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. 
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References

SHOWING 1-10 OF 26 REFERENCES
Explicit substitutions
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
TLDR
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
Preservation of strong normalisation in named lambda calculi with explicit substitution and garbage collection
TLDR
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
TLDR
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
  • H. Barendregt
  • Mathematics
    Studies in logic and the foundations of mathematics
  • 1985
Higher-order critical pairs
  • T. Nipkow
  • Computer Science
    [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science
  • 1991
TLDR
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
  • Z. M. AriolaJ. Klop
  • Computer Science
    Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science
  • 1994
TLDR
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.
...
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