POLARIS · Reachability in Infinite Systems at High Resolution

The reachability problem is central in computer science. One of the first steps towards understanding computation was showing undecidability of the problem for Turing machines. Recently with co-authors we achieved another milestone, which opens new horizons in the area: we determined the complexity of reachability for Vector Addition Systems with States (VASS) to be Ackermann-complete. The aim of this project is to obtain new milestones in understanding of the reachability problem and the related separability problem for computation models of concurrency and recursion.(1) The first task focuses on the reachability problem for VASS and its extensions. Despite recent progress, the community's understanding of VASS reachability is still limited. The goals of the task are divided into two groups:(a) I plan to design novel techniques that help us better understand the structure of VASS runs. Two concrete goals are settling the (currently poorly understood) complexity of the following natural VASS problems. A prime example can be observed with 3-dimensional VASS: reachability is between PSpace and Tower. Furthermore, the gap for reachability in fixed VASS is in between PSpace and Ackermann.(b) Reachability for VASS with a pushdown is not known to be decidable, while in the 1-dimensional case the problem is known to be PSpace-hard and decidable. I plan to develop techniques able to deal with the reachability problem for VASS extended by adding some recursive mechanisms in a substantially more efficient way than the current state-of-art.(2) The second task concerns the separability problem. It asks whether two input sets can be separated by a set that is a member of a specified family. A lot of research has been devoted to the case when either the two input sets are regular languages, or the separating family is regular. The main goal is to solve separability questions for non-regular input languages and separating families inside regular languages.