, on -terms by the following rule: fi N o : (x:A)B ! (B=x)A, where B can be instantiated ... |

22 |
The Church-Rosser theorem in orthogonal combinatory reduction systems. Rapports de Recherche 1825, INRIA-Rocquencourt
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Citation Context ...roperty is called strong normalization in [BI94]. showing that the latter are UN, and Khasidashvili [Kha94c] generalized the Conservation Theorem to all orthogonal Expression Reduction Systems (ERSs) =-=[Kha92]-=-, by proving that all non-erasing redexes are perpetual in orthogonal ERSs. For orthogonal Term Rewriting Systems (OTRSs), a very powerful perpetuality criterion was obtained by Klop [Klo92] in terms ... |

21 | The Clausal Theory of Types, Cambridge Tracts - Wolfram - 1993 |

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Citation Context ... ) L (P=Q)=Q 0 . Theorem 2.5 (Strong Church-Rosser [KO95]) For any finite co-initial reductionssP and Q in an OCERS, P + (Q=P ) L Q+ (P=Q). 2.2 Similarity of redexes The idea of similarity of redexes =-=[Kha94]-=- u and v is that u and v are weakly similar, i.e., match the same rewrite rule, and quantifiers in the pattern of u and v bind `similarly ' in the corresponding arguments. Consequently, for any pair o... |

19 | The longest perpetual reductions in orthogonal expression reduction systems
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Citation Context ...ed Church's Theorem to all non-erasing orthogonal Combinatory Reduction Systems (CRSs) by 1 The UN property is called strong normalization in [BI94]. showing that the latter are UN, and Khasidashvili =-=[Kha94c]-=- generalized the Conservation Theorem to all orthogonal Expression Reduction Systems (ERSs) [Kha92], by proving that all non-erasing redexes are perpetual in orthogonal ERSs. For orthogonal Term Rewri... |

19 |
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Citation Context ...tual strategy: simply reduce a redex which does not erase a potentially infinite subterm. It is easy to check that the (maximal) perpetual strategies of Barendregt et al [BBKV76, Bar84] and de Vrijer =-=[dVr87]-=-, and in general, the limit perpetual strategy of Khasidashvili [Kha94b, Kha94c], are special cases, as these strategies contract redexes whose arguments are in normal form, and no (sub)terms can be s... |

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Citation Context ...rule since j-redexes are non-erasing) using van Oostrom and van Raamsdonk's technique for simulating fij reductions with fi-reductions [OR94]. 5.5 Honsell &Lenisa's fi N o-calculus Honsell and Lenisa =-=[HL93]-=- define a similar reduction, fi N o -reduction, on -terms by the following rule: fi N o : (x:A)B ! (B=x)A, where B can be instantiated to a closed fi-normal form. We have immediately from Corollary 4.... |

14 |
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Citation Context ...e behaviour not characteristic of the -calculi. In order to prove our perpetuality criteria, we first generalize the constrictings(or zoom-in) perpetual strategy, independently discovered by Plaisted =-=[Pla93]-=-, Srensen [Sr95], Gramlich [Gra96], and Melli`es [Mel96] (with small differences), from term rewriting and the -calculus to OCERSs. These strategies specify a construction of infinite reductions (when... |

13 |
Strong normalization and perpetual reductions in the lambda calculus
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(Show Context)
Citation Context ..., stating that thesI -calculus is uniformly normalizing, and the Conservation Theoremsof Barendregt et al [BBKV76, Bar84], stating that fi I -redexes are perpetual in the -calculus. Bergstra and Klop =-=[BK82]-=- give a sufficient and necessary criterion for perpetuality of fi K -redexes in every context. Klop [Klo80] generalized Church's Theorem to all non-erasing orthogonal Combinatory Reduction Systems (CR... |

12 | Oostrom V., Context-sensitive conditional expression reduction systems
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Citation Context ...lts of this paper, and we will demonstrate their usefulness in applications. We obtain our results in the framework of Orthogonal (Context-sensitive) Conditional Expression reduction Systems (OCERSs) =-=[KO95]-=-. CERS is a format for higherorder rewriting, or to be precise, second-order rewriting, which extends ERSs [Kha92] by allowing restrictions both on arguments of redexes and on the contexts in which th... |

10 | Perpetuality and strong normalization in orthogonal term rewriting systems - Khasidashvili - 1994 |

8 |
The Ant-Lion paradigm for strong normalization
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Citation Context ...flict because of the conditions on bound variables). Using this result, the author proves strong normalization of a number of typed -calculi. 5.4 Bohm &Intrigila's -ffi k -calculus Bohm and Intrigila =-=[BI94]-=- introduced the -ffi k -calculus in order to study UN solutions to fixed point equations, in the j-calculus. Since the K-redexes are the source of failure of the UN property in the (j)-calculus, they ... |

7 | Some notes on lambda-reduction, In: Degrees, reductions, and representability in the lambda calculus - Barendregt, Bergstra, et al. - 1976 |

7 |
Some Problems of the Notation Theory (in Russian
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Citation Context ...tensive survey of the relationship between various formats of higher-order rewriting, such as [Klo80, Kha92, Wol93, Nip93, OR94]. Restricted rewriting systems with substitutions were first studied in =-=[Pkh77]-=- and [Acz78]. We refer to [Klo92] for a survey of results concerning conditional TRSs. Terms in CERSs are built from the alphabet like in the first order case. The symbols having binding power (likesi... |

5 | Minimal relative normalization in orthogonal expression reduction systems
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(Show Context)
Citation Context ...f partial recursive functions by UN-terms only, in the -calculus. 1 The UN property is clearly useful as then all strategies are normalizing, and in particular, there is more room for optimality (cf. =-=[GK96]-=-). It is easy to see that a rewriting system is UN iff all of its redexes are perpetual. These are redexes that reduce terms having an infinite reduction, which we call 1-terms, to 1-terms. Therefore,... |

5 |
Properties of infinite reduction paths in untyped - calculus
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(Show Context)
Citation Context ...haracteristic of the -calculi. In order to prove our perpetuality criteria, we first generalize the constrictings(or zoom-in) perpetual strategy, independently discovered by Plaisted [Pla93], Srensen =-=[Sr95]-=-, Gramlich [Gra96], and Melli`es [Mel96] (with small differences), from term rewriting and the -calculus to OCERSs. These strategies specify a construction of infinite reductions (whenever possible) s... |

2 |
call-by-value and the -calculus
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(Show Context)
Citation Context ..., including the simply typed -calculus and the system F [Bar92], can directly be encoded as OCERSs (see also [KOR93]); --calculi with specific reduction strategies (such as the call-byvalues-calculus =-=[Plo75]-=-) can also be naturally encoded as OCERSs. ERSs are very close to the more familiar format of CRSs of Klop [Klo80], and we claim that all our results are valid for orthogonal CRSs as well (see [Raa96]... |