A New Parallel Closed Condition for Church-Rossser of Left-Linear Term Rewriting Systems

G.Huet (1980) showed that a left-linear term-rewriting system (TRS) is Church-Rosser (CR) if for every critical pair where is a parallel reduction from P to Q. But, it remains open whether it is CR when for every critical pair . In this paper, we give a partial solution to this problem, that is, a left-linear TRS is CR if for every critical pair where 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. Furthermore, a left-linear TRS is CR if or \(P\xrightarrow{\varepsilon }Q\)for every critical pair where W satisfies the same condition as the above and \(P\xrightarrow{\varepsilon }Q\) is a reduction whose redex occurrence is e (i.e., the root).