As a window into the subject, we recount some of the history (and geography) of two mature, challenging, partially open, partially closed problems in the theory of rewriting (numbers 13 and 21 from the original
RTA List of Open Problems
). One problem deals with (criteria for left-linear) confluence and the other with termination (of one linear or string rule), the two paradigmatic properties of interest for rewrite systems of any flavor. Both problems were formulated a relatively long time ago, have seen considerable progress, but remain open. We also venture to contemplate the future evolution and impact of these investigations.
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