Joël Ouaknine

Piotr Bacik, Joël Ouaknine, David Purser et al.

The Skolem Problem asks to determine whether a given linear recurrence sequence (LRS) has a zero term. Showing decidability of this problem is equivalent to giving an effective proof of the Skolem-Mahler-Lech Theorem, which asserts that a non-degenerate LRS has finitely many zeros. The latter result was proven over 90 years ago via an ineffective method showing that such an LRS has only finitely many $p$-adic zeros. In this paper we consider the problem of determining whether a given LRS has a $p$-adic zero, as well as the corresponding function problem of computing exact representations of all $p$-adic zeros. We present algorithms for both problems and report on their implementation. The output of the algorithms is unconditionally correct, and termination is guaranteed subject to the $p$-adic Schanuel Conjecture (a standard number-theoretic hypothesis concerning the $p$-adic exponential function). While these algorithms do not solve the Skolem Problem, they can be exploited to find natural-number and rational zeros under additional hypotheses. To illustrate this, we apply our results to show decidability of the Simultaneous Skolem Problem (determine whether two coprime linear recurrences have a common natural-number zero), again subject to the $p$-adic Schanuel Conjecture.

Nathanaël Fijalkow, Joël Ouaknine, Amaury Pouly et al.

We consider the decidability of state-to-state reachability in linear time-invariant control systems over discrete time. We analyse this problem with respect to the allowable control sets, which in general are assumed to be defined by boolean combinations of linear inequalities. Decidability of the version of the reachability problem in which control sets are affine subspaces of $\mathbb{R}^n$ is a fundamental result in control theory. Our first result is that reachability is undecidable if the set of controls is a finite union of affine subspaces. We also consider versions of the reachability problem in which (i)~the set of controls consists of a single affine subspace together with the origin and (ii)~the set of controls is a convex polytope. In these two cases we respectively show that the reachability problem is as hard as Skolem's Problem and the Positivity Problem for linear recurrence sequences (whose decidability has been open for several decades). Our main contribution is to show decidability of a version of the reachability problem in which control sets are convex polytopes, under certain spectral assumptions on the transition matrix.

Piotr Bacik, Joris Nieuwveld, Joël Ouaknine et al.

We consider expansions of Presburger arithmetic with families of monadic polynomial predicates. (Examples of such predicates are the set of perfect squares, or the set of integers of the form $2n^3-5n+3$, etc.) Although the full attendant first-order theories are well known to be undecidable, very little is known when one restricts the number of variables. In the case of single-variable theories, we obtain positive results for the following two families of predicates: (i) for perfect fixed powers, decidability ofthe corresponding theory follows from the solvability of hyperellipticDiophantine equations; and (ii) for polynomials of degree at most three, we establish decidability by relying on the low genus of the resulting algebraic curves. Finally, we discuss limitations and hardness results (via encodings of longstanding open Diophantine problems) as soon as any of the above restrictions are lifted.