# Uniform Normalisation beyond Orthogonality

@inproceedings{Khasidashvili2001UniformNB, title={Uniform Normalisation beyond Orthogonality}, author={Zurab Khasidashvili and Mizuhito Ogawa and Vincent van Oostrom}, booktitle={RTA}, year={2001} }

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 λ/-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…

## 14 Citations

### Vicious circles in rewriting systems

- Mathematics
- 2004

textabstractWe continue our study of the difference between Weak Normalisation (WN) and Strong Normalisation (SN). We extend our earlier result that orthogonal TRSs with the property WN do not admit…

### Equivalence of Reductions in Higher-Order Rewriting

- Mathematics, Computer Science
- 2008

The main result of this dissertation is that for local, orthogonal HRSs the three notions of equivalence coincide, and the proof that H RSs enjoy finite family developments, meaning that in infinite reductions, there is no bound on the number of steps which were involved in creating a given symbol.

### Perpetuality for Full and Safe Composition (in a Constructive Setting)

- MathematicsICALP
- 2008

The strong normalisation proof is based on implicit substitution rather than explicit substitution, so that it turns out to be modular w.r.t. the well-known proofs for typed lambda-calculus.

### The Theory of Calculi with Explicit Substitutions Revisited

- Computer ScienceCSL
- 2007

Very simple technology is used to establish a general theory of explicit substitutions for the lambda-calculus which enjoys fundamental properties such as simulation of one-step beta-reduction, confluence on metaterms, preservation of beta-strong normalisation, strong normalisation of typed terms and full composition.

### Weak Convergence and Uniform Normalization in Infinitary Rewriting

- MathematicsRTA
- 2010

It is shown that if a weakly convergent reduction is not strongly convergent, it contains a term that reduces to itself in one step (but the step itself need not be part of the reduction), and it is proved that for any orthogonal system with finitely many rules, the system is weakly normalizing under weak convergence.

### Extending the Explicit Substitution Paradigm

- EconomicsRTA
- 2005

We present a simple term language with explicit operators for erasure, duplication and substitution enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic's…

### Delimiting diagrams

- Mathematics
- 2004

We propose a way in which a strategy may exceed another one. In particular, we say that a strategy uniformly exceeds another one, if every maximal reduction for the former is as least as long as…

### Metamathematics in Coq

- Mathematics
- 2003

Chapter 1: Automated Proof Construction in Type Theory using Resolution.
We describe techniques to integrate resolution logic in type
theory. Refutation proofs obtained by resolution are translated…

### From Higher-Order to First-Order Rewriting

- Computer ScienceRTA
- 2001

A characterization of the class of higher-order rewriting systems which can be encoded by first- order rewriting modulo an empty theory (that is, Ɛ = θ), which includes of course the λ-calculus.

## References

SHOWING 1-10 OF 40 REFERENCES

### Development Closed Critical Pairs

- Computer ScienceHOA
- 1995

This work extends results from Huet and Toyama on left-linear first-order term rewriting systems by replacing the parallel closed condition by a development closed condition, yielding a confluence criterion for Klop's combinatory reduction systems), Khasidashvili's expression reduction systems, and Nipkow's higher-order pattern rewriting systems.

### Perpetuality and Uniform Normalization in Orthogonal Rewrite Systems

- MathematicsInf. Comput.
- 2001

We study perpetuality of reduction steps, as well as perpetuality of redexes, in orthogonal rewrite systems. A perpetual step is a reduction step which retains the possibility of infinite reductions.…

### On the longest perpetual reductions in orthogonal expression reduction systems

- MathematicsTheor. Comput. Sci.
- 2001

### Strong sequentiality of left-linear overlapping term rewriting systems

- Mathematics[1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science
- 1992

It is shown that index reduction is normalizing for the class of strongly sequential left-linear term rewriting systems in which every critical pair can be joined with root balanced reductions.

### Normalisation in Weakly Orthogonal Rewriting

- MathematicsRTA
- 1999

(Infinitary) normalisation is established and a counterexample against head-normalisation is given for the larger class of weakly orthogonal rewrite systems.

### The Longest Perpetual Reductions in Orthogonal Expression Reduction Systems

- Computer ScienceLFCS
- 1994

A strategy is designed that for any given term t constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise, and the Conservation Theorem for OERSs follows easily from the properties of the strategy.

### Conflunt reductions: Abstract properties and applications to term rewriting systems

- Computer Science18th Annual Symposium on Foundations of Computer Science (sfcs 1977)
- 1977

This paper gives new results, and presents old ones in a unified formalism, concerning Church-Rosser theorems for rewriting systems, and shows how these results yield efficient methods for the mechanization of equational theories.

### lambda-nu, A Calculus of Explicit Substitutions which Preserves Strong Normalisation

- Computer ScienceJ. Funct. Program.
- 1996

Abstract Explicit substitutions were proposed by Abadi, Cardelli, Curien, Hardin and Lévy to internalise substitutions into λ-calculus and to propose a mechanism for computing on substitutions. λν is…

### Perpetuality in a named lambda calculus with explicit substitutions

- MathematicsMathematical Structures in Computer Science
- 2001

To complete the study of Fes, the property of subject reduction is shown to hold by extending type assignments of the typing rules to allow non-pure types (types with possible occurrences of the type substitution operator).

### On Termination and Confluence Properties of Disjoint and Constructor-Sharing Conditional Rewrite Systems

- MathematicsTheor. Comput. Sci.
- 1996