# On Stepwise Explicit Substitution

@article{Kamareddine1993OnSE, title={On Stepwise Explicit Substitution}, author={Fairouz Kamareddine and R. P. Nederpelt}, journal={Int. J. Found. Comput. Sci.}, year={1993}, volume={4}, pages={197-240} }

This paper starts by setting the ground for a lambda calculus notation that strongly mirrors the two fundamental operations of term construction, namely abstraction and application. In particular, we single out those parts of a term, called items in the paper, that are added during abstraction and application. This item notation proves to be a powerful device for the representation of basic substitution steps, giving rise to different versions of β-reduction including local and global…

## 54 Citations

Reductions, intersection types, and explicit substitutions

- MathematicsMathematical Structures in Computer Science
- 2003

It is shown that the terms that normalise by leftmost reduction and the terms by head reduction can each be characterised as the terms typable in a certain system, and refine and strengthen the classical normalisation theorems.

Generalized -reduction and Explicit Substitutions

- Mathematics
- 1996

Extending the-calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently which still has many open problems. Due to this reason, the…

A Flexible Framework for Visualisation of Computational Properties of General Explicit Substitutions Calculi

- Computer ScienceLSFA
- 2010

Comparing and implementing calculi of explicit substitutions with eta-reduction

- Computer ScienceAnn. Pure Appl. Log.
- 2005

Preservation of strong normalisation in named lambda calculi with explicit substitution and garbage collection

- Computer Science
- 1995

It is shown that xgc is a conservative extension which preserves strong normalisation (PSN) of the untyped-calculus, which has two distinguishing features: rst, it retains the use of traditional variable names, specifying terms modulo renaming; this simpliies the reduction system.

Extending a λ-calculus with explicit substitution which preserves strong normalisation into a confluent calculus on open terms

- MathematicsJournal of Functional Programming
- 1997

This paper extends the λ-calculus with explicit substitutions to a confluent calculus on open terms and considers that this extended method is a useful tool to obtain confluence when strong normalisation of the subcalculus of substitutions is not available.

Comparing Calculi of Explicit Substitutions with Eta-reduction

- MathematicsElectron. Notes Theor. Comput. Sci.
- 2002

A useful ?-notation

- Computer Science
- 1996

This article shows that deenitions such as compatibility, the heart of a term and-redexes become simpler in item notation and shows that with this item notation, reduction can be generalised in a nice way.

Explicit Substitution On the Edge of Strong Normalization

- Computer ScienceTheor. Comput. Sci.
- 1999

## References

SHOWING 1-10 OF 36 REFERENCES

Generalizing Automath by means of a lambda-typed lambda calculus

- Computer Science, Mathematics
- 1987

Some category theoretical properties related to a model for a polymorphic lambda-calculus

- Mathematics
- 1989

H.M.M. Ten Eikelder C. Hemerik A model for a second order polymorphic lambda calculus is sketched. Some category theoretical questions appearing in the model construction are extensively treated.…

The mathematical language AUTOMATH, its usage, and some of its extensions

- Computer Science
- 1970

The possibilities of superimposed languages, automatic theorem proving, and extensions of Automath are described in this chapter.

Semantics in a frege structure

- Philosophy
- 1988

A perspective in which the semantics of natural language constructs are unpacked in terms of Peter Aczel's Frege structures is offered and is shown to provide promising results for both nominalisation and intensionality.

The lambda calculus - its syntax and semantics

- MathematicsStudies in logic and the foundations of mathematics
- 1985