## Axiomatic Rewriting Theory I - A Diagrammatic Standardization Theorem (2001)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Mellies01axiomaticrewriting,

author = {Paul-Andre Mellies and Equipe Preuves},

title = {Axiomatic Rewriting Theory I - A Diagrammatic Standardization Theorem},

year = {2001}

}

### OpenURL

### Abstract

Machine translation ## -calculus interpretation ## -calculus Formally, the -calculus contains two classes of objects: terms and substitutions. Terms are written in the de Bruijn notation.

### Citations

1131 |
The Lambda-Calculus, its Syntax and Semantics
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- 1984
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Citation Context ...ransition system, in order to capture the analogy. The refinement is based on the concept of redex permutation introduced by J.-J. Lévy in his work on the�-calculus and on term rewriting systems, see =-=[24, 18, 3]-=-. Permuting redexes inside rewriting paths enables to express by local transformations that two different rewriting paths compute the same events, but in a different order. Typically, the transition s... |

958 |
Term Rewriting and All That
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Citation Context ...der. The permutation is thus neutral from the point of view of standardization. ���¡¡� ¡� �����¡¡� ��� ��� ��� �������¡¡� ¡� ���������¡¡� ������ �� �������� ��� ��� ��� ������ �� �������� Permutation =-=[2]-=- is called irreversible because it replaces the “inside-out” computation ¡�¡� or��¡� by its “outside-in” equivalent�or� — thus strictly improving the computation from the point of view of standardizat... |

937 |
Categories for the working mathematicians
- Lane
- 1971
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Citation Context ...een the two rewriting paths� and�. In this section is to show that all tilings� ���from a path� to its standard path�, are equivalent in an intuitive sense. We refer the reader to the last chapter of =-=[25]-=- (second edition) for a nice and motivated introduction to 2-categories. 5.1 Tiling graph, tiling paths, and partial injections To every 2-dimensional transition system����� we associate a tiling grap... |

569 | Term rewriting systems
- Klop
- 1990
(Show Context)
Citation Context ...dardization axioms hold in������� for the same reasons as in the hierarchical case. TERM REWRITING SYSTEMS. The reader interested in term rewriting systems will find an introduction to the subject in =-=[21, 19, 2, 11]-=- and a comprehensive study of standardization in [35]. Here, we recall only that 1. a term rewriting system is a pair¦ ���������������� where� is the signature of an algebra and every�� is a rewriting... |

393 | Explicit substitutions
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- 1991
(Show Context)
Citation Context ...ormations that two different rewriting paths compute the same events, but in a different order. Typically, the transition system of the and������ terms���¡¡� may be equipped with the two permutations =-=[1]-=- and [2] indicated below: ���¡¡� ¡� �����¡¡� ��� ��� ��� �������¡¡� ¡� ���������¡¡� ��� ��� ��� � ��� � ������ �� �������� ��� ��� ��� ������ �� �������� � �� ��� ��� � ��� � Consider for instance the... |

369 |
Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems
- Huet
- 1980
(Show Context)
Citation Context ...eason diagrammatically instead of syntactically, and to develop a syntax-free Rewriting Theory, based on a 2-dimensional refinement of the traditional notion of Abstract Rewriting System developed in =-=[32, 17, 21]-=-. (VAR) ����� (BETA) �������� (APP) (XI) ������ ����������� ������������ �������������� ����� ����������� ����� Fig. 1. An inductive definition of Curry and Feys’ leftmost outermost strategy. 6 (5)s1.... |

264 | The revised report on the syntactic theories of sequential control and
- Felleisen, Hieb
- 1992
(Show Context)
Citation Context ...s the ten N-axioms formulated in Section 6. The resulting standardization theorem, which is non-trivial to prove directly on the syntax, leads to Plotkin’s formalization of Landin’s SECD machine, see =-=[12]-=- for instance. EXPLICIT SUBSTITUTIONS. The usual�-reduction�������s������� copies its argument� as many times as the variable�occurs in�. This is fine theoretically, but inefficient if one wants to im... |

263 | Models for Concurrency
- Winskel, Nielsen
- 1995
(Show Context)
Citation Context ...US TRANSITION SYSTEMS. Asynchronous transition systems extend both non-deterministic transition systems, and Mazurkiewicz trace languages. They were introduced independently in [4] and [39], see also =-=[33]-=-. An asynchronous transition system� is a quintuple� ��������������� where – � is a set of states with initial state�, – � is a set of events, – ������¢�¢� is the transition relation, – ���¢� is an ir... |

123 |
On theories with a combinatorial definition of ”equivalence
- Newman
- 1942
(Show Context)
Citation Context ...eason diagrammatically instead of syntactically, and to develop a syntax-free Rewriting Theory, based on a 2-dimensional refinement of the traditional notion of Abstract Rewriting System developed in =-=[32, 17, 21]-=-. (VAR) ����� (BETA) �������� (APP) (XI) ������ ����������� ������������ �������������� ����� ����������� ����� Fig. 1. An inductive definition of Curry and Feys’ leftmost outermost strategy. 6 (5)s1.... |

74 |
1936], `Some properties of conversion
- Church, Rosser
(Show Context)
Citation Context ...s by far the most studied rewriting system in history. A remarkable illustration is the confluence theorem. The theorem was formulated by A. Church and J.B. Rosser in the early years of the�-calculus =-=[7]-=-. The theorem was then generalized and applied extensively to other rewriting systems. It became eventually an object of study in itself, in a line of research pioneered by H.-B. Curry and R. Feys in ... |

73 | Confluence for Abstract and Higher-Order Rewriting - Oostrom - 1994 |

70 |
Higher Operads, Higher Categories
- Leinster
- 2004
(Show Context)
Citation Context ...proach to Rewriting Theory is possible and fruitful. It also opens a series of interesting research directions, at the frontier of Rewriting Theory and Higher-Dimensional Categories, see for instance =-=[23]-=- and [31]. More specifically, we would like to capture properly the causal principles underlying Rewriting Systems like the�-calculus with�-reduction and�expansion, the non left-linear term rewriting ... |

45 |
Concurrent machines
- Shields
- 1985
(Show Context)
Citation Context ...lems ASYNCHRONOUS TRANSITION SYSTEMS. Asynchronous transition systems extend both non-deterministic transition systems, and Mazurkiewicz trace languages. They were introduced independently in [4] and =-=[39]-=-, see also [33]. An asynchronous transition system� is a quintuple� ��������������� where – � is a set of states with initial state�, – � is a set of events, – ������¢�¢� is the transition relation, –... |

41 |
Modèles complètement adéquats et stables des lambda-calculs typés, Thèse de doctorat d’état, Université Paris VII
- Berry
- 1979
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Citation Context ...low as its contrapose. The axiom states that the characteristic function of the event of creating the�-redex��� (or equivalently the�-redex��, or the�-redex��) is stable in the sense of G. Berry, see =-=[5]-=-. Axiom reversible-stability repeats the axiom in the reversible case. Axiom 7 (Stability) We ask that every diagram � �� ����� ����� ��� � �� �� ��� �� �� � ���������� � ����� ��� ��� ������� where��... |

35 |
Call by need computations in non-ambiguous linear term rewriting systems
- Huet, Lévy
- 1979
(Show Context)
Citation Context ...general case is even worse. A rewriting system does not enjoy any uniform orientation in general, and finding the “judicious” strategy, even if we know that it exists, is a non decidable problem, see =-=[18]-=-. Despite the apparent mess, we will initiate in this article a generic theory of orientations and causality in rewriting systems. But on what foundations? Obviously, we need to abstract away from syn... |

34 |
Confluence results for the pure strong categorical logic CCL. *-calculi as subsystems
- Hardin
- 1989
(Show Context)
Citation Context ...retation from terms to computations, and to project every ��-rewriting path� �� to a�-rewriting path���� ����� (modulo equivalence� though). Properties of the interpretation are studied thoroughly in =-=[14, 9, 40, 28]-=-. The ��-calculus is kind of hybrid between deterministic and non-deterministic rewriting systems. As a fibered system over the�-calculus, it satisfies many properties of conflict-free rewriting syste... |

31 |
Description abstraite des systèmes de réécriture
- Melliès
- 1996
(Show Context)
Citation Context ...shape again. This was a completely independent discovery originating from a long and obsessive reflexion on the diagrammatic presentation of [13]. Already in germ there and in the author’s PhD thesis =-=[27]-=- the idea emerged finally that the standardization mechanism described by G. Huet and J.-J. Lévy reduces to distinguishing two classes of permutations: – the reversible permutations — for instance, pe... |

19 | Functional back-ends within the lambda-sigma calculus
- Hardin, Maranget, et al.
- 1996
(Show Context)
Citation Context ...mentations. In the��-calculus, substitutions are explicit, they can be delayed and stored just like closures. This enables to factorize many translations from abstract machines to the �-calculus, see =-=[15]-=-. ����������� Abstract Machine ����-calculus �����������������-calculus Formally, the ��-calculus contains two classes of objects: terms and substitutions. Terms are written in the de Bruijn notation.... |

17 |
R'eductions correctes et optimales dans le -calcul. Th`ese de doctorat d'etat, Universit'e Paris VII
- L'evy
- 1978
(Show Context)
Citation Context ...ransition system, in order to capture the analogy. The refinement is based on the concept of redex permutation introduced by J.-J. Lévy in his work on the�-calculus and on term rewriting systems, see =-=[24, 18, 3]-=-. Permuting redexes inside rewriting paths enables to express by local transformations that two different rewriting paths compute the same events, but in a different order. Typically, the transition s... |

17 |
Typed lambda-calculi with explicit substitutions may not terminate
- Melliès
- 1995
(Show Context)
Citation Context ...lculus with conflicts. Besides, to add some spice, its evaluation mechanism may behave counter-intuitively, as witnessed by the author’s non-termination example of a simply-typed��-term, presented in =-=[26]-=-. For all these reasons, the��-calculus has been our training partner since the early days of the axiomatic theory. Many fundamental ideas of the theory (e.g. factorization, stability) originate from ... |

13 | Termination of term rewriting by interpretation
- ZANTEMA
- 1993
(Show Context)
Citation Context ...retation from terms to computations, and to project every ��-rewriting path� �� to a�-rewriting path���� ����� (modulo equivalence� though). Properties of the interpretation are studied thoroughly in =-=[14, 9, 40, 28]-=-. The ��-calculus is kind of hybrid between deterministic and non-deterministic rewriting systems. As a fibered system over the�-calculus, it satisfies many properties of conflict-free rewriting syste... |

11 |
Computational semantics of term rewriting systems. Algebraic methods in semantics
- Boudol
- 1986
(Show Context)
Citation Context ...ns operating in [13] was certainly too complicated. Slowly, the 2-dimensional presentation emerged, leading the author to the elementary axiomatics of this article. Twenty-five years ago, the work of =-=[18, 6]-=- on term rewriting systems revealed that the “conflict-free left-regular” rewriting systems considered earlier was the emerged part of the much wider and exciting world of causal computations. This is... |

11 | Rewrite Proofs and Computations
- Jouannaud
- 1995
(Show Context)
Citation Context ...dardization axioms hold in������� for the same reasons as in the hierarchical case. TERM REWRITING SYSTEMS. The reader interested in term rewriting systems will find an introduction to the subject in =-=[21, 19, 2, 11]-=- and a comprehensive study of standardization in [35]. Here, we recall only that 1. a term rewriting system is a pair¦ ���������������� where� is the signature of an algebra and every�� is a rewriting... |

10 |
Towards a proof theory of rewriting: the simply typed 2λ-calculus
- Hilken
- 1996
(Show Context)
Citation Context ...normal form (=standard path) in a way similar to B. Hilken when he relaxes the definition of 1-dimensional normal form, in order to characterize the��-long normal forms of simply-typed�-calculus, see =-=[16, 28]-=- and the paragraph below. 77s�-CALCULUS [ETA-EXPANSION]. B. Hilken considers the following permutation in simply-typed�-calculus with�-reduction and�-expansion, see [16]: �������������� ��� ��� � ����... |

10 | Stability and sequentiality in data flow networks - Panangaden, Shanbhogue - 1990 |

8 |
Strong normalization of substitutions
- Curien, Hardin, et al.
- 1992
(Show Context)
Citation Context ...retation from terms to computations, and to project every ��-rewriting path� �� to a�-rewriting path���� ����� (modulo equivalence� though). Properties of the interpretation are studied thoroughly in =-=[14, 9, 40, 28]-=-. The ��-calculus is kind of hybrid between deterministic and non-deterministic rewriting systems. As a fibered system over the�-calculus, it satisfies many properties of conflict-free rewriting syste... |

8 |
Axiomatic rewriting theory VI: Residual theory revisited
- Melliès
- 2002
(Show Context)
Citation Context ... Rewriting Theory is possible and fruitful. It also opens a series of interesting research directions, at the frontier of Rewriting Theory and Higher-Dimensional Categories, see for instance [23] and =-=[31]-=-. More specifically, we would like to capture properly the causal principles underlying Rewriting Systems like the�-calculus with�-reduction and�expansion, the non left-linear term rewriting systems, ... |

8 | Confluence and Normalization for Higher-Order Rewriting - Raamsdonk - 1996 |

4 | Event structures and non-orthogonal term graph rewriting
- Clark, Kennaway
- 1996
(Show Context)
Citation Context ... and establish in this way a normalization theorem for the needed strategies of the��-calculus, see [28]. DAGS. The definition of a rewriting system¦ on directed acyclic graphs (dags) may be found in =-=[8]-=-. We interpret any dag rewriting system¦ as the following axiomatic rewriting system��¦��¦�. The graph�¦ has dags and redexes of¦ as vertices and edges. Two paths� and� are related as��¦� in two cases... |

4 | Axiomatic Rewriting Theory IV: A stability theorem in Rewriting Theory
- Melliès
- 1998
(Show Context)
Citation Context ...formative in������� because every permutation being reversible, all paths are standard. However, the axiomatics itself ensures that every asynchronous system satisfies the stability theorem stated in =-=[30]-=- which describes the structure of its successful runs. PETRI NETS. The theory of Petri nets illustrates nicely the notion of asynchronous transition system. A Petri net is a quintuple� �������� pre� p... |

3 |
P.A.Mellies, \An Abstract Standardization Theorem
- Gonthier
- 1992
(Show Context)
Citation Context ...tical details of [18] that the 2-dimensional approach took shape again. This was a completely independent discovery originating from a long and obsessive reflexion on the diagrammatic presentation of =-=[13]-=-. Already in germ there and in the author’s PhD thesis [27] the idea emerged finally that the standardization mechanism described by G. Huet and J.-J. Lévy reduces to distinguishing two classes of per... |

2 |
Rewrite systems. Chap
- Dershowitz, Jouannaud
- 1990
(Show Context)
Citation Context ...dardization axioms hold in������� for the same reasons as in the hierarchical case. TERM REWRITING SYSTEMS. The reader interested in term rewriting systems will find an introduction to the subject in =-=[21, 19, 2, 11]-=- and a comprehensive study of standardization in [35]. Here, we recall only that 1. a term rewriting system is a pair¦ ���������������� where� is the signature of an algebra and every�� is a rewriting... |

2 |
Axiomatic Rewriting Theory II: The lambda-sigma-calculus enjoys finite normalisation cones
- Melliès
- 2000
(Show Context)
Citation Context |

2 | Axiomatic Rewriting Theory III: A factorisation theorem in Rewriting Theory
- Melliès
- 1997
(Show Context)
Citation Context ...¡����� ��������� ������� ������������� ��� � �������������s�s� ��� ������������� ��� Fig. 7. The 11 critical pairs of the��-calculus factorization and stability theorems established in later articles =-=[29, 30]-=-. We believe that this series of structure theorems play the same regulating role for the��-calculus as the Church-Rosser property plays traditionnaly for the�-calculus. For instance, we were able to ... |

1 |
Categories of asynchronous systems
- Bednarczyck
- 1988
(Show Context)
Citation Context ...pen Problems ASYNCHRONOUS TRANSITION SYSTEMS. Asynchronous transition systems extend both non-deterministic transition systems, and Mazurkiewicz trace languages. They were introduced independently in =-=[4]-=- and [39], see also [33]. An asynchronous transition system� is a quintuple� ��������������� where – � is a set of states with initial state�, – � is a set of events, – ������¢�¢� is the transition re... |

1 |
Combinatory Reduction Systems. Thèse de l’Université d’Utrecht, Pays-Bas
- Klop
- 1980
(Show Context)
Citation Context ...use the notation:������. Each order induces in turn its own permutation relation�����,����� and���� on the transition system��. The order considered in the literature is generally the left-order, see =-=[10, 24, 20]-=-. However, we prefer to study here the tree-order, because this seems the most natural choice after the work by G. Huet and J.-J. Lévy on term rewriting systems [18]. The two alternative orders����� a... |

1 |
Chapter 4 in the book Term Rewriting System, edited by TeReSe, Cambridge Tracts
- Orthogonality
- 2003
(Show Context)
Citation Context ...tial, and����������, 3. the�-redex�� is the (unique) residual of�after�, and the path�develops the (possibly) several residuals of� after�. [For a definition of residual and complete development, see =-=[10, 24, 18, 3, 21, 22]-=- or Section 6.] � � ��� ��� ����� ���� � � ��� Thus, every permutation������� is of the form: ���¡�� ���¡� where�and� are different�-redexes,�� is a�-redex and�is a path. The three paradigmatic exampl... |

1 |
de Vrijer. Equivalence of reductions. Chapter 8 in the book Term Rewriting System, edited by TeReSe, Cambridge Tracts
- Oostrom, R
- 2003
(Show Context)
Citation Context ... hierarchical case. TERM REWRITING SYSTEMS. The reader interested in term rewriting systems will find an introduction to the subject in [21, 19, 2, 11] and a comprehensive study of standardization in =-=[35]-=-. Here, we recall only that 1. a term rewriting system is a pair¦ ���������������� where� is the signature of an algebra and every�� is a rewriting rule on this algebra. 2. a rewriting rule����� is a ... |

1 |
Call-by-name, call-by-value, and the�-calculus
- Plotkin
- 1975
(Show Context)
Citation Context ...us — just like the treeorder����� and the left-order�����. �-CALCULUS [CALL-BY-VALUE]. A value of the�-calculus is defined either as a variable or as a�-term of the form����. G. Plotkin introduces in =-=[38]-=- the call-by-value �-calculus, whose unique��-reduction������� ������� is the�-rule restricted to value arguments� . It is not difficult to show that the��-calculus — interpreted as an axiomatic nesti... |

1 | Rewrite systems. Chap. 6 of Handbook of Theoretical Computer Science B: Formal Methods and Semantics - Dershowitz, Jouannaud - 1990 |

1 | the Association for Computing Machinery vol. 27, No 4 - Huet, Lévy - 1980 |