Abstract
It is known that the left-conjugacy problem and the conjugacy problem are decidable for each finite, length-reducing, and complete rewriting system. This does not hold for the problem of cyclic equality as is shown here by presenting a finite, length-reducing, and complete rewriting system with an undecidable problem of cyclic equality. Further, a finite complete rewriting system T is constructed such that the left-conjugacy problem and the conjugacy problem are undecidable for T.
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