## Optimal Normalization in Orthogonal Term Rewriting Systems (1993)

Venue: | In: Proc. of the 5 th International Conference on Rewriting Techniques and Applications, RTA'93 |

Citations: | 24 - 20 self |

### BibTeX

@INPROCEEDINGS{Khasidashvili93optimalnormalization,

author = {Zurab Khasidashvili},

title = {Optimal Normalization in Orthogonal Term Rewriting Systems},

booktitle = {In: Proc. of the 5 th International Conference on Rewriting Techniques and Applications, RTA'93},

year = {1993},

pages = {243--258},

publisher = {Springer}

}

### OpenURL

### Abstract

. We design a normalizing strategy for orthogonal term rewriting systems (OTRSs), which is a generalization of the call-by-need strategy of Huet-L'evy [4]. The redexes contracted in our strategy are essential in the sense that they have "descendants" under any reduction of a given term. There is an essential redex in any term not in normal form. We further show that contraction of the innermost essential redexes gives an optimal reduction to normal form, if it exists. We classify OTRSs depending on possible kinds of redex creation as non-creating, persistent, inside-creating, non-left-absorbing, etc. All these classes are decidable. TRSs in these classes are sequential, but they do not need to be strongly sequential. For non-creating and persistent OTRSs, we show that our optimal strategy is efficient as well. 1 Introduction In this paper, we study correct and optimal computations in Orthogonal Term Rewriting Systems (OTRSs). We only consider one-step rewriting strategies, which selec...

### Citations

578 | Term Rewriting Systems
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Citation Context ... the optimality of the graph implementation of call-by-need. Kennaway [5] showed that every OTRS admits a computable correct strategy, and not only strongly sequential systems. In Klop and Middeldorp =-=[10, 11]-=-, simpler proofs of decidability of strong sequentiality are given. In Thatte [18] the notion of left-sequentiality is introduced. In Oyamaguchi [15] a class of sufficiently sequential OTRSs, which pr... |

88 |
Optimal reductions in the lambda calculus, in To H
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Citation Context ...eps it comprises, which is also the number of redexes it contracts. Clearly, such a cost notion assumes a term implementation of term rewriting. By contrast, the more sophisticated notions of cost of =-=[1, 12, 13]-=- take the contraction of one set of similar redexes, a so-called redex family, as the unitary cost. Thus, in a realistic implementation, redex families must be implemented as a single shared object, s... |

87 | Computing in Systems Described by Equations - O’Donnell - 1977 |

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Citation Context ...nay, France Zurab.Khasidashvili@inria.fr Abstract. We design a normalizing strategy for orthogonal term rewriting systems (OTRSs), which is a generalization of the call-by-need strategy of Huet-L'evy =-=[4]. The rede-=-xes contracted in our strategy are essential in the sense that they have "descendants" under any reduction of a given term. There is an essential redex in any term not in normal form. We fur... |

43 |
Recursive applicative program schemes
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Citation Context ...n persistent systems, creation depends on rewrite rules only; the actual arguments and the context in which reduction takes place do not matter. In particular, Recursive (Applicative) Program Schemes =-=[2]-=- are persistent. In inside-creating systems, the arguments, but not the context, can also take part in creation; any created redex is inside the contractum of a creating redex. In outside-creating sys... |

41 |
Minimal and optimal computations of recursive programs
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Citation Context ...eps it comprises, which is also the number of redexes it contracts. Clearly, such a cost notion assumes a term implementation of term rewriting. By contrast, the more sophisticated notions of cost of =-=[1, 12, 13]-=- take the contraction of one set of similar redexes, a so-called redex family, as the unitary cost. Thus, in a realistic implementation, redex families must be implemented as a single shared object, s... |

41 |
Sequentiality in orthogonal term rewriting systems
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(Show Context)
Citation Context ... the optimality of the graph implementation of call-by-need. Kennaway [5] showed that every OTRS admits a computable correct strategy, and not only strongly sequential systems. In Klop and Middeldorp =-=[10, 11]-=-, simpler proofs of decidability of strong sequentiality are given. In Thatte [18] the notion of left-sequentiality is introduced. In Oyamaguchi [15] a class of sufficiently sequential OTRSs, which pr... |

38 |
Term rewriting systems: a tutorial
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Citation Context ... reductions in the "Notation Theory" of Sh. Pkhakadze [16]. The results were generalized for OTRSs and reported in [8] after I became acquainted with the TRS theory by J. W. Klop's introduct=-=ory paper [9]-=-, pointed to me out by G. Mints. I am grateful to them, as well as to H. Barendregt and J.-J. L'evy, for constant support. I thank H. Barendregt, P.- L. Curien, H. Ganzinger, D. Kesner, J. W. Klop, G.... |

30 |
Jouannaud J.P.: Rewrite Systems
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Citation Context ...endent subterms. In section 7 we establish the relation between neededness and essentiality, and in section 8 we study optimal reductions in Persistent TRSs. 2 Strict Confluence for OTRSs We refer to =-=[3]-=- and [10] for introductions in the TRS theory. Below we consider only OTRSs. Notation We use R for OTRSs, t; s; e; o for terms, r for rewrite rules, u; v; w for redexes, and P; Q for reductions. We wr... |

30 | Sequential Evaluation Strategies for Parallel-Or and Related Reduction Systems - Kennaway - 1989 |

24 |
Strong Sequentiality of Left-Linear Overlapping Term Rewriting Systems
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(Show Context)
Citation Context ...ciently sequential OTRSs, which properly contains the class of strongly sequential OTRSs, is shown to be decidable. In Kesner [7] sequentiality is studied in OTRSs with order-sorted alphabets. Toyama =-=[19]-=- generalized some results from [4] to left-linear overlapping TRSs. The essential strategy, as defined in this paper, is based on a notion of descendant, which allows the tracing of all subterms, incl... |

17 |
Neededness is hypernormalizing in regular combinatory reduction systems. Report. University of East Anglia
- Kennaway, Sleep
- 1989
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Citation Context ...to find a needed redex in any term not in normal form. They also proved that strong sequentiality is a decidable property. This fundamental work was later extended by many authors. Kennaway and Sleep =-=[6]-=- gave a proof of correctness of the call-by-need strategy for all orthogonal combinatory reduction systems using labeling. Maranget [13] studied the optimality of the graph implementation of call-by-n... |

14 |
A refinement of strong sequentiality for term rewriting with constructors
- Thatte
- 1987
(Show Context)
Citation Context ...every OTRS admits a computable correct strategy, and not only strongly sequential systems. In Klop and Middeldorp [10, 11], simpler proofs of decidability of strong sequentiality are given. In Thatte =-=[18]-=- the notion of left-sequentiality is introduced. In Oyamaguchi [15] a class of sufficiently sequential OTRSs, which properly contains the class of strongly sequential OTRSs, is shown to be decidable. ... |

7 |
Some Problems of the Notation Theory (in Russian
- Sh
- 1977
(Show Context)
Citation Context ...m 5.1. Acknowledgements The correctness and optimality results were first obtained under the supervision of Sh. Pkhakadze and Kh. Rukhaia for some reductions in the "Notation Theory" of Sh. =-=Pkhakadze [16]-=-. The results were generalized for OTRSs and reported in [8] after I became acquainted with the TRS theory by J. W. Klop's introductory paper [9], pointed to me out by G. Mints. I am grateful to them,... |

3 |
La strat'egie paresseuse", These de l'Universit'e' de Paris VII
- Maranget
- 1992
(Show Context)
Citation Context ...eps it comprises, which is also the number of redexes it contracts. Clearly, such a cost notion assumes a term implementation of term rewriting. By contrast, the more sophisticated notions of cost of =-=[1, 12, 13]-=- take the contraction of one set of similar redexes, a so-called redex family, as the unitary cost. Thus, in a realistic implementation, redex families must be implemented as a single shared object, s... |

3 |
Sufficient sequentiality: a decidable condition for call-by-need computations in term rewriting systems
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(Show Context)
Citation Context ...gly sequential systems. In Klop and Middeldorp [10, 11], simpler proofs of decidability of strong sequentiality are given. In Thatte [18] the notion of left-sequentiality is introduced. In Oyamaguchi =-=[15]-=- a class of sufficiently sequential OTRSs, which properly contains the class of strongly sequential OTRSs, is shown to be decidable. In Kesner [7] sequentiality is studied in OTRSs with order-sorted a... |

3 |
A simple proof of confluence for weakly orthogonal combinatory reduction systems
- Raamsdonk
- 1992
(Show Context)
Citation Context ... non-absorbing, non-left-absorbing, and non-right-absorbing systems. All these classes have different normalization behavior. For example, in outside -creating systems, all reductions are terminating =-=[17]-=-. In persistent systems, normalization and termination are decidable. In non-absorbing systems, all outermost redexes are essential, etc. Furthermore, these classes are easily decidable. In fact, the ... |

2 |
Minimal Normalizing Strategy for Orthogonal and Persistent Term Rewriting Systems (short version
- Khasidashvili
- 1990
(Show Context)
Citation Context ...s were first obtained under the supervision of Sh. Pkhakadze and Kh. Rukhaia for some reductions in the "Notation Theory" of Sh. Pkhakadze [16]. The results were generalized for OTRSs and re=-=ported in [8]-=- after I became acquainted with the TRS theory by J. W. Klop's introductory paper [9], pointed to me out by G. Mints. I am grateful to them, as well as to H. Barendregt and J.-J. L'evy, for constant s... |

1 |
sequentiality in orthogonal order-sorted rewriting systems with constructors
- Free
- 1992
(Show Context)
Citation Context ...n of left-sequentiality is introduced. In Oyamaguchi [15] a class of sufficiently sequential OTRSs, which properly contains the class of strongly sequential OTRSs, is shown to be decidable. In Kesner =-=[7]-=- sequentiality is studied in OTRSs with order-sorted alphabets. Toyama [19] generalized some results from [4] to left-linear overlapping TRSs. The essential strategy, as defined in this paper, is base... |