# Combinatory Reduction Systems with Explicit Substitution that Preserve Strong Nomalisation

@inproceedings{Bloo1996CombinatoryRS, title={Combinatory Reduction Systems with Explicit Substitution that Preserve Strong Nomalisation}, author={Roel Bloo and Kristoffer H{\o}gsbro Rose}, booktitle={RTA}, year={1996} }

We generalise the notion of explicit substitution from the λ-calculus to higher order rewriting, realised by combinatory reduction systems (CRSs). For every confluent CRS, R, we construct an explicit substitution variant, Rx, which we prove confluent.

## 30 Citations

X.R.S : Explicit Reduction Systems - A First-Order Calculus for Higher-Order Calculi

- Mathematics, Computer ScienceCADE
- 1998

The σ⇑-calculus is used as the substitution mechanism of general higher-order systems which the authors will name Explicit Reduction Systems and general conditions to define a confluent XRS are given.

Explicit Substitutions for the Lambda-Calculus

- MathematicsALP/HOA
- 1997

It is shown that λΔexp preserves strongnormalisation, which provides the first example -moreover a very natural one indeed of explicit substitution calculus which is not structure-preserving and has the preservation of strong normalisation property.

Expression Reduction Systems and Extensions: An Overview

- MathematicsProcesses, Terms and Cycles
- 2005

The technique develops an isomorphic model of ERSs with variable names, based on de Bruijn indices, which is translated into equational first-order rewriting.

Director Strings Revisited: A Generic Approach to the Efficient Representation of Free Variables in Higher-order Rewriting

- Computer ScienceJ. Log. Comput.
- 2005

This work gives an innovative, although very natural, representation of variables abstract enough to fit in many different frameworks and more satisfactory from an operational perspective than usual representations.

Explicit substitutions for control operators ?

- Mathematics
- 1997

The-calculus is a-calculus with a local operator closely related to normal-isation procedures in classical logic and control operators in functional programming. We introduce exp, an explicit…

CINNI – A Generic Calculus of Explicit Substitutions and its Application to λ-, ς- and π-Calculi

- Computer Science
- 2005

The solution is based on CINNI, a new calculus of explicit substitutions that makes use of a term representation that contains both the standard named notation and de Bruijn’s indexed notation as special subcases.

Comparing and implementing calculi of explicit substitutions with eta-reduction

- Computer ScienceAnn. Pure Appl. Log.
- 2005

Pattern matching as cut elimination

- Computer ScienceProceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158)
- 1999

We present a typed pattern calculus with explicit pattern matching and explicit substitutions, where both the typing rules and the reduction rules are modeled on the same logical proof system, namely…

Explicit Substitutions for Objects and Functions

- Computer ScienceJ. Funct. Log. Program.
- 1999

This paper proposes an implementation of objects and functions via a calculus with explicit substitutions which is confluent and preserves strong normalization. The source calculus corresponds to the…

## References

SHOWING 1-10 OF 26 REFERENCES

Explicit substitutions

- MathematicsPOPL '90
- 1989

The λ&sgr;-calculus is a refinement of the λ-calculus where substitutions are manipulated explicitly. The λ&sgr;-calculus provides a setting for studying the theory of substitutions, with pleasant…

Explicit Cyclic Substitutions

- Computer ScienceCTRS
- 1992

It is demonstrated how this may be used to describe standard binding constructions (let and letrec)—directly using substitution and fixed point induction as well as using ‘small-step’ rewriting semantics where substitution is interleaved with the mechanics of the following β-reductions.

Combinatory Reduction Systems: Introduction and Survey

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 1993

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.

Lambda Calculus with Explicit Recursion

- Computer ScienceInf. Comput.
- 1997

A family of cyclic?-graph calculi that support the letrec, a feature that is essential to reason about time and space behavior of functional languages and also about compilation and optimizations of programs.

The lambda calculus - its syntax and semantics

- MathematicsStudies in logic and the foundations of mathematics
- 1985

Higher-order critical pairs

- Computer Science[1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science
- 1991

The notion of critical pair is generalized to higher-order rewrite systems, and the analog of the critical pair lemma is proved.

Cyclic lambda graph rewriting

- Computer ScienceProceedings Ninth Annual IEEE Symposium on Logic in Computer Science
- 1994

This paper indicates how Wadsworth's (1978) interpreter can be simulated in the /spl lambda/-graph rewrite rules that are proposed, and is not concerned with optimality questions for acyclic /spl Lambda/-reduction.