On the Open Problems Concerning Church-Rosser of Left-Linear Term Rewriting Systems(<Special Section>Foundations of Computer Science)

概要

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G. Huet (1980) showed that a left-linear term-rewriting system (TRS) is Church-Rosser (CR) if P 〓 Q for every critical pair <P, Q> where P 〓 Q is a parallel reduction from P to Q. But, it remains open whether it is CR when Q 〓 P for every critical pair <P, Q>. In this paper, we give a partial solution to this problem, that is, a left-linear TRS is CR if Q 〓^^W P for every critical pair <P, Q> where Q 〓^^W P is a parallel reduction with the set W of redex occurrences satisfying that if the critical pair is generated from two rules overlapping at an occurrence u, then the length |w| ≤ |u| for every w ∈ W. We also show that a left-linear TRS is CR if Q →^^<≯u>__= P for every critical pair <P, Q> generated from two rules overlapping at an occurrence u. Here, Q →^^<≯u>__= P is either Q = P or a reduction whose redex occurrence is not greater than u. This condition is incomparable with the previous one and a partial solution to the open problem of ascertaining whether it is CR when Q →__= P for every critical pair <P, Q>.
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2004-02-01