The Orbit Problem consists of determining, given a matrix
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
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 succinct invariants of polynomial size.
Our results imply that the class of closed semialgebraic invariants is closure-complete: there exists a closed semialgebraic invariant if and only if y is not in the topological closure of the orbit of x under A.
Theory of Computing Systems, 2019. 23 pages.
© 2019 Nathanaël Fijalkow, Pierre Ohlmann, Joël
Ouaknine, Amaury Pouly, and James Worrell.