Reachability in fixed dimension vector addition systems with states

The reachability problem is a central decision problem in verification of vector addition systems with states (VASS). In spite of recent progress, the complexity of the reachability problem remains unsettled, and it is closely related to the lengths of shortest VASS runs that witness reachability. We obtain three main results for VASS of fixed dimension. For the first two, we assume that the integers in the input are given in unary, and that the control graph of the given VASS is flat (i.e., without nested cycles). We obtain a family of VASS in dimension 3 whose shortest runs are exponential, and we show that the reachability problem is NP-hard in dimension 7. These results resolve negatively questions that had been posed by the works of Blondin et al. in LICS 2015 and Englert et al. in LICS 2016, and contribute a first construction that distinguishes 3-dimensional flat VASS from 2-dimensional ones. Our third result, by means of a novel family of products of integer fractions, shows that 4-dimensional VASS can have doubly exponentially long shortest runs. The smallest dimension for which this was previously known is 14

Reachability in Two-Dimensional Vector Addition Systems with States: One Test Is for Free

Vector addition system with states is an ubiquitous model of computation with extensive applications in computer science. The reachability problem for vector addition systems is central since many other problems reduce to that question. The problem is decidable and it was recently proved that the dimension of the vector addition system is an important parameter of the complexity. In fixed dimensions larger than two, the complexity is not known (with huge complexity gaps). In dimension two, the reachability problem was shown to be PSPACE-complete by Blondin et al. in 2015. We consider an extension of this model, called 2-TVASS, where the first counter can be tested for zero. This model naturally extends the classical model of one counter automata (OCA). We show that reachability is still solvable in polynomial space for 2-TVASS. As in the work Blondin et al., our approach relies on the existence of small reachability certificates obtained by concatenating polynomially many cycles

Reachability in Continuous Pushdown VASS

Pushdown Vector Addition Systems with States (PVASS) consist of finitely many control states, a pushdown stack, and a set of counters that can be incremented and decremented, but not tested for zero. Whether the reachability problem is decidable for PVASS is a long-standing open problem. We consider continuous PVASS, which are PVASS with a continuous semantics. This means, the counter values are rational numbers and whenever a vector is added to the current counter values, this vector is first scaled with an arbitrarily chosen rational factor between zero and one. We show that reachability in continuous PVASS is NEXPTIME-complete. Our result is unusually robust: Reachability can be decided in NEXPTIME even if all numbers are specified in binary. On the other hand, NEXPTIME-hardness already holds for coverability, in fixed dimension, for bounded stack, and even if all numbers are specified in unary

Improved Ackermannian Lower Bound for the Petri Nets Reachability Problem

Petri nets, equivalently presentable as vector addition systems with states, are an established model of concurrency with widespread applications. The reachability problem, where we ask whether from a given initial configuration there exists a sequence of valid execution steps reaching a given final configuration, is the central algorithmic problem for this model. The complexity of the problem has remained, until recently, one of the hardest open questions in verification of concurrent systems. A first upper bound has been provided only in 2015 by Leroux and Schmitz, then refined by the same authors to non-primitive recursive Ackermannian upper bound in 2019. The exponential space lower bound, shown by Lipton already in 1976, remained the only known for over 40 years until a breakthrough non-elementary lower bound by Czerwi?ski, Lasota, Lazic, Leroux and Mazowiecki in 2019. Finally, a matching Ackermannian lower bound announced this year by Czerwi?ski and Orlikowski, and independently by Leroux, established the complexity of the problem. Our primary contribution is an improvement of the former construction, making it conceptually simpler and more direct. On the way we improve the lower bound for vector addition systems with states in fixed dimension (or, equivalently, Petri nets with fixed number of places): while Czerwi?ski and Orlikowski prove F_k-hardness (hardness for kth level in Grzegorczyk Hierarchy) in dimension 6k, our simplified construction yields F_k-hardness already in dimension 3k+2

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

The reachability problem in vector addition systems is a central question, not only for the static verification of these systems, but also for many inter-reducible decision problems occurring in various fields. The currently best known upper bound on this problem is not primitive-recursive, even when considering systems of fixed dimension. We provide 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. While this does not match the currently best known TOWER lower bound for reachability, it is optimal for related problems

Integer Vector Addition Systems with States

This paper studies reachability, coverability and inclusion problems for Integer Vector Addition Systems with States (ZVASS) and extensions and restrictions thereof. A ZVASS comprises a finite-state controller with a finite number of counters ranging over the integers. Although it is folklore that reachability in ZVASS is NP-complete, it turns out that despite their naturalness, from a complexity point of view this class has received little attention in the literature. We fill this gap by providing an in-depth analysis of the computational complexity of the aforementioned decision problems. Most interestingly, it turns out that while the addition of reset operations to ordinary VASS leads to undecidability and Ackermann-hardness of reachability and coverability, respectively, they can be added to ZVASS while retaining NP-completness of both coverability and reachability.Comment: 17 pages, 2 figure

Reachability problems for products of matrices in semirings

We consider the following matrix reachability problem: given $r$ square matrices with entries in a semiring, is there a product of these matrices which attains a prescribed matrix? We define similarly the vector (resp. scalar) reachability problem, by requiring that the matrix product, acting by right multiplication on a prescribed row vector, gives another prescribed row vector (resp. when multiplied at left and right by prescribed row and column vectors, gives a prescribed scalar). We show that over any semiring, scalar reachability reduces to vector reachability which is equivalent to matrix reachability, and that for any of these problems, the specialization to any $r\geq 2$ is equivalent to the specialization to $r=2$. As an application of this result and of a theorem of Krob, we show that when $r=2$, the vector and matrix reachability problems are undecidable over the max-plus semiring $(Z\cup\{-\infty\},\max,+)$. We also show that the matrix, vector, and scalar reachability problems are decidable over semirings whose elements are ``positive'', like the tropical semiring $(N\cup\{+\infty\},\min,+)$.Comment: 21 page

Reachability in Biochemical Dynamical Systems by Quantitative Discrete Approximation (extended abstract)

In this paper, a novel computational technique for finite discrete approximation of continuous dynamical systems suitable for a significant class of biochemical dynamical systems is introduced. The method is parameterized in order to affect the imposed level of approximation provided that with increasing parameter value the approximation converges to the original continuous system. By employing this approximation technique, we present algorithms solving the reachability problem for biochemical dynamical systems. The presented method and algorithms are evaluated on several exemplary biological models and on a real case study.Comment: In Proceedings CompMod 2011, arXiv:1109.104