## The Longest Perpetual Reductions in Orthogonal Expression Reduction Systems (1994)

Venue: | In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, A. Nerode and Yu.V. Matiyasevich, eds., Springer LNCS |

Citations: | 18 - 8 self |

### BibTeX

@INPROCEEDINGS{Khasidashvili94thelongest,

author = {Zurab Khasidashvili},

title = {The Longest Perpetual Reductions in Orthogonal Expression Reduction Systems},

booktitle = {In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, A. Nerode and Yu.V. Matiyasevich, eds., Springer LNCS},

year = {1994},

pages = {191--203},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the -calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. The Conservation Theorem for OERSs follows easily from the properties of the strategy. We develop a method for computing the length of a longest reduction starting from a strongly normalizable term. We study properties of pure substitutions and several kinds of similarity of redexes. We apply these results to construct an algorithm for computing lengths of longest reductions in strongly persistent OERSs that does not require actual transformation of the input term. As a corollary, we have an algorithm for computing lengths of longest developments in OERSs. 1 Introduction A strategy is perpetual if, given a term t, it constructs an infinit...

### Citations

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Citation Context ...culus, Combinatory Logic, and all left-normal systems are nonleft -absorbing. Therefore the perpetual strategy of Barendregt et al. [2] is a limit strategy. The Conservation Theorem for the -calculus =-=[2, 1]-=- states that non-erasing redexes are perpetual. Its proof remains valid for OERSs if one uses the limit strategy instead of the perpetual strategy of [2]. Our method for proving that the reductions co... |

751 | Rewrite systems
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Citation Context ...ds: Rewrite systems, -calculus, perpetual reductions, strong normalization. 1 Introduction O'Donnell [48] showed that the innermost strategy is perpetual for orthogonal Term Rewriting Systems (OTRSs) =-=[13,39,3]-=-. This means that a repeated contraction of innermost redexes in a term yields an infinite reduction whenever the term has an infinite reduction. In fact, any strategy that contracts only the redexes ... |

565 | Term Rewriting Systems - Klop - 1992 |

320 |
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Citation Context ...le also occur in the corresponding right-hand side. 2. A TRS is called weakly innermost normalizing [48] if each term has a normal form reachable by an innermost reduction. 13 Corollary 22 1. (Church =-=[9]-=-,Klop [39]) 5 Let R be a non-erasing OTRS. Then R is weakly normalizing iff it is strongly normalizing. 2. (O'Donnell [48]) Let R be an OTRS. Then R is strongly normalizing iff it is weakly innermost ... |

265 | Orderings for term rewriting systems
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- 1982
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Citation Context ... !t 2 u2 ! : : :, where u i is an r i -redex). And the converse can be shown by employing the above-mentioned correspondence between PTRSs and RPSs and applying the method of Recursive Path Orderings =-=[12]-=- (since in any RPS, a well-founded order on rules implies a well-founded order on the corresponding unknown symbols). By combining this criterion with Theorem 30 and Lemmas 37-39, we obtain the follow... |

245 |
Combinatory reduction systems
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Citation Context ... strategy of [2]. Our method for proving that the reductions constructed according to our perpetual strategy are the longest, and for computing their lengths, is a refinement of Nederpelt-Klop method =-=[14, 10]-=-, used to reduce proofs of strong normalization to proofs of weak normalization. For any OERS R, we define the corresponding non-erasing OERS Rs, called the -extension of R. We add fresh function symb... |

183 | The Lambda Calculus
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Citation Context ...e redexes whose erased arguments are strongly normalizing (Klop [39]). For the lambda-calculus, a more subtle computable perpetual strategy, F1 , was invented by Barendregt, Bergstra, Klop and Volken =-=[5,4]-=-. This strategy reduces the leftmost fi-redex that is not contained in the opPreprint submitted to Elsevier Preprint 12 July 2000 erator of another redex and that is either an I-redex or a K-redex who... |

86 |
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Citation Context ...ategy) implies strong normalization of the initial term. For orthogonal (left-linear and non-overlapping) TRSs a very simple perpetual strategy exists --- just contract any innermost redex (O'Donnell =-=[15]-=-). In fact, any complete strategy, i.e., a strategy that in each term contracts a redex that does not erase any other redex, is perpetual. Moreover, one can even reduce redexes all erased arguments of... |

84 |
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- 1991
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Citation Context ...nditions if in each term s we contract a limit redex which is defined as follows: Choose in s an external redex u 1 (i.e., a redex whose descendants never appear inside the arguments of other redexes =-=[19]-=-); choose an erased argument s 1 of u 1 that is not in normal form; choose in s 1 an external redex u 2 , and so on as long as possible. The last redex chosen is a limit redex of s. This strategy, whi... |

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54 |
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Citation Context ... strategy of [2]. Our method for proving that the reductions constructed according to our perpetual strategy are the longest, and for computing their lengths, is a refinement of Nederpelt-Klop method =-=[14, 10]-=-, used to reduce proofs of strong normalization to proofs of weak normalization. For any OERS R, we define the corresponding non-erasing OERS Rs, called the -extension of R. We add fresh function symb... |

50 | Term Rewriting and All, That Cambridge - Baader, Nipkow - 1999 |

50 |
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Citation Context ... with OTRSs [13,39,3] and not interested in higher-order rewriting can skip the next section, and refer to it only for the notation (Notation 3), for an informal description of the descendant concept =-=[8,27]-=-, and for the concept of similarity of redexes (Definition 6). The results on the longest perpetual reductions have been reported previously in [29,31,32,34]. Here we simplify and correct several conc... |

38 |
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- 1978
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Citation Context ...a uniform framework for reductions with substitutions like that in the -calculus [4] and its extensions. Restricted rewriting systems with substitutions were first studied by Pkhakadze [52] and Aczel =-=[1]-=-. Several interesting formalisms were introduced later [24,47,68,51]. See [57] for a survey. We will refer to such systems using a collective name `higher-order rewriting'. Here we use Expression Redu... |

32 | P.: The Conservation Theorem Revisited
- Groote
- 1993
(Show Context)
Citation Context ...f a term t in an OERS R has an infinite reduction and t u !s, where u is a non-erasing redex, then s has also an infinite reduction. An extension of the Conservation Theorem can be found in de Groote =-=[4]-=-. It is a neat translation of Klop's Lemma 4.1 by simulatingswith some reductions on -terms. Unfortunately, this is not possible for arbitrary OERSs. As shown above (Lemmas 4.1 and 4.3), a term in an ... |

30 |
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Citation Context ... uniform framework for reductions with substitutions (also referred to as higher order rewriting) as in the -calculus [1]. Several other formalisms have been introduced later. We refer to Klop et al. =-=[12]-=- for a survey. Here we use a system of higher order rewriting, Expression Reduction Systems (ERSs), defined in [6] (under the name of CRSs). Definition 2.1 (1) Let \Sigma be an alphabet, comprising va... |

29 |
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Citation Context ...erms [57,60,58]. Termination proofs for higher-order rewriting: Van de Pol and Schwichtenberg [54--56] developed a method, similar to the one used by de Vrijer [66] and based on earlier work by Gandy =-=[16]-=-, for strong normalization proofs with upper bound information in higher-order rewrite systems and typed -calculi. Van Oostrom proved termination of reductions in which contraction of redexes with a l... |

28 | Expression reduction systems
- Khasidashvili
- 1990
(Show Context)
Citation Context ...,31,32,34]. Here we simplify and correct several concepts and some proofs. Some of the termination proofs employing the memory method and establishing exact upper bounds were obtained much earlier in =-=[22,24]-=-. 2 Preliminaries: Orthogonal Expression Reduction Systems Klop introduced Combinatory Reduction Systems (CRSs) [38] to provide a uniform framework for reductions with substitutions like that in the -... |

24 |
J.J.: Computations in orthogonal rewriting systems
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Citation Context ... o coincides with the number of -occurrences in the Rs-normal form of o. Unabsorbed redexes exist in any term not in a normal form, but they can be found efficiently only in strongly sequential OERSs =-=[5]-=-. Therefore, our method is applicable to all strongly sequential OERSs that are strongly normalizing. Several typed -calculi and typed OTRSs are such OERSs. Moreover, it is not always necessary to do ... |

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Citation Context ...rties of substitutions. We call a subterm s in t essential (written ES(s; t)) if s has at least one descendant under any reduction starting from t and call it inessential (written IE(s; t)) otherwise =-=[7]-=-. The notion of essentiality is a generalization of the notion of neededness [5, 13] in a way that it works for all subterms, bound variables in particular. Notation Below FV (t) (resp. EFVR (t)) deno... |

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19 |
Surjective Pairing and Strong Normalization: Two Themes
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- 1987
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14 |
strat'egie paresseuse. Th`ese de l'Universit'e de Paris VII
- La
- 1992
(Show Context)
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13 |
Strong normalization and perpetual reductions in the lambda calculus
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- 1982
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Citation Context ...nclusions We presented an algorithm for constructing longest reductions and computing their lengths in OERSs, and used the algorithm to generalize the Conservation Theorem to OERSs. Bergstra and Klop =-=[3]-=- gave a characterization of erasing fi-redexes (K-redexes) for which the Conservation Theorem in -calculus still is valid. We leave this question for OERSs to a future investigation. De Vrijer [17] ga... |

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Citation Context ...in -calculus still is valid. We leave this question for OERSs to a future investigation. De Vrijer [17] gave another characterization of lengths of longest reductions in a typed -calculus. Van de Pol =-=[16]-=- used a semantic method for strong normalization proofs in higher order rewriting systems. Acknowledgments I enjoyed discussions with H. Barendregt, J. W. Klop, J.-J. L'evy, L. Maranget, G. Mints, V. ... |

11 |
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Citation Context ...a strategy was found by Barendregt et al. [2]. In this paper, we design a perpetual strategy for all OERSs. It is a generalization of the perpetual strategy of the -calculus and of the limit strategy =-=[8]-=-. Our aim is not only to construct an infinite reduction of any given term t whenever it exists, but also to construct a longest possible one if t is strongly normalizable and to find a method to char... |

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- 1994
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8 |
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Citation Context ...,51], even non-erasing redexes may not be perpetual. A strictly stronger criterion of perpetuality of fi-redexes is obtained by Honsell and Lenisa [18] by using semantical methods. Bohm and Intrigila =-=[7]-=- showed that any step is perpetual in the -ffi k - 34 calculus, which is obtained from thesI -calculus by adding a delta rule for a `restricted K combinator' ffi K : ffi K AB ! A, where B can be insta... |

7 |
Some notes on lambda-reduction, In: Degrees, reductions, and representability in the lambda calculus
- Barendregt, Bergstra, et al.
- 1976
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Citation Context ... OERSs. Therefore one can erase only strongly normalizable arguments in which no substitution of external subterms is possible. For the lambda-calculus, such a strategy was found by Barendregt et al. =-=[2]-=-. In this paper, we design a perpetual strategy for all OERSs. It is a generalization of the perpetual strategy of the -calculus and of the limit strategy [8]. Our aim is not only to construct an infi... |

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4 |
A λ-to-CL Translation for Strong Normalization
- Akama
- 1997
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Citation Context ... Thus (*) is valid and the lemma is proved. The following lemma is a main one. It states that any Rs-step s v !s 0 corresponding to a limit step t u !t 0 in an OTRS R preserves the properties [1] and =-=[2]-=- of Definition 16. Clearly, if s satisfies [2], then the step s v !s 0 increases the number of -occurrences in s exactly by 1. We can conclude by transitivity that if P : t 0 !! t n is a normalizing l... |

3 |
Perpetual reductions in orthogonal combinatory reduction systems
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- 1993
(Show Context)
Citation Context ...edexes in OERSs. The above result relies on the fact that strongly similar redexes generate the same number of strongly similar redexes. The results of this paper with complete proofs are reported in =-=[9]-=-. 2 Orthogonal Expression Reduction Systems Klop introduced Combinatory Reduction Systems (CRSs) in [10] to provide a uniform framework for reductions with substitutions (also referred to as higher or... |

3 |
Combinatory Logic. North-Holland
- Curry, Feys
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(Show Context)
Citation Context ...rnal redex can be found effectively in any reducible term. The left-normal OTRSs, where function symbols precede variables in left-hand sides of rewrite rules, and in particular the Combinatory Logic =-=[11]-=-, are strongly sequential as the leftmostoutermost redex is external in every term. Further, note that the limit strategy is non-deterministic because neither external redexes nor erased arguments are... |

3 |
conservation and preservation of strong normalization for generalized reduction
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- 2000
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Citation Context ...typed -calculi by first proving weak normalization of some auxiliary reductions: instead of keeping erased subterms, they postpone contractions of erasing redexes for as long as possible. Kamareddine =-=[20]-=- showed that postponement of erasing steps can also be achieved by using the generalized fi-reduction. Srensen [63] and Xi [69] independently developed methods for reducing fi-strong normalization pro... |

3 | On the equivalence of persistent term rewriting systems and recursive program schemes
- Khasidashvili
- 1993
(Show Context)
Citation Context ...let r 2 R. Let us call an r-chain a maximal sequence of R-rules r = r 1 ; r 2 ; r 3 ; : : : such that an r i+1 -redex has an occurrence in the right-hand side of r i (i = 1; 2; : : :). It is shown in =-=[28]-=- that a term t in R is strongly normalizing iff all chains of the rules whose corresponding redexes occur in t are finite. Instead of recalling the proof from [28] here, we remark that the sufficiency... |

3 | Perpetual reductions and strong normalization in orthogonal term rewriting systems
- Khasidashvili
- 1993
(Show Context)
Citation Context ...an informal description of the descendant concept [8,27], and for the concept of similarity of redexes (Definition 6). The results on the longest perpetual reductions have been reported previously in =-=[29,31,32,34]-=-. Here we simplify and correct several concepts and some proofs. Some of the termination proofs employing the memory method and establishing exact upper bounds were obtained much earlier in [22,24]. 2... |

2 |
Strong normalization and perpetual reductions
- Bergstra, Klop
- 1982
(Show Context)
Citation Context ...petual -- that is, they preserve the possibility of an infinite reduction -- and, similarly, for fully-extended OERSs it states that non-erasing redexes are perpetual. Klop [39] and Bergstra and Klop =-=[6]-=- gave characterizations of perpetual redexes (implying conservation) in OTRSs and the -calculus, respectively, and similar perpetuality criteria for fully-extended orthogonal CCERSs were obtained by K... |

2 |
Semantical analysis of perpetual strategies
- Honsell, Lenisa
- 1999
(Show Context)
Citation Context ...ewriting where function variables can be bound [47,68,51], even non-erasing redexes may not be perpetual. A strictly stronger criterion of perpetuality of fi-redexes is obtained by Honsell and Lenisa =-=[18]-=- by using semantical methods. Bohm and Intrigila [7] showed that any step is perpetual in the -ffi k - 34 calculus, which is obtained from thesI -calculus by adding a delta rule for a `restricted K co... |

1 |
Some properties of reconstruction of forms from contracted forms
- Khasidashvili
- 1985
(Show Context)
Citation Context ...ral. We develop a method for proving that the reductions constructed according to our perpetual strategy are indeed the longest and for computing their lengths. Our method, developed independently in =-=[22]-=-, can be viewed as a refinement of the Nederpelt-Klop method, which reduces proving strong normalization in a typed -calculus [46] and orthogonal Combinatory Reduction Systems (OCRSs) in general [38] ... |