## Oostrom, Uniform normalisation beyond orthogonality (2001)

Venue: | Proceedings of the Twelfth International Conference on Rewriting Techniques and Applications (RTA ’01), Lecture Notes in Computer Science (2001 |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Khasidashvili01oostrom,uniform,

author = {Zurab Khasidashvili and Mizuhito Ogawa and Vincent Van Oostrom},

title = {Oostrom, Uniform normalisation beyond orthogonality},

booktitle = {Proceedings of the Twelfth International Conference on Rewriting Techniques and Applications (RTA ’01), Lecture Notes in Computer Science (2001},

year = {2001},

pages = {122--136},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A well-known fact is uniform normalisation of orthogonal non-erasing term rewrite systems, e.g. the λI-calculus. In the present paper both restrictions are analysed. Orthogonality is seen to pertain to the linear part and non-erasingness to the non-linear part of rewrite steps. Based on this analysis, a modular proof method for uniform normalisation is presented which allows to go beyond orthogonality. The method is shown applicable to biclosed first- and second-order term rewrite systems as well as to a λ-calculus with explicit substitutions. 1

### Citations

1119 | The Lambda Calculus: Its Syntax and Semantics - Barendregt - 1984 |

567 | Term Rewriting Systems
- Klop
- 1992
(Show Context)
Citation Context ...sed systems, the results obtained are not novel (cf. [15, Lem. 8.11.3.2] and [9, Sect. 3.3]). The reader is assumed to be familiar with first-order term rewrite systems (TRSs) as can be found in e.g. =-=[15]-=- or [1]. We summarise some aberrations and additional concepts: Definition 6. – A term is linear if any variable occurs at most once in it. Let ϱ : l → r be a TRS rule. It is left-linear ( right-linea... |

365 | Con reductions: Abstract properties and applications to term rewriting systems - Huet - 1980 |

323 | The Calculi of Lambda Conversion - Church - 1941 |

244 |
Combinatory Reduction Systems
- Klop
- 1980
(Show Context)
Citation Context ...an be found as [12, Thm. 60] and [14, Cor. II.5.9.4], respectively. The reader is assumed to be familiar with secondorder term rewrite systems be it in the form of combinatory reduction systems (CRSs =-=[14]-=-), expression reduction systems (ERSs [13]), or higher-order pattern rewrite systems (PRSs [17]). We employ PRSs as defined in [17], but will write x.s instead of λx.s, thereby freeing the λ for usage... |

99 |
λυ, a calculus of explicit substitutions which preserves strong normalisation
- Benaissa, Briaud, et al.
- 1993
(Show Context)
Citation Context ...res. Since x is terminating S must contain infinitely many Beta-steps. Since S is pretty S↓ x is an infinite β-reduction from s by (the remark after) Lem. 5. ⊓⊔ Our method relates to closure-tracking =-=[3]-=- as preventing to curing. Trying to apply it to prove [4, Conj. 6.45], stating that explicification of redex preserving CRSs is PSN, led to the following counterexample. Example 3. Consider the term s... |

84 | Computations in orthogonal rewriting systems - Huet, Lévy - 1991 |

73 |
Confluence for Abstract and Higher-Order Rewriting
- Oostrom
- 1994
(Show Context)
Citation Context ...Note that f(x) ∈ SN, but by contracting the 〈 := 〉-redex a is substituted for x and f(a) ∈ ∞. Definition 12. An occurrence of (the head symbol of) a subterm is potentially infinite if some descendant =-=[20]-=- of it along some reduction is in ∞. A step is ∞-erasing if it erases all potentially infinite subterms in its arguments. For TRSs this notion of ∞-erasingness coincides with the one of Def. 9. Coroll... |

63 | T.: Higher-order rewrite systems and their confluence
- Mayr, Nipkow
- 1998
(Show Context)
Citation Context ... familiar with secondorder term rewrite systems be it in the form of combinatory reduction systems (CRSs [14]), expression reduction systems (ERSs [13]), or higher-order pattern rewrite systems (PRSs =-=[17]-=-). We employ PRSs as defined in [17], but will write x.s instead of λx.s, thereby freeing the λ for usage as a function symbol. Definition 10. – The order of a rewrite rule is the maximal order of the... |

54 | From proof nets to interaction nets
- Lafont
- 1995
(Show Context)
Citation Context ...al, the results in this section and their proofs form the heart of the following sections. Moreover, they are applicable to various concrete (linear) rewrite systems, for instance to interaction nets =-=[16]-=-. The reader is assumed to be familiar with abstract rewrite systems (ARSs, [15, Chap. 1] or [1, Chap. 2]). Definition 1. Let a be an object of an abstract rewrite system. a is terminating (strongly n... |

41 | Preservation of Termination for Explicit Substitution
- Bloo
- 1997
(Show Context)
Citation Context ... and the proof of F 2TP goes through unchanged. ⊓⊔s132 Z. Khasidashvili, M. Ogawa, and V. van Oostrom 5 λx − In this section familiarity with the nameful λ-calculus with explicit substitutions λx− of =-=[4]-=- is assumed. We define it as a P2RS and establish the fundamental theorem of perpetuality for λx− : Theorem 11 (FxTP). Non-erasing steps are perpetual in λx− . Definition 13. The alphabet of λx− [4] c... |

32 |
Confluence and Normalisation for higher-order rewriting. Thèse de doctorat, Vrije Universiteit
- Raamsdonk
- 1996
(Show Context)
Citation Context ... 3 The restriction to P2RSs entails no restriction w.r.t. the other formats, since both CRSs and ERSs can be embedded into P2RSs, by coding metavariables in rules as free variables of type o→ ...→o→o =-=[23]-=-. To adapt the proof of F1TP to P2RSs, we review its two main ingredients. The first one was a notion of simultaneous reduction, extending one-step reduction such that: – The residual of a non-erasing... |

31 | Description abstraite des systèmes de réécriture - Melliès - 1996 |

25 | Termination and confluence properties of structured rewrite systems
- Gramlich
- 1996
(Show Context)
Citation Context ...tter case, si → si+1 may take place in an erased argument, and si+1 →p ti+1 = ti. But since all arguments to l are SN, this can happen only finitely often and eventually the first case applies. ⊓⊔ In =-=[9]-=- a uniform normalisation result not requiring left-linearity, but having a critical pair condition incomparable to biclosedness was presented. 4 Second-Order Term Rewriting In this section, the fundam... |

24 | Strong sequentiality of left-linear overlapping term rewriting systems - Toyama - 1992 |

18 | The longest perpetual reductions in orthogonal expression reduction systems - Khasidashvili - 1994 |

10 |
Paul-André Melliès. An abstract standardisation theorem
- Gonthier, Lévy
- 1992
(Show Context)
Citation Context ...transformation preserves infiniteness. Proof. The first part of the theorem was shown to hold for orthogonal TRSs in [11, Thm. 3.19] and extended to left-linear TRSs possibly having critical pairs in =-=[8]-=-. That standardisation preserves infiniteness follows from the fact that at some moment along an infinite reduction S : s0 → s1 → ... a redex at minimal position p w.r.t. the prefix order ≤ [1, Def. 3... |

8 |
Term Rewriting and All That. CUP
- Baader, Nipkow
- 1998
(Show Context)
Citation Context ...ems, the results obtained are not novel (cf. [15, Lem. 8.11.3.2] and [9, Sect. 3.3]). The reader is assumed to be familiar with first-order term rewrite systems (TRSs) as can be found in e.g. [15] or =-=[1]-=-. We summarise some aberrations and additional concepts: Definition 6. – A term is linear if any variable occurs at most once in it. Let ϱ : l → r be a TRS rule. It is left-linear ( right-linear) ifl ... |

8 |
The Ant-Lion paradigm for strong normalization
- Bohm, Intrigila
- 1994
(Show Context)
Citation Context ... perpetual in biclosed P2RSs. Many variations of this result are possible. We mention two. First, the motivation for this paper originates with [13, Sect. 6.4], where we failed to obtain: Theorem 9. (=-=[5]-=-) λ-δK-calculus is uniformly normalising. Proof. By Cor. 8, since λ-δK-calculus is weakly orthogonal. ⊓⊔ Second, we show that non-fully-extended P2RSs may have uniform normalisation. By the same metho... |

8 | Normalisation in weakly orthogonal rewriting - Oostrom - 1999 |

7 | Oostrom. Perpetuality and uniform normalization in orthogonal rewrite systems
- Khasidashvili, Ogawa, et al.
- 2001
(Show Context)
Citation Context ...alising [7, p. 20, 7 XXV], and – non-erasing steps are perpetual in orthogonal TRSs [14, Thm. II.5.9.6]. In previous work we have put these results and many variations on them in a unifying framework =-=[13]-=-. At the heart of that paper is the result (Thm. 3.16) that a term s not in normal form contains a redex which is external for any reduction from s. 1 Since external redexes need not exist in rewrite ... |

5 |
Perpetuality in a named lambda calculus with explicit substitutions
- Bonelli
(Show Context)
Citation Context ...for biclosed rewrite systems (e.g. Cor. 2, 5, 6, and 8). In Sect. 5 uniform normalisation for λx − , a prototypical λ-calculus with explicit substitutions, is shown to hold, extending earlier work of =-=[6]-=- who only shows it for the explicit substitution part x of the calculus. The proof boils down to an analysis of the (only) critical pair of λx − and uses a particularly simple proof of preservation of... |

5 | Development closed critical pairs - Oostrom - 1995 |

5 | Effective Longest and Infinite Reduction Paths in Untyped λ-Calculi - Sørensen - 1996 |

4 | Standardization and evaluation in Combinatory Reduction Systems
- Wells, Muller
- 2000
(Show Context)
Citation Context ...mation preserves infiniteness. Proof. The proof of the second part of the theorem is as for TRSs. For a proof of the first part for left-linear fully-extended (orthogonal) CRSs see [18, Sect. 7.7.3] (=-=[26]-=-). By the correspondence between CRSs and P 2RSs this suffices for our purposes. (STD even holds for PRSs [22, Cor. 1.5].) ⊓⊔ Proof. (of Thm. 6) Replace in the proof of Thm. 2 everywhere −→ � by −→. ◦... |

3 | Higher-order rewrite systems and their con - Mayr, Nipkow - 1998 |

2 | Termination and con properties of structured rewrite systems - Gramlich - 1996 |

1 | Eective longest and in reduction paths in untyped lambdacalculi - Srensen - 1996 |