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Combinatory Reduction Systems: introduction and survey
 THEORETICAL COMPUTER SCIENCE
, 1993
"... Combinatory Reduction Systems, or CRSs for short, were designed to combine the usual firstorder format of term rewriting with the presence of bound variables as in pure λcalculus and various typed calculi. Bound variables are also present in many other rewrite systems, such as systems with simpl ..."
Abstract

Cited by 84 (9 self)
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Combinatory Reduction Systems, or CRSs for short, were designed to combine the usual firstorder format of term rewriting with the presence of bound variables as in pure λcalculus and various typed calculi. Bound variables are also present in many other rewrite systems, such as systems with simplification rules for proof normalization. The original idea of CRSs is due to Aczel, who introduced a restricted class of CRSs and, under the assumption of orthogonality, proved confluence. Orthogonality means that the rules are nonambiguous (no overlap leading to a critical pair) and leftlinear (no global comparison of terms necessary). We introduce the class of orthogonal CRSs, illustrated with many examples, discuss its expressive power, and give an outline of a short proof of confluence. This proof is a direct generalization of Aczel's original proof, which is close to the wellknown confluence proof for λcalculus by Tait and MartinLof. There is a wellknown connection between the para...
Four Equivalent Equivalences of Reductions
, 2002
"... Two coinitial reductions in a term rewriting system are said to be equivalent if they perform the same steps, albeit maybe in a di#erent order. We present four characterisations of such a notion of equivalence, based on permutation, standardisation, labelling and projection, respectively. We prove ..."
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Two coinitial reductions in a term rewriting system are said to be equivalent if they perform the same steps, albeit maybe in a di#erent order. We present four characterisations of such a notion of equivalence, based on permutation, standardisation, labelling and projection, respectively. We prove that the characterisations all yield the same notion of equivalence, for the class of firstorder leftlinear term rewriting systems. A crucial role in our development is played by the notion of a proof term. 1