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# Challenging Problems in Infinite-State Systems

## Periodic Reporting for period 2 - INFSYS (Challenging Problems in Infinite-State Systems)

Okres sprawozdawczy: 2022-09-01 do 2024-02-29

The project INFSYS (Challenging Problems in Infinite-State Systems) focuses on a class of computational models called infinite-state systems. Similarly as in economy market models help understanding future prices or meteorological models predict future weather in computer science one uses computation models. This approach was for example applied by Intel company to check whether the processors they design do not have some fundamental flaws. However, the main motivation behind the project is a deep believe that understanding basics of nature may have a profound impact on the way people live. Therefore the INFSYS project aims at solving the most natural problems in the area of infinite-state systems with a lot of focus on mathematical elegance, which be belive is a kind of a witness that our research touches a fundamental topic.

There are three main topics of the project: reachability problems, separability problem and unambiguous systems. Below I focus on explaining the reachability problems, this is the largest part of the project and also the one where we managed to achieve major contributions.

The reachability problem asks whether in an infinite graph with a distinguished inital and final vertex there is a path between these vertices. If the graph is a model of a program, which starts in the initial vertex and the final vertex is some erroneous configuration then existence of a path means that there is a possible error in the program. Therefore the reachability problem can be relevant in applications. On the other hand it is the most fundamental problem one can ask for a class of systems. Gaining understanding on the complexity of it helps a lot for investigation of other problems.

In the project we work a lot on the reachability problem for vector addition systems. Vector addition systems (VAS) are computation models, which focus on modelling counters. On one hand the model of VAS is almost equivalent to the model of Petri nets, which is widely used in practial modelling of systems. On the other hand the mathematics of VAS is very interesting and elegant, but still very far from being understood. There are many fundamental problems in this field, which despite of a pretty simple formulations, are waiting to be solved for decades. In the INFSYS project we want to find novel techniques which will accelerate the progress on understanding the structure of VAS and in consequence deliver new notions and new algorithms for various fundamental problems.
There is a succesfull line of research in the INFSYS project concerning vector addition systems (VAS). Together with Łukasz Orlikowski we have solved a 45-year old open problem regarding the complexity of the reachability problem for VAS. In the paper "Reachability in Vector Addition Systems is Ackermann-complete" we proved that the problem is Ackermann-hard, which is a very high complexity. This result was published on one of two best conferences for theoretical computer science: the FOCS conference. The other member of our team: Sławomir Lasota was also working with us and contributed to other results on the way to the main one. The reachability problem was mentioned as one of the main challenges in the proposal for INFSYS and we are happy that we were able to achieve this goal.

Another important result was achieved by Filip Mazowiecki and his co-authors. In the paper "Coverability in VASS Revisited: Improving Rackoff's Bound to Obtain Conditional Optimality" they have closed a gap concerning the length of the shortest path in the coverability problem. The covarability problem is very similar to the reachability problem, instead of asking whether one can reach a certain configuration it asks whether one can reach a certain set of configurations which has a nice shape (is upward-closed). Since the 70-ties it was no clear whether this length for d-dimensional VAS is 2^{2^O(d)} or 2^{2^O(d log d)}. The article by Filip Mazowiecki and co-authors showed the 2^{2^d} upper bound thus closing the gap and was awarded the Best Paper Award on a top TCS conference ICALP 2023.

Beside these two major achievements we performed also another research directions on the area of vector addition systems resulting in several other papers. The above line of research turned out to be very succesfull and now it is a very interesting time for the community working on VAS. Many new results and new techniques are appearing. This is why a lot of the effort in the INFSYS project was devoted for this line of research.
There are several problems on which we are currently working and in my opinion we have good chances to obtain interesting contributions there. On top of the results mentioned above we hope to gain more understanding on a few other fields of research.

One such field is understanding vector addition systems with states in low dimension. We pursue a few projects, which concern VAS in dimension 2, 3, or other small fixed dimensions and we gain understanding in this respect. I expect that in this line of research we may have interesting results until the of the project.

Yet another topic is the reachability problem for vector addition systems equipped additionally with a pushdown. This is a very involved model, for which the reachability problem is not even known to be decidable. We are considering this question for subclasses of VAS with a pushdown and there is hope we may get some results in the coming years.