Combinatory Reduction Systems with Explicit Substitution that Preserve Strong Nomalisation

  title={Combinatory Reduction Systems with Explicit Substitution that Preserve Strong Nomalisation},
  author={Roel Bloo and Kristoffer H{\o}gsbro Rose},
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. 
X.R.S : Explicit Reduction Systems - A First-Order Calculus for Higher-Order Calculi
  • B. Pagano
  • Mathematics, Computer Science
  • 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
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
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
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 ?
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
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.
Pattern matching as cut elimination
  • S. CerritoD. Kesner
  • Computer Science
    Proceedings. 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
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
A rho-Calculus of Explicit Constraint Application


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
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
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
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
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  • 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
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
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.