| 总结: | <p>The Orbit Problem consists of determining, given a linear transformation A on Qd, together with vectors x and y, whether the orbit of x under repeated applications of A can ever reach y. This problem was famously shown to be decidable by Kannan and Lipton in the 1980s.</p> <br/> <p>In this paper, we are concerned with the problem of synthesising suitable invariants P ⊆ Rd, i.e., sets that are stable under A and contain x and not y, thereby providing compact and versatile certificates of non-reachability. We show that whether a given instance of the Orbit Problem admits a semialgebraic invariant is decidable, and moreover in positive instances we provide an algorithm to synthesise suitable invariants of polynomial size.</p> <br/> <p>It is worth noting that the existence of semilinear invariants, on the other hand, is (to the best of our knowledge) not known to be decidable.</p>
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