On the Positivity Problem for Simple Linear Recurrence Sequences

Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem} asks whether all terms of the sequence are positive. We show that, for simple LRS (those whose characteristic polynomial has no repeated roots) of order 9 or less, Positivity is decidable, with complexity in the Counting Hierarchy.Comment: arXiv admin note: substantial text overlap with arXiv:1307.277

The Parametric Ordinal-Recursive Complexity of Post Embedding Problems

Post Embedding Problems are a family of decision problems based on the interaction of a rational relation with the subword embedding ordering, and are used in the literature to prove non multiply-recursive complexity lower bounds. We refine the construction of Chambart and Schnoebelen (LICS 2008) and prove parametric lower bounds depending on the size of the alphabet.Comment: 16 + vii page

Sequential Relational Decomposition

The concept of decomposition in computer science and engineering is considered a fundamental component of computational thinking and is prevalent in design of algorithms, software construction, hardware design, and more. We propose a simple and natural formalization of sequential decomposition, in which a task is decomposed into two sequential sub-tasks, with the first sub-task to be executed before the second sub-task is executed. These tasks are specified by means of input/output relations. We define and study decomposition problems, which is to decide whether a given specification can be sequentially decomposed. Our main result is that decomposition itself is a difficult computational problem. More specifically, we study decomposition problems in three settings: where the input task is specified explicitly, by means of Boolean circuits, and by means of automatic relations. We show that in the first setting decomposition is NP-complete, in the second setting it is NEXPTIME-complete, and in the third setting there is evidence to suggest that it is undecidable. Our results indicate that the intuitive idea of decomposition as a system-design approach requires further investigation. In particular, we show that adding a human to the loop by asking for a decomposition hint lowers the complexity of decomposition problems considerably

Cyclic-routing of Unmanned Aerial Vehicles

© 2019 Various missions carried out by Unmanned Aerial Vehicles (UAVs) are concerned with permanent monitoring of a predefined set of ground targets under relative deadline constraints, i.e., the targets have to be revisited ‘indefinitely’ and there is an upper bound on the time between two consecutive successful scans of each target. A solution to the problem is a set of routes—one for each UAV—that jointly satisfy these constraints. Our goal is to find a solution with the least number of UAVs. We show that the decision version of the problem (given k, is there a solution with k UAVs?) is PSPACE-complete. On the practical side, we propose a portfolio approach that combines the strengths of constraint solving and model checking. We present an empirical evaluation of the different solution methods on several hundred randomly generated instances

The pseudo-reachability problem for diagonalisable linear dynamical systems

We study fundamental reachability problems on pseudo-orbits of linear dynamical systems. Pseudo-orbits can be viewed as a model of computation with limited precision and pseudo-reachability can be thought of as a robust version of classical reachability. Using an approach based on o-minimality of Rexp we prove decidability of the discrete-time pseudo-reachability problem with arbitrary semialgebraic targets for diagonalisable linear dynamical systems. We also show that our method can be used to reduce the continuous-time pseudo-reachability problem to the (classical) time-bounded reachability problem, which is known to be conditionally decidable

Robust Model-Checking of Linear-Time Properties in Timed Automata

International audienceFormal verification of timed systems is well understood, but their \emphimplementation is still challenging. Recent works by Raskin \emphet al. have brought out a model of parameterized timed automata that can be used to prove \emphimplementability of timed systems for safety properties. We define here a more general notion of robust model-checking for linear-time properties, which consists in verifying whether a property still holds even if the transitions are slightly delayed or expedited. We provide PSPACE algorithms for the robust model-checking of Büchi-like and LTL properties. We also verify bounded-response-time properties