# Lower Bounds for the Reachability Problem in Fixed Dimensional VASSes

@article{Czerwinski2022LowerBF, title={Lower Bounds for the Reachability Problem in Fixed Dimensional VASSes}, author={Wojciech Czerwi'nski and Lukasz Orlikowski}, journal={Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science}, year={2022} }

We study the complexity of the reachability problem for Vector Addition Systems with States (VASSes) in fixed dimensions. We provide four lower bounds improving the currently known state-of-the-art: 1) NP-hardness for unary flat 4-VASSes (VASSes in dimension 4), 2) PSpace-hardness for unary 5-VASSes, 3) ExpSpace-hardness for binary 6-VASSes and 4) Tower-hardness for unary 8-VASSes.

## 2 Citations

### Long Runs Imply Big Separators in Vector Addition Systems

- Mathematics, Computer ScienceArXiv
- 2021

Improved the complexity of the reachability problem (for any subclass) using the separators approach may not be simpler than using the short run approach, and it is proved that a few known examples of involved VASSes fulﬁl the mentioned conditions.

### Involved VASS Zoo

- Computer Science
- 2022

We briefly describe recent advances on understanding the complexity of the reachability problem for vector addition systems (or equivalently for vector addition systems with states - VASSes). We…

## References

SHOWING 1-10 OF 25 REFERENCES

### Reachability in Two-Dimensional Vector Addition Systems with States Is PSPACE-Complete

- Mathematics2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science
- 2015

The reach ability problem for vector addition systems with states with states (VASS) is shown PSPACE-complete in the two-dimensional case, vastly improving on the doubly exponential time bound established in 1986.

### Reachability in fixed dimension vector addition systems with states

- Mathematics, Computer ScienceCONCUR
- 2020

This work obtains a family of VASS in dimension 3 whose shortest reachability witnessing runs are exponential, and shows that the reachability problem is NP-hard in dimension 7, and contributes a first construction that distinguishes 3-dimensional flat VASS from 2-dimensional VASS.

### Reachability in Vector Addition Systems is Primitive-Recursive in Fixed Dimension

- Computer Science, Mathematics2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2019

This work provides significant refinements to the classical decomposition algorithm of Mayr, Kosaraju, and Lambert and to its termination proof, which yield an ACKERMANN upper bound in the general case, and primitive-recursive upper bounds in fixed dimension.

### Reachability in Two-Dimensional Unary Vector Addition Systems with States is NL-Complete*

- Mathematics, Computer Science2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2016

This work answers positively the main question left open by Blondin et al, namely establish that reachability witnesses of pseudo-polynomial length always exist, and shows that when the input vectors are given in unary, the improved guess-and-verify algorithm requires only logarithmic space.

### The Reachability Problem for Two-Dimensional Vector Addition Systems with States

- Mathematics
- 2021

We prove that the reachability problem for two-dimensional vector addition systems with states is NL-complete or PSPACE-complete, depending on whether the numbers in the input are encoded in unary ...

### On Flatness for 2-Dimensional Vector Addition Systems with States

- Computer ScienceCONCUR
- 2004

2-dim VASS are flat (i.e. they “intrinsically” contain no nested loops), and it is obtained that – forward, backward and binary – reachability sets are effectively semilinear for the class of 2-Dim VASS, and that these sets can be computed using generic acceleration techniques.

### Reachability in Succinct and Parametric One-Counter Automata

- Computer Science, MathematicsCONCUR
- 2009

One of the main results of this paper is to show that the reachability problem for parametric one-counter automata is in fact in NP, and is thus NP -complete.

### Reachability in Vector Addition Systems is Ackermann-complete

- Computer Science2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

It is proved that the problem is $\mathcal{F}_{k}$-hard for Vector Addition Systems with States in dimension 6k, where k is the $k$-th complexity class from the hierarchy of fast-growing complexity classes.

### The Reachability Problem for Petri Nets is Not Primitive Recursive

- Computer Science2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

We present a way to lift up the Tower complexity lower bound of the reachability problem for Petri nets to match the Ackermannian upper bound closing a long standing open problem. We also prove that…

### Improved Ackermannian Lower Bound for the Petri Nets Reachability Problem

- Computer ScienceSTACS
- 2022

An improvement of the former construction of Petri nets with fixed number of places, making it conceptually simpler and more direct, and improving the lower bound for vector addition systems with states in fixed dimension.