Decidable/Undecidable Properties of Conditional Term Rewriting Systems

Decidability/undecidability of properties in subclasses of conditional term rewriting systems (CTRSs, for short) of natural, join, and oriented types are studied. The properties focused are convertibility, reachability, joinability, con uence, and termination. In particular, it is shown that (1) all of the properties listed above are undecidable for left-linear right-ground CTRSs of each types having at most one leftlinear right-ground condition for each rules; (2) all of the properties listed above are decidable for left-linear right-ground CTRSs of each types having ground conditions; (3) termination (con uence) is decidable for right-ground (left-linear right-ground, respectively) CTRSs of join and oriented types having at most one left-linear right-ground constructor condition for each rules; and (4) termination is decidable for left-linear right-ground growing CTRSs of oriented type having at most one left-linear right-ground condition for each rules. These results are closely related to known decidability/undecidability results in (conditional) term rewriting.