## Definitional Trees (1992)

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Venue: | In Proc. of the 3rd International Conference on Algebraic and Logic Programming |

Citations: | 152 - 39 self |

### BibTeX

@INPROCEEDINGS{Antoy92definitionaltrees,

author = {Sergio Antoy},

title = {Definitional Trees},

booktitle = {In Proc. of the 3rd International Conference on Algebraic and Logic Programming},

year = {1992},

pages = {143--157},

publisher = {Springer LNCS}

}

### Years of Citing Articles

### OpenURL

### Abstract

. Rewriting is a computational paradigm that specifies the actions, but not the control. We introduce a hierarchical structure representing, at a high level of abstraction, a form of control. Its application solves a specific problem arising in the design and implementation of inherently sequential, lazy, functional programming languages based on rewriting. For example, we show how to extend the expressive power of Log(F ) and how to improve the efficiency of an implementation of BABEL. Our framework provides a notion of degree of parallelism of an operation and shows that the elements of a necessary set of redexes are related by an and-or relation. Both concepts find application in parallel implementations of rewriting. In an environment in which computations can be executed in parallel we are able to detect sequential computations in order to minimize overheads and/or optimize execution. Conversely, we are able to detect when inherently sequential computations can be executed in para...

### Citations

752 | Rewrite systems
- Dershowitz, Jouannaud
- 1990
(Show Context)
Citation Context ...ly orthogonal a left-linear system whose critical pairs are all trivial, that is, if ht; t 0 i is a critical pair, then t is syntactically equal to t 0 . For related concepts and terminology see also =-=[5, 9, 21]-=-. Lemma 5. Any inductively sequential rewrite system R is orthogonal. Lemma 6. If an inductively sequential rewrite system is complete, then a ground term is a normal form if and only if it is a value... |

566 | Term rewriting systems
- Klop
- 1992
(Show Context)
Citation Context ...te an inductively sequential rewrite system R if and only if any operation of R is complete. We now state two simple results about inductively sequential rewrite systems. We follow Klop's terminology =-=[12]-=- and call orthogonal a left-linear, non-overlapping rewrite system, and weakly orthogonal a left-linear system whose critical pairs are all trivial, that is, if ht; t 0 i is a critical pair, then t is... |

365 |
Confluent reductions: Abstract properties and applications to term rewriting systems
- Huet
- 1980
(Show Context)
Citation Context ...if and only if T is a ppdt with f(X 1 ; : : : ; Xn ) as pattern argument, where n is the arity of f and X 1 ; : : : ; Xn are new variables. The next results entail the concept of subsumption ordering =-=[8], a quasi-ordering relation on-=- terms denoted by the infix operator "" and its derivatives "!", "j", etc. We say that tst 0 when t 0 is an instance of t, that is, there exists a substitution oe such th... |

156 |
Logic Programming with Functions and Predicates: The language Babel
- Moreno-Navarro, Rodríguez-Artalejo
- 1992
(Show Context)
Citation Context ...565. non-parallel form of definitional trees. This class is interesting in its own right, since it properly contains the rewrite systems underlying both DF [17, 18, 19] and an implementation of BABEL =-=[14, 15, 16]-=-. Our results extend the expressive power of the first and improve an implementation of the second. Then, we consider a more general, parallel form of definitional trees which are able to contain the ... |

86 |
Computing in Systems Described by Equations
- O’DONNELL
- 1977
(Show Context)
Citation Context ...pproach is concerned with the trade-off between the generality of a class of rewrite systems and the efficiency of a normalizing strategy for systems in the class. Our effort belongs to this category =-=[7, 9, 10, 13, 20, 21, 22, 23]-=-. We introduce a hierarchical structure, called definitional tree, containing rewrite rules and show some of its applications to the control aspects of a rewrite strategy. First, we consider a class o... |

80 |
Equational Logic as a Programming Language
- O’Donnell
- 1985
(Show Context)
Citation Context ...pproach is concerned with the trade-off between the generality of a class of rewrite systems and the efficiency of a normalizing strategy for systems in the class. Our effort belongs to this category =-=[7, 9, 10, 13, 20, 21, 22, 23]-=-. We introduce a hierarchical structure, called definitional tree, containing rewrite rules and show some of its applications to the control aspects of a rewrite strategy. First, we consider a class o... |

64 |
Computations in orthogonal term rewriting systems
- Huet, Lévy
- 1991
(Show Context)
Citation Context ...ly orthogonal a left-linear system whose critical pairs are all trivial, that is, if 〈t, t ′ 〉 is a critical pair, then t is syntactically equal to t ′ . For related concepts and terminology see also =-=[5,8,21]-=-. Lemma 5. Any inductively sequential rewrite system R is orthogonal. Lemma 6. If an inductively sequential rewrite system is complete, then a ground term is a normal form if and only if it is a value... |

50 |
Computational semantics of term rewriting systems
- Boudol
- 1985
(Show Context)
Citation Context ...ategies appears in [9, 20]. The problem is further analyzed and/or summarized in [4, 13, 21]. The difficulty of the normalization problem led to a number of variations of the concept of sequentiality =-=[3, 7, 13, 23]-=-. The comparison of inductive sequentiality with these approaches is outside the scope of this note. Rather, we focus on a few specific issues. We first discuss sequential applications. For example, t... |

42 | Programming in equational logic: Beyond strong sequentiality
- Sekar, Ramakrishnan
- 1993
(Show Context)
Citation Context ...pproach is concerned with the trade-off between the generality of a class of rewrite systems and the efficiency of a normalizing strategy for systems in the class. Our effort belongs to this category =-=[7, 9, 10, 13, 20, 21, 22, 23]-=-. We introduce a hierarchical structure, called definitional tree, containing rewrite rules and show some of its applications to the control aspects of a rewrite strategy. First, we consider a class o... |

36 | Lazy Narrowing in a Graph Machine
- Kuchen, Loogen, et al.
- 1990
(Show Context)
Citation Context ...565. non-parallel form of definitional trees. This class is interesting in its own right, since it properly contains the rewrite systems underlying both DF [17, 18, 19] and an implementation of BABEL =-=[14, 15, 16]-=-. Our results extend the expressive power of the first and improve an implementation of the second. Then, we consider a more general, parallel form of definitional trees which are able to contain the ... |

35 |
Call by need computations in non-ambiguous linear term rewriting systems
- Huet, Lévy
- 1979
(Show Context)
Citation Context |

31 | Sequential evaluation strategies for parallel-or and related reduction systems - Kennaway - 1989 |

15 |
A refinement of strong sequentiality for term rewriting with constructor. ic
- Thatte
- 1987
(Show Context)
Citation Context ...ategies appears in [9, 20]. The problem is further analyzed and/or summarized in [4, 13, 21]. The difficulty of the normalization problem led to a number of variations of the concept of sequentiality =-=[3, 7, 13, 23]-=-. The comparison of inductive sequentiality with these approaches is outside the scope of this note. Rather, we focus on a few specific issues. We first discuss sequential applications. For example, t... |

14 |
Non-Determinism and Lazy Evaluation in Logic Programming
- Antoy
- 1991
(Show Context)
Citation Context ... branch(1; hrule(0s! true); branch(2; hrule(s( )s0 ! false); rule(s(X)ss(Y ) ! XsY )ii)) branch(1; hrule(half (0) ! 0); branch(1\Delta1; hrule(half (s(0)) ! 0); rule(half (s(s(X))) ! s(half (X)))ii)) =-=(2)-=- Figure 1 shows the same trees in pictorial representation. An operation may have distinct definitional trees. We do not address this problem per se. In the final part of this note we will show an ope... |

13 |
Implementing algebraic specifications
- Bouma, Walters
(Show Context)
Citation Context ...Related Work and Applications The notion of sequentiality has been widely investigated. Seminal work on normalizing strategies appears in [9, 20]. The problem is further analyzed and/or summarized in =-=[4, 13, 21]-=-. The difficulty of the normalization problem led to a number of variations of the concept of sequentiality [3, 7, 13, 23]. The comparison of inductive sequentiality with these approaches is outside t... |

13 |
Strandh. Implementation of an interpreter for abstract equations
- Hoffman, O'Donnel, et al.
- 1985
(Show Context)
Citation Context |

13 |
Implementing parallel rewriting
- Kirchner, Viry
- 1992
(Show Context)
Citation Context ...erging approaches to efficient parallel rewriting emphasize respectively "quantity" and "quality". The first one attempts to rewrite simultaneously a large, possibly maximal, set o=-=f redexes of a term [6, 11]-=-. This approach has greater potential, but has the drawback of requiring specialized architectures. The second approach is concerned with the trade-off between the generality of a class of rewrite sys... |

12 |
Klop and Aart Middeldorp. Sequentiality in orthogonal term rewriting systems
- Willem
- 1991
(Show Context)
Citation Context ...nical, of descendants of a subterm s in a term t after another (not necessarily distinct) subterm of t is rewritten, can be found in [8]. A less formal, but more intuitive, formulation is proposed in =-=[13]-=-. Suppose we have a rewrite sequence t ∗ → t ′ and a subterm s of t. The descendants of s in t ′ are computed as follows. 148sUnderline the root of s and perform the rewrite sequence t ∗ → t ′ . The d... |

7 |
LOG(F): An Optimal Combination of Logic Programming, Rewrite Rules and Lazy Evaluation
- Narain
- 1988
(Show Context)
Citation Context ...tional Science Foundation Grant No. CCR-8908565. non-parallel form of definitional trees. This class is interesting in its own right, since it properly contains the rewrite systems underlying both DF =-=[17, 18, 19]-=- and an implementation of BABEL [14, 15, 16]. Our results extend the expressive power of the first and improve an implementation of the second. Then, we consider a more general, parallel form of defin... |

5 |
Lazy evaluation in logic programming
- Narain
- 1990
(Show Context)
Citation Context ...tional Science Foundation Grant No. CCR-8908565. non-parallel form of definitional trees. This class is interesting in its own right, since it properly contains the rewrite systems underlying both DF =-=[17, 18, 19]-=- and an implementation of BABEL [14, 15, 16]. Our results extend the expressive power of the first and improve an implementation of the second. Then, we consider a more general, parallel form of defin... |

4 |
Design strategies for rewrite rules
- Antoy
- 1990
(Show Context)
Citation Context ...tial applications. For example, the system Log(F ) [17, 18, 19] is concerned with a class DF of rewrite systems and a lazy rewrite strategy for these systems. DF bans operations with non-pure columns =-=[1]-=- to guarantee termination. Briefly, for any i, the i-th argument in the left sides of all the rewrite rules defining an operation is either always or never a variable. Further, non-variable arguments ... |

3 |
Optimization by Non-Deterministic, Lazy Rewriting
- Narain
- 1989
(Show Context)
Citation Context ...tional Science Foundation Grant No. CCR-8908565. non-parallel form of definitional trees. This class is interesting in its own right, since it properly contains the rewrite systems underlying both DF =-=[17, 18, 19]-=- and an implementation of BABEL [14, 15, 16]. Our results extend the expressive power of the first and improve an implementation of the second. Then, we consider a more general, parallel form of defin... |

2 |
The rewrite rule machine project
- Goguen, Meseguer, et al.
- 1987
(Show Context)
Citation Context ...erging approaches to efficient parallel rewriting emphasize respectively "quantity" and "quality". The first one attempts to rewrite simultaneously a large, possibly maximal, set o=-=f redexes of a term [6, 11]-=-. This approach has greater potential, but has the drawback of requiring specialized architectures. The second approach is concerned with the trade-off between the generality of a class of rewrite sys... |

2 |
Graphbased implementation of a functional language
- Kuchen, Loogen, et al.
- 1990
(Show Context)
Citation Context ...565. non-parallel form of definitional trees. This class is interesting in its own right, since it properly contains the rewrite systems underlying both DF [17, 18, 19] and an implementation of BABEL =-=[14, 15, 16]-=-. Our results extend the expressive power of the first and improve an implementation of the second. Then, we consider a more general, parallel form of definitional trees which are able to contain the ... |

1 |
Klop and Aart Middledorp. Sequentiality in orthogonal term rewriting systems
- Willem
- 1989
(Show Context)
Citation Context |

1 |
Building definitional trees. Master's project
- Yan
- 1992
(Show Context)
Citation Context ... or infer some aspects of the control. The problem of building definitional trees from rewrite systems has been addressed only marginally in Theorem 19. Some algorithms for this task are presented in =-=[24]-=-. We have shown the application definitional trees to two interesting classes of rewrite systems, one appropriate for sequential computations, the other for parallel ones. An application of our result... |

1 | Previous version: Term rewriting systems - Press - 1992 |

1 |
Building definitional trees
- Yan
- 1992
(Show Context)
Citation Context ... or infer some aspects of the control. The problem of building definitional trees from rewrite systems has been addressed only marginally in Theorem 19. Some algorithms for this task are presented in =-=[24]-=-. We have shown the application definitional trees to two interesting classes of rewrite systems, one appropriate for sequential computations, the other for parallel ones. An application of our result... |