## Higher-Order Rewriting (1999)

Venue: | 12th Int. Conf. on Rewriting Techniques and Applications, LNCS 2051 |

Citations: | 20 - 1 self |

### BibTeX

@INPROCEEDINGS{Raamsdonk99higher-orderrewriting,

author = {Femke Van Raamsdonk},

title = {Higher-Order Rewriting},

booktitle = {12th Int. Conf. on Rewriting Techniques and Applications, LNCS 2051},

year = {1999},

pages = {261--275},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag.

### Citations

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Simple word problems in universal algebras
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Citation Context ...nce of critical pairs. A natural next step is to investigate to what extent the condition on critical pairs can be relaxed while maintaining confluence, or a weaker version of it. Huet [11] (see also =-=[17]-=-) shows that a first-order term rewriting system is locally confluent (that is, t 0 i r j t whenever t 0 / s ! t) if all its critical pairs are confluent. A critical pair (s; s 0 ) is said to be confl... |

364 |
Confluent reductions: Abstract properties and applications to term rewriting systems: Abstract properties and applications to term rewriting systems
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Citation Context ...royed by contraction of the leftmost redex a. This kind of interference may cause a rewriting system to be non-confluent: in the term rewriting system ff(x; x) ! a; f(x; g(x)) ! b; c ! g(c)g given in =-=[11]-=-, we have both f(c; c) ! a and f(c; c) i b. Another example, given in [16], is the term rewriting system fd(x; x) ! e; c(x) ! d(x; c(x)); a ! c(a)g. We have c(a) i e and c(a) i c(e). The source of non... |

287 | A logic programming language with lambda-abstraction, function variables, and simple unification
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Citation Context ...n in a higher-order rewriting system. For the definition of a rewrite rule we first need to introduce the notion of a rule-pattern, which is a slight adaptation of the notion of pattern introduced in =-=[21]-=-. A rule-pattern is a term of the form x:f(s) such that, first, every x i 2 x occurs free in f(s), and second, every x i 2 x occurs only in subterms of the form x i (y 1 ; : : : ; ym ) with y 1 ; : : ... |

286 |
The Lambda-Calculus, its Syntax and Semantics, volume 103
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Citation Context ...xtensions of lambda-calculus are studied. Examples of such extensions are lambda-calculus extended with infinitely many ffi-rules that test for equality of closed normal forms, defined by Church (see =-=[3]-=-), lambda-calculus with rules for surjective pairing, and the class of (a)-reductions which consists of lambdacalculus with constants and rewrite rules for these constants [10]. Higher-order rewrite s... |

265 | Orderings for term rewriting systems
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Citation Context ...z 0 ]](0) + 1 = [[forall(x:and(z; z 0 (x)))]]: 4.2 Recursive Path Ordering An important method to prove termination of a first-order term rewriting system is the one using the recursive path ordering =-=[6]-=-. This ordering is roughly speaking defined by extending a well-founded ordering on function symbols in a recursive way. The two key results concerning the recursive path ordering are that it is a wel... |

124 |
Higher-order critical pairs
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Citation Context ...h the work by Klop, who introduces in [16] the class of combinatory reduction systems. Another impulse for the study of rewriting with bound variables originates in the work by Nipkow, who defines in =-=[22]-=- the class of higher-order rewrite systems. Combinatory reduction systems form a generalisation of the class of contraction schemes defined in [1], and can more generally be seen as standing in a trad... |

95 | Equations and rewrite rules - a survey
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Citation Context ...lgebraic side is formulated in terms of valuations; that condition implies the last requirement as above. The second requirement yields that [[C[s]]] ? [[C[t]]] whenever [[s]] ? [[t]]. Huet and Oppen =-=[12]-=- and Zantema [37] prove that a first-order term rewriting system is terminating if and only if it has a termination model. This result provides a complete but not algorithmic method for proving termin... |

88 | Conditional rewrite rules: Confluence and termination - Bergstra, Klop - 1982 |

86 |
Computing in Systems Described by Equations
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Citation Context ...ot in normal form. A strategy is said to be normalising if repeatedly contracting redex occurrences selected by the strategy yields a normal form whenever the initial term has one. O'Donnell shows in =-=[24]-=- that the parallel-outermost strategy, which contracts all redexes that are outermost in a term in one step, is normalising for first-order term rewriting systems that are left-linear, and where trivi... |

85 | Termination of term rewriting: interpretation and type elimination
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Citation Context ...formulated in terms of valuations; that condition implies the last requirement as above. The second requirement yields that [[C[s]]] ? [[C[t]]] whenever [[s]] ? [[t]]. Huet and Oppen [12] and Zantema =-=[37]-=- prove that a first-order term rewriting system is terminating if and only if it has a termination model. This result provides a complete but not algorithmic method for proving termination. Van De Pol... |

76 | Higher-Order Narrowing with Definitional Trees
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Citation Context ...ing redexes is ruled out by disallowing the non-occur check as present in the example. Formally, this is done by requiring the rewrite rules to be fully extended, a notion which is defined as follows =-=[9]-=-. Definition 4. 1. Let z:l be a rule-pattern. An occurrence of z 2 z in l is fully extended if it is of the form z(x 1 ; : : : ; xn ) with x 1 ; : : : ; xn all the variables bound above it. 2. A rewri... |

74 |
Confluence for Abstract and Higher-Order Rewriting
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Citation Context ...r-order rewrite systems were introduced with the aim to study the meta-theory of systems like Isabelle and -Prolog. The presentation of higher-order rewriting systems in this paper is mainly based on =-=[25, 32]-=-, which builds on [16, 22]. However, I would like to stress that the actual format of higher-order rewriting is not of the utmost importance here, since the paper is informal in nature. Moreover, the ... |

71 |
Some properties of conversion
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Citation Context ...adapted to situations where the rewriting system is not quite orthogonal or where the structures that are rewritten are not quite terms. Developments are used to show confluence of lambda-calculus in =-=[5]-=- and to show confluence of combinatory reduction systems in [16]. We follow here basically the account presented in [25, 26]. The idea is to define a relation ) with ! ` ) ` i that satisfies moreover ... |

71 |
Tree manipulation systems and Church-Rosser theorems
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Citation Context ...ence of higher-order term rewriting. 3.1 Orthogonality A first general confluence result for first-order term rewriting systems is the one stating that orthogonal term rewriting systems are confluent =-=[33]-=-. Orthogonality of a rewriting system roughly speaking expresses that rewrite steps are independent. In the setting of first-order term rewriting, it is usually formalised by two requirements imposed ... |

62 | Higher-order rewrite systems and their confluence
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Citation Context ...han that of first-order terms; it still decidable [21] and moreover has linear complexity [31]. Here we do not give a completely formal definition of critical pair, which can be found for instance in =-=[22, 18]-=-. Definition 1. 1. A rewrite rule x:l ! x:r of a higher-order rewriting system is left-linear if every variable x 2 x occurs exactly once in l. 2. Let C[ls] = gt indicate a most general overlap betwee... |

44 | The higher-order recursive path ordering
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(Show Context)
Citation Context ...define a well-founded ordering on the function symbols and show that in its extension to the recursive path ordering we have l ? r for every rewrite rule. Jouannaud and Rubio define in a recent paper =-=[13]-=- a generalisation of the recursive path ordering to higher-order terms, and show that it provides a method to prove termination of higher-order term rewriting systems. The remarkable feature of their ... |

42 | Programming in equational logic: Beyond strong sequentiality
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(Show Context)
Citation Context ... sequence f(g(a)) ! f(g(a)) ! f(g(a)) ! : : :. The proof presented in [27] is abstract in nature and applies also to the case of higher-order rewriting. It makes use of ideas that are also present in =-=[34, 20, 8]-=-. For proofs according to the sketch given above, see [24, 4, 32]. Theorem 6. Outermost-fair rewriting is normalising for higher-order rewriting systems that are weakly orthogonal and fully extended. ... |

40 | Call by Need Computations to Root-Stable Form
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Citation Context ... sequence f(g(a)) ! f(g(a)) ! f(g(a)) ! : : :. The proof presented in [27] is abstract in nature and applies also to the case of higher-order rewriting. It makes use of ideas that are also present in =-=[34, 20, 8]-=-. For proofs according to the sketch given above, see [24, 4, 32]. Theorem 6. Outermost-fair rewriting is normalising for higher-order rewriting systems that are weakly orthogonal and fully extended. ... |

39 | Interaction systems I: The theory of optimal reductions
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Citation Context ...introduced in the literature. To mention a two of them: expression reduction systems [15] are similar to combinatory reduction systems, but have been introduced independently, and interaction systems =-=[2]-=- form a subclass of combinatory reduction systems that is introduced for the study of optimality. It is not hard to adapt the presentation of these two classes of systems, and of the one of combinator... |

38 |
A general Church-Rosser theorem
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Citation Context ...les originates in the work by Nipkow, who defines in [22] the class of higher-order rewrite systems. Combinatory reduction systems form a generalisation of the class of contraction schemes defined in =-=[1]-=-, and can more generally be seen as standing in a tradition where extensions of lambda-calculus are studied. Examples of such extensions are lambda-calculus extended with infinitely many ffi-rules tha... |

34 | Solving higher-order equations: From logic to programming. Birkhäuser PCTS Series - Prehofer - 1997 |

32 |
Confluence and Normalisation for higher-order rewriting. Thèse de doctorat, Vrije Universiteit
- Raamsdonk
- 1996
(Show Context)
Citation Context ...r-order rewrite systems were introduced with the aim to study the meta-theory of systems like Isabelle and -Prolog. The presentation of higher-order rewriting systems in this paper is mainly based on =-=[25, 32]-=-, which builds on [16, 22]. However, I would like to stress that the actual format of higher-order rewriting is not of the utmost importance here, since the paper is informal in nature. Moreover, the ... |

31 |
Description abstraite des systèmes de réécriture
- Melliès
- 1996
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Citation Context ...tional reasoning and narrowing for higher-order rewriting are referred to [23, 30] and the literature mentioned there. Further, results concerning confluence and termination can be found in detail in =-=[16, 25, 19, 29, 32]-=-. Acknowledgements. I am grateful to Jan Willem Klop and Vincent van Oostrom for discussions that in the course of the last years always have been, and still are, a source of inspiration. I wish to th... |

29 |
Proofs of strong normalization
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- 1980
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Citation Context ...hod for proving termination. Van De Pol presents in [28, 29] a generalisation of this result to the higherorder case. This generalisation makes use of ideas that are also present in the work by Gandy =-=[7]-=- and De Vrijer [35, 36] on termination of simply typed - calculus. For every type A, the set of functionals of type A is defined by induction on the definition of simple types: a base type is interpre... |

28 | Expression reduction systems
- Khasidashvili
- 1990
(Show Context)
Citation Context ...the details of the chosen format. For various purposes other classes of rewriting systems with bound variables are introduced in the literature. To mention a two of them: expression reduction systems =-=[15]-=- are similar to combinatory reduction systems, but have been introduced independently, and interaction systems [2] form a subclass of combinatory reduction systems that is introduced for the study of ... |

21 |
Combinatory Reduction Systems, volume 127
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(Show Context)
Citation Context ... proofs should be closed under substitution. This theme plays a role throughout the paper. A systematic study of rewriting systems with bound variables starts with the work by Klop, who introduces in =-=[16]-=- the class of combinatory reduction systems. Another impulse for the study of rewriting with bound variables originates in the work by Nipkow, who defines in [22] the class of higher-order rewrite sys... |

19 |
de Pol. Termination of Higher-order Rewrite Systems
- van
- 1996
(Show Context)
Citation Context ...tional reasoning and narrowing for higher-order rewriting are referred to [23, 30] and the literature mentioned there. Further, results concerning confluence and termination can be found in detail in =-=[16, 25, 19, 29, 32]-=-. Acknowledgements. I am grateful to Jan Willem Klop and Vincent van Oostrom for discussions that in the course of the last years always have been, and still are, a source of inspiration. I wish to th... |

19 |
Surjective Pairing and Strong Normalization: Two Themes
- Vrijer
- 1987
(Show Context)
Citation Context ...rmination. Van De Pol presents in [28, 29] a generalisation of this result to the higherorder case. This generalisation makes use of ideas that are also present in the work by Gandy [7] and De Vrijer =-=[35, 36]-=- on termination of simply typed - calculus. For every type A, the set of functionals of type A is defined by induction on the definition of simple types: a base type is interpreted as some fixed nonem... |

17 | Relative normalization in deterministic residual structures
- Glauert, Khasidashvili
- 1996
(Show Context)
Citation Context ... sequence f(g(a)) ! f(g(a)) ! f(g(a)) ! : : :. The proof presented in [27] is abstract in nature and applies also to the case of higher-order rewriting. It makes use of ideas that are also present in =-=[34, 20, 8]-=-. For proofs according to the sketch given above, see [24, 4, 32]. Theorem 6. Outermost-fair rewriting is normalising for higher-order rewriting systems that are weakly orthogonal and fully extended. ... |

17 | Developing developments
- Oostrom
- 1997
(Show Context)
Citation Context ... not quite terms. Developments are used to show confluence of lambda-calculus in [5] and to show confluence of combinatory reduction systems in [16]. We follow here basically the account presented in =-=[25,-=- 26]. The idea is to define a relation ) with ! ` ) ` i that satisfies moreover the following diagram: // // // Confluence follows then by an easy induction, since the diagram above and the fact tha... |

16 |
Exactly estimating functionals and strong normalization, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen
- Vrijer
- 1987
(Show Context)
Citation Context ...rmination. Van De Pol presents in [28, 29] a generalisation of this result to the higherorder case. This generalisation makes use of ideas that are also present in the work by Gandy [7] and De Vrijer =-=[35, 36]-=- on termination of simply typed - calculus. For every type A, the set of functionals of type A is defined by induction on the definition of simple types: a base type is interpreted as some fixed nonem... |

14 |
Reductions of residuals are finite
- Hindley
- 1978
(Show Context)
Citation Context ..., defined by Church (see [3]), lambda-calculus with rules for surjective pairing, and the class of (a)-reductions which consists of lambdacalculus with constants and rewrite rules for these constants =-=[10]-=-. Higher-order rewrite systems were introduced with the aim to study the meta-theory of systems like Isabelle and -Prolog. The presentation of higher-order rewriting systems in this paper is mainly ba... |

14 |
Higher-Order Rewriting and Equational Reasoning
- Nipkow, Prehofer
- 1998
(Show Context)
Citation Context ...s nevertheless by no means possible to give a complete overview in the present paper. Readers interested in the theory of equational reasoning and narrowing for higher-order rewriting are referred to =-=[23, 30]-=- and the literature mentioned there. Further, results concerning confluence and termination can be found in detail in [16, 25, 19, 29, 32]. Acknowledgements. I am grateful to Jan Willem Klop and Vince... |

14 |
Unification of higher-order patterns in linear time and
- Qian
- 1996
(Show Context)
Citation Context ...r is technically more complicated. Also unification of higher-order patterns is technically more complicated than that of first-order terms; it still decidable [21] and moreover has linear complexity =-=[31]-=-. Here we do not give a completely formal definition of critical pair, which can be found for instance in [22, 18]. Definition 1. 1. A rewrite rule x:l ! x:r of a higher-order rewriting system is left... |

13 | Termination Proofs for Higher-Order Rewrite Systems
- Pol
- 1994
(Show Context)
Citation Context ... first-order term rewriting system is terminating if and only if it has a termination model. This result provides a complete but not algorithmic method for proving termination. Van De Pol presents in =-=[28, 29]-=- a generalisation of this result to the higherorder case. This generalisation makes use of ideas that are also present in the work by Gandy [7] and De Vrijer [35, 36] on termination of simply typed - ... |

3 |
Oostrom. Normalisation in weakly orthogonal rewriting
- van
- 1999
(Show Context)
Citation Context ...pp(z; x)) ! z are not fully extended. Recall that z:f(x:z(x; y)) and z:f(x:z(x; x)) are not even rule-patterns. Now we come back to the restriction imposed on critical pairs. Recently, it is shown in =-=[27]-=- that outermost-fair rewriting is normalising for weakly orthogonal rewriting systems, where arbitrary trivial pairs are allowed. It seems that the restriction to trivial critical pairs cannot be rela... |

1 |
Termination proofs in an abstract setting
- Kahrs
(Show Context)
Citation Context ...ve the following [28, 29]. Theorem 5. A higher-order rewriting system is terminating if it has a termination model. The reverse of the statement does not hold; a simplification of a counterexample in =-=[14]-=- is given in [29]. The restriction that left-hand sides of the rewrite rules should be patterns is not necessary for the proof of the theorem. Further, a corollary is that the combination of a termina... |