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

1. Introduction

Vector addition systems with states

(VASS) are basically finite state systems with vectors of integers as transition weights, as depicted in Fig. 1.

Figure 1. A vector addition system with states.

Their semantics, starting from an initial vector of natural numbers, simply adds component-wise the weights of the successive transitions, but the current values should remain non-negative at all times on every coordinate. For instance, in the three-dimensional system of Fig. 1,

qin(0,0,2)\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha1$−−−−−−−−−−−−−−−−−−−−−−−→$qin(0,2,2)\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha1$−−−−−−−−−−−−−−−−−−−−−−−→$qin(0,4,2)\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha3$−−−−−−−−−−−−−−−−−−−−−−−→$qout(1,4,2)\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha6$−−−−−−−−−−−−−−−−−−−−−−−→$qout(2,3,2)\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha7$−−−−−−−−−−−−−−−−−−−−−−−→$q(3,2,0)\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$% \boldmatha8$−−−−−−−−−−−−−−−−−−−−−−−→$q(1,1,0)\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha9$−−−−−−−−−−−−−−−−−−−−−−−→$qout(1,1,0)

is a path witnessing that $q_{o u t} (1, 1, 0)$ can be reached from $q_{i n} (0, 0, 2)$ , but for instance $q(1,1,0)\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha8$−−−−−−−−−−−−−−−−−−−−−−−→$q(−1,0,0)$ is not a valid execution step due to the negative value in the first coordinate.

Vector addition systems with states are equivalent to Petri nets, and well-suited whenever one needs to model discrete resources, for instance threads in concurrent computations, molecules in chemical reactions, organisms in biological processes, etc. They are also a crucial ingredient in many algorithms. In particular, the decidability of their reachability problem [21, 13, 14, 17] is the cornerstone of many decidability results—see for instance [28, Sec. 5] for a large sample of problems inter-reducible with VASS reachability in logic, formal languages, verification, etc.

In spite of its relevance to a wide range of problems, the complexity of the VASS reachability problem is still not well understood. Indeed, it turns out that this seemingly simple problem is both conceptually and computationally very complex.

On a conceptual level,

the 1981 decidability proof by Mayr [21] was the culmination of more than a decade of research in the topic and is considered as one of the great achievements of theoretical computer science. Both Mayr’s decomposition algorithm and its proof are however quite intricate. Kosaraju [13] and Lambert [14] contributed several simplifications of Mayr [21]’s original arguments and Leroux and Schmitz [18] recast the decomposition algorithm in a more abstract framework based on well-quasi-order ideals, while Leroux [17] provides a very simple algorithm with a short but non constructive proof, but none of these developments can be called ‘easy’ and the problem seems inherently involved.

On a computational level,

on the one hand, the best known lower bound—which was from 1976 until very recently -hardness [19]—is now TOWER-hardness [6]. This new lower bound puts the problem firmly in the realm of non-elementary complexity. In this realm, complexity is measured using the ‘fast-growing’ complexity classes $(\ComplexityFontFα)α$ from [27], which form a strict hierarchy indexed by ordinals. The already mentioned $\ComplexityFontTOWER=\ComplexityFontF3$ corresponds to problems solvable in time bounded by a tower of exponentials; each $\ComplexityFontFk$ for a finite $k$ is primitive recursive, and $\ComplexityFontACKERMANN=\ComplexityFontFω$ corresponds to problems solvable with Ackermannian resources (see Fig. 2). On the other hand, due to the intricacy of the decomposition algorithm, it eluded analysis for a long time until a ‘cubic Ackermann’ upper bound was obtained in [18] at level $\ComplexityFontFω3$ , with a slightly improved $\ComplexityFontFω2$ upper bound in [29].

Figure 2. Pinpointing

\ComplexityFontFω=\ComplexityFontACKERMANN

among the complexity classes beyond ELEMENTARY [27].

This leaves a gigantic gap between the known lower and upper bounds. This is however mitigated by the fact that the decomposition algorithm, on which the upper bounds were obtained, provably has a non primitive-recursive complexity. This was already observed by Müller [22], due to the algorithm’s reliance on Karp and Miller trees [12]. Moreover, the full decomposition produced by the algorithm contains more information than just the existence of a reachability witness (which exists if and only if the full decomposition is not empty). For instance, Lambert [14] exploits the full decomposition to derive a pumping lemma for labelled VASS languages, Habermehl et al. [10] further show that one can compute a finite-state automaton recognising the downward-closure of a labelled VASS language with respect to the scattered subword ordering, and Czerwiński et al. [5] show how to exploit the decomposition for deciding language boundedness properties. In particular, the result of Habermehl et al. means that one can decide, given two labelled VASS, whether an inclusion holds between the downward-closures of their languages, which is an ACKERMANN-hard problem [32]. Thus any algorithm that returns such a full decomposition must be non primitive-recursive.

Contributions.

In this paper, we show that VASS reachability is in $\ComplexityFontACKERMANN=\ComplexityFontFω$ , and more precisely in $\ComplexityFontFd+4$ when the dimension $d$ of the system is fixed. This improvement over the bound $\ComplexityFontFω2$ (resp. $\ComplexityFontFω⋅(d+1)$ in fixed dimension) shown in [29] is obtained by analysing a decomposition algorithm similar to those of Mayr [21], Kosaraju [13], and Lambert [14]. In a nutshell, a decomposition algorithm defines both

a structure (resp. ‘regular constraint graphs’ for Mayr, ‘generalised VASSes’ for Kosaraju, and ‘marked graph-transition sequences’ for Lambert)—see Sec. 3—and
a condition on this structure that ensures there is an execution witnessing reachability (resp. ‘consistent marking’, ‘property $\uptheta$ ’, and ‘perfectness’)—see Sec. 4.3.3.

The algorithms compute a decomposition by successive refinements of the structure until the condition is fulfilled, by which time the existence of an execution becomes guaranteed—see Sec. 4.

We work in this paper with a decomposition algorithm quite similar to that of Kosaraju [13], for which the reader will find good expositions for instance in [22, 25, 15]. We benefit however from two key insights (which in turn require significant adaptations throughout the algorithm).

The first key insight is a new termination argument for the decomposition process, based on the dimensions of the vector spaces spanned by the cycles of the structure (see Sec. 3.2). On its own, this new termination argument would already be enough to yield $\ComplexityFontACKERMANN$ upper bounds and primitive-recursive ones in fixed dimension.

The second key insight lies within the decomposition process itself, where we show using techniques inspired by Rackoff [24] that we can eschew the computation of Karp and Miller’s coverability trees, and therefore the worst-case Ackermannian blow-up that arises from their use [3]—see Sec. 4.2.1. In itself, this new decomposition algorithm would not bring the complexity below the previous bounds, but combined with the first insight, it yields rather tight upper bounds, at level $\ComplexityFontFd+4$ in fixed dimension $d$ —see Sec. 5.

In fact, the new upper bounds apply to other decision problems. As we discuss in Sec. 6, Zetzsche’s $\ComplexityFontACKERMANN$ lower bound [32] can be refined to prove that the inclusion problem between the downward-closures of two labelled VASS languages is $\ComplexityFontFd$ -hard in fixed dimension $d \geq 3$ , thus close to matching the $\ComplexityFontFd+4$ upper bound one obtains by applying the results of Habermehl et al. [10] to our decomposition algorithm.

We start in Sec. 2 by recalling basic definitions and notations on vector addition systems. The full proofs for the decomposition algorithm are presented in Appendices A to C.

2. Background

Notations

Let $Nω\raisebox−0.43pt$\raisebox−1.075pt\scalebox.5def=$N⊎{ω}$ extend the set of natural numbers with an infinite element $ω$ with $n < ω$ for all $n \in N$ . We also use the partial order $⊑$ over $N_{ω}$ defined by $x ⊑ y$ if $y \in {x, ω}$ .

Let $d \in N$ be a dimension. The relations $\leq$ and $⊑$ are extended component-wise to vectors in $N_{ω}^{d}$ . The components of a vector that are equal to $ω$ intuitively denote arbitrarily large values; we call a vector in $N^{d}$ finite. Given a vector $\boldmathx\boldmathx∈Ndω$ and a subset $I \subseteq {1, \dots, d}$ of the components, we denote by $\boldmathx\boldmathx|I$ the vector obtained from $x$ by replacing components not in $I$ by $ω$ . Note that $\boldmathx\boldmathx⊑\boldmathy$ implies $\boldmathx\boldmathx≤\boldmathy\boldmathy$ and that $\boldmathx\boldmathx⊑\boldmathx|I$ for all $\boldmathx\boldmathx,\boldmathy\boldmathy∈Ndω$ and $I \subseteq {1, \dots, d}$ . For instance, for $d = 3$ , $(3, 2, 1) \leq (4, ω, 1)$ but $(3, 2, 1) ⋢ (4, ω, 1)$ ; if $I = {2, 3}$ , then $(3, 2, 1) |_{I} = (ω, 2, 1)$ and $(4, ω, 1) |_{I} = (ω, ω, 1)$ , and then $(ω, 2, 1) ⊑ (ω, ω, 1)$ . We let $0$ denote the zero vector and $ω$ the vector with $\boldmathω(i)\raisebox−0.43pt$\raisebox−1.075pt\scalebox.5def=$ω$ for all $1 \leq i \leq d$ . Observe that $\boldmathx\boldmathx⊑\boldmathω$ for all $\boldmathx\boldmathx∈Ndω$ .

For a vector $\boldmathx\boldmathx∈Ndω$ , its norm $∥\boldmathx% ∥$ is defined over its finite components as $∑1≤i≤d∣\boldmathx(i)<ω\boldmathx(i)$ (a sum over an empty set is zero); for a vector $\boldmathx\boldmathx∈Zd$ , we let as usual $∥\boldmathx% ∥\raisebox−0.43pt$\raisebox−1.075pt\scalebox.5def=$∑1≤i≤d|\boldmathx(i)|$ . For instance, $∥ (3, ω, 1) ∥ = 4$ and $∥ (- 4, 2, 1) ∥ = 7$ .

Vector Addition Systems

While we focus in this paper on reachability in vector addition systems with a finite set of control states, we also rely on notations for the simpler case of vector addition systems.

A vector addition system (VAS) [12] of dimension $d \in N$ is a finite set $\boldmathA\boldmathA⊆Zd$ of vectors called actions. The semantics of a VAS is defined over configurations in $N_{ω}^{d}$ . We associate to an action $\boldmatha\boldmatha∈\boldmathA\boldmathA$ the binary relation $\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha$−−−−−−−−−−−−−−−−−−−−−−→$$ over configurations by $\boldmathx\boldmathx\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha$−−−−−−−−−−−−−−−−−−−−−−→$\boldmathy$ if $\boldmathy\boldmathy=\boldmathx\boldmathx+\boldmatha\boldmatha$ , where addition is performed component-wise with the convention that $ω + z = ω$ for every $z \in Z$ . Given a finite word $σ=\boldmatha1…\boldmathak∈\boldmathA∗$ of actions we also define the binary relation $\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$σ$−−−−−−−−−−−−−−−−−−−−→$$ over configurations by $\boldmathx\boldmathx\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$σ$−−−−−−−−−−−−−−−−−−−−→$\boldmathy$ if there exists a sequence $\boldmathc\boldmathc0,…,\boldmathck$ of configurations such that

\boldmathx\boldmathx=\boldmathc\boldmathc0\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha1$−−−−−−−−−−−−−−−−−−−−−−−→$\boldmathc1⋯\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$% \boldmathak$−−−−−−−−−−−−−−−−−−−−−−−→$\boldmathc\boldmathck=\boldmathy.

The VAS reachability problem consists in deciding given two finite configurations $\boldmathc\boldmathcin,\boldmathcout$ in $N^{d}$ and a VAS $A$ whether there exists a word $σ∈\boldmathA∗$ such that $\boldmathc\boldmathcin\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$σ$−−−−−−−−−−−−−−−−−−−−→$\boldmathcout$ .

Vector Addition Systems with States

A vector addition system with states (VASS) [11] of dimension $d \in N$ is a triple $G = (Q, q_{i n}, q_{o u t}, T)$ where $Q$ is a non-empty finite set of states, $q_{i n} \in Q$ is the input state, $q_{o u t} \in Q$ is the output state, and $T$ is a finite set of transitions in $Q \times Z^{d} \times Q$ ; $\boldmathA\boldmathA\raisebox−0.43pt$\raisebox−1.075pt\scalebox.5def=${\boldmatha∣∃p,q∈Q.(p,\boldmatha\boldmatha,q)∈T}$ is the associated set of actions.

Example 2.1.

Figure 1 depicts the VASS $G_{e x} = (Q_{e x}, q_{i n}, q_{o u t}, T_{e x})$ of dimension $3$ where $Q_{e x} = {q_{i n}, q_{o u t}, p, q}$ and $T_{e x} = {t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}, t_{7}, t_{8}, t_{9}}$ with

$t_{1}$	$= (q_{i n}, (0, 2, 0), q_{i n}),$	$t_{2}$	$= (q_{i n}, (2, 2, - 1), p),$
$t_{3}$	$= (q_{i n}, (1, 0, 0), q_{o u t}),$	$t_{4}$	$= (q_{i n}, (1, 0, - 2), q_{o u t}),$
$t_{5}$	$= (p, (1, 0, - 2), q_{i n}),$	$t_{6}$	$= (q_{o u t}, (1, - 1, 0), q_{o u t}),$
$t_{7}$	$= (q_{o u t}, (1, - 1, - 2), q),$	$t_{8}$	$= (q, (- 2, - 1, 0), q),$
$t_{9}$	$= (q, (0, 0, 0), q_{o u t}) .$

We focus on VASSes in this paper rather than VASes, because we exploit the properties of their underlying directed graphs. A path $π$ in a VASS $G$ from a state $p$ to a state $q$ labelled by a word $\boldmatha\boldmatha1…\boldmathak$ of actions is a word of transitions of $G$ of the form $(p1,\boldmatha1,q1)…(pk,\boldmathak,qk)$ with $p_{0} = p$ , $q_{k} = q$ , and $q_{j} = p_{j + 1}$ for all $1 \leq j < k$ . Such a path is complete if $p = q_{i n}$ and $q = q_{o u t}$ are the input and output states of $G$ . A cycle on a state $q$ is a path from $q$ to $q$ .

Example 2.2.

For instance, in Ex. 2.1, the execution presented in the introduction corresponds to the path $π_{e x} = t_{1} t_{1} t_{3} t_{6} t_{7} t_{8} t_{9}$ labelled by $σex=% \boldmatha1\boldmatha1\boldmatha3\boldmatha6\boldmatha\boldmatha7\boldmatha% 8\boldmatha9$ , and is complete.

We write $p \equiv_{G} q$ if there exists a path from $p$ to $q$ and a path from $q$ to $p$ ; this defines an equivalence relation whose equivalence classes are called the strongly connected components of $G$ . In Ex. 2.1, the strongly connected components are ${q_{i n}, p}$ and ${q, q_{o u t}}$ . A VASS $G = (Q, q_{i n}, q_{o u t}, T)$ is said to be strongly connected if $Q$ is a strongly connected component of $G$ .

The Parikh image of a path $π$ is the function $ϕ : T \to N$ that maps each transition $t \in T$ to its number of occurrences in $π$ . The displacement of a path $π$ labelled by a word $\boldmatha\boldmatha1…\boldmathak$ of actions is the vector $Δ(π)\raisebox−0.43pt$\raisebox−1.075pt\scalebox.5def=$∑kj=1\boldmathaj$ ; note that this is equal to $Δ(ϕ)\raisebox−0.43pt$\raisebox−1.075pt\scalebox.5def=$∑t=(p,\boldmatha,q)∈Tϕ(t)⋅\boldmatha\boldmatha$ if $ϕ$ is the Parikh image of $π$ .

Example 2.3.

For the example path $π_{e x}$ from Ex. 2.2, $ϕ_{e x} = (2, 0, 1, 0, 0, 1, 1, 1, 1)$ and $Δ (π_{e x}) = (1, 1, - 2)$ .

A state-configuration of a VASS $G = (Q, q_{i n}, q_{o u t}, T)$ is a pair $(q,\boldmathx)∈Q×Ndω$ denoted by $q(\boldmathx% )$ in the sequel. Given an action $a$ we define the step relation $\raisebox0.0pt[10.0pt][1.72pt]$[−−−−−−−−−−−−→\raisebox2.0pt[0.0pt0.0pt]$G$]\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha\boldmatha$$$ over state-configurations by $p(\boldmathx% )\raisebox0.0pt[10.0pt][1.72pt]$[−−−−−−−−−−−−→\raisebox2.0pt[0.0pt0.0pt]$G$]\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha\boldmatha$$q(\boldmathy)$ if $(p,\boldmatha,q)∈T$ and $\boldmathx\boldmathx\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha$−−−−−−−−−−−−−−−−−−−−−−→$\boldmathy$ . By extension, given a word $σ$ of actions $σ=\boldmatha1…\boldmathak$ , $p(\boldmathx% )\raisebox0.0pt[10.0pt][1.72pt]$[−−−−−−−−−−−−→\raisebox2.0pt[0.0pt0.0pt]$G$]\raisebox−1.5pt[0.0pt][1.29pt]$σ$$q(\boldmathy)$ if there exists a sequence $q0(\boldmathc0),…,qk(\boldmathck)$ of state-configurations such that

p(\boldmathx)=q0(\boldmathc0)\raisebox0.0pt[10.0pt][1.72pt]$[−−−−−−−−−−−−→\raisebox2.0pt[0.0pt0.0pt]$G$]\raisebox−1.5pt[0.0pt][1.29pt]$\boldmatha1$$q1(\boldmathc1)⋯\raisebox0.0pt[10.0pt][1.72pt]$[−−−−−−−−−−−−→\raisebox2.0pt[0.0pt0.0pt]$G$]\raisebox−1.5pt[0.0pt][1.29pt]$\boldmathak$$qk(\boldmathck)=q(\boldmathy).

Notice that $p(\boldmathx% )\raisebox0.0pt[10.0pt][1.72pt]$[−−−−−−−−−−−−→\raisebox2.0pt[0.0pt0.0pt]$G$]\raisebox−1.5pt[0.0pt][1.29pt]$σ$$q(\boldmathy)$ if, and only if, there exists a path in $G$ from $p$ to $q$ labelled by $σ$ such that $\boldmathx\boldmathx\raisebox0.0pt[10.0pt][1.72pt]$\raisebox−1.5pt[0.0pt][1.29pt]$σ$−−−−−−−−−−−−−−−−−−−−→$\boldmathy$ . In Ex. 2.1, $qin((0,0,2))\raisebox0.0pt[10.0pt][1.72pt]$[−−−−−−−−−−−−→\raisebox2.0pt[0.0pt0.0pt]$Gex$]\raisebox−1.5pt[0.0pt][1.29pt]$σex$$qout((1,1,0))$ . Finally, we write $p(\boldmathx% )\raisebox0.0pt[10.0pt][1.72pt]$[−−−−−−−−−−−−→\raisebox2.0pt[0.0pt0.0pt]$G$]\raisebox−1.5pt[0.0pt][1.29pt]$∗$$q(\boldmathy)$ if there exists $σ∈\boldmathA∗$ such that $p(\boldmathx% )\raisebox0.0pt[10.0pt][1.72pt]$[−−−−−−−−−−−−→\raisebox2.0pt[0.0pt0.0pt]$G$]\raisebox−1.5pt[0.0pt][1.29pt]$σ$$q(\boldmathy)$ .

Reachability

We focus in this paper on the following decision problem.

Problem: 1 (VASS reachability).

: []
input:: a VASS $G = (Q, q_{i n}, q_{o u t}, T)$ of dimension $d$ and two finite configurations $\boldmathc\boldmathcin,\boldmathcout∈Nd$
question:: does $qin(\boldmath% cin)\raisebox0.0pt[10.0pt][1.72pt]$[−−−−−−−−−−−−→\raisebox2.0pt[0.0pt0.0pt]$G$]\raisebox−1.5pt[0.0pt][1.29pt]$∗$$qout(\boldmathcout)$ hold?

The previously mentioned VAS reachability problem reduces to Problem: 1 (VASS reachability).: given a VAS $A$ and two finite configurations $\boldmathc\boldmathcin,\boldmathcout$ , it suffices to consider the Problem: 1 (VASS reachability). problem with input $({q},q,q,{q}×% \boldmathA×{q})$ and the same configurations $\boldmathc\boldmathcin,\boldmathcout$ . A converse reduction is possible by encoding the states, at the expense of increasing the dimension by three [11].

3. Decomposition Structures

The version of the decomposition algorithm we present in Sec. 4 proceeds globally as the ones of Mayr, Kosaraju, and Lambert, and we call the underlying structures KLM sequences after them.

3.1. KLM Sequences

A KLM sequence $ξ$ of dimension $d$ is a sequence

ξ=(\boldmathx0G0\boldmathy0)\boldmatha1(\boldmathx1G1\boldmathy1)…\boldmatha\boldmathak(\boldmathxkGk\boldmathyk)

(1)

where $\boldmathx\boldmathx0,\boldmathy0,…,\boldmathxk,\boldmathyk$ are configurations, $G_{0}, \dots, G_{k}$ are VASSes of dimension $d$ , and $\boldmatha\boldmatha1,…,\boldmathak$ are actions. KLM sequences are essentially the same as Kosaraju’s ‘generalised VASSes’ [13], except that we do not require $G_{0}, \dots, G_{k}$ to be strongly connected.

The action language of a KLM sequence $ξ$ is the set $L_{ξ}$ of words of actions of the form $σ0\boldmatha1σ1…\boldmathakσk$ such that $σ_{j}$ is the label of a complete path of $G_{j}$ for every $j$ , and such that there exists a sequence $\boldmathm\boldmathm0,\boldmathn0,…,\boldmathmk,\boldmathnk$ of configurations in $N^{d}$ such that

\boldmathm\boldmathm0σ0−→\boldmathn0\boldmatha1−−−−−−−→⋯\boldmathm\boldmathmkσk−→\boldmathnk

(2)

where $\boldmathm\boldmathmj⊑\boldmathxj$ and $\boldmathn\boldmathnj⊑\boldmathyj$ for every $0 \leq j \leq k$ .

Note that the reachability problem for a VASS $G$ and two finite configurations $\boldmathc\boldmathcin,\boldmathc% out∈Nd$ reduces to the non-emptiness of the action language of the KLM sequence $(\boldmathcinG\boldmathcout)$ . In fact, in that case, the action language is the set of words $σ∈\boldmathA∗$ such that $qin(\boldmath% cin)\raisebox0.0pt[10.0pt][1.72pt]$[−−−−−−−−−−−−→\raisebox2.0pt[0.0pt0.0pt]$G$]\raisebox−1.5pt[0.0pt][1.29pt]$σ$$qout(\boldmathcout)$ .

Example 3.1.

In Ex. 2.1, $ξ_{e x} = ((0, 0, 2) G_{e x} (1, 1, 0))$ is a KLM sequence with action language

	$L_{ξ_{e x}}$	$={\boldmatha2+3n1\boldmatha% 3\boldmatha1+4n6\boldmatha\boldmatha7\boldmatha1+2n8\boldmatha9∣n∈N}$
		$∪{\boldmatha2+3n1\boldmatha% 3\boldmatha4n6\boldmatha7\boldmatha\boldmatha1+2n8\boldmatha9\boldmatha6∣n∈N}.$

3.2. Ranks and Sizes

Vector Spaces

We associate to a transition $t$ of a VASS $G$ the vector space $\boldmathV\boldmathVG(t)⊆Qd$ spanned by the displacements of the cycles that contain $t$ . The following lemma shows that this vector space only depends on the strongly connected components of $G$ .

Lemma 3.2.

Let $t$ be a transition of a strongly connected VASS $G$ . Then the vector space $\boldmathV\boldmathVG(t)$ is equal to the vector space spanned by the displacements of the cycles of $G$ .

Proof.

Let $V$ be the vector space spanned be the displacements of the cycles of $G$ . Naturally, we have $\boldmathV\boldmathVG(t)⊆\boldmath% V$ . For the converse, let us consider a sequence $θ_{1}, \dots, θ_{k}$ of cycles such that $θ_{j}$ is a cycle on a state $q_{j}$ for every $1 \leq j \leq k$ , and such that $Δ (θ_{1}), \dots, Δ (θ_{k})$ span the vector space $V$ . Since $G$ is strongly connected, there exists a path $π_{j}$ from $q_{j - 1}$ to $q_{j}$ for every $j \in {1, \dots, k}$ with $q0\raisebox−0.43pt$\raisebox−1.075pt\scalebox.5def=$qk$ . Moreover, we can assume without loss of generality that $t$ occurs in the cycle $θ\raisebox−0.43pt$\raisebox−1.075pt\scalebox.5def=$π1…πk$ . Let $θ_{j}^{'}$ be the cycle obtained from $θ$ by inserting $θ_{j}$ in $q_{j}$ and formally defined as $θ′j\raisebox−0.43pt$\raisebox−1.075pt\scalebox.5def=$π1…πjθjπj+1…πk$ . Observe that $Δ (θ)$ and $Δ (θ_{j}^{'})$ are both in $\boldmathV\boldmathVG(t)$ since $t$ occurs in the cycles $θ$ and $θ_{j}^{'}$ . As $Δ (θ_{j}) = Δ (θ_{j}^{'}) - Δ (θ)$ , it follows that $Δ(θj)∈% \boldmathVG(t)$ . We derive that the vector space spanned by $Δ (θ_{1}), \dots, Δ (θ_{k})$ is included in $\boldmathV\boldmathVG(t)$ . Hence $\boldmathV\boldmathV⊆\boldmathVG(t)$ . ∎

As a corollary, if two transitions $t$ and $t^{'}$ are induced by the same strongly connected component of a VASS $G$ , then $\boldmathV\boldmathVG(t)=\boldmathVG(t′)$ .

Ranks

The rank of a VASS $G$ is the tuple $rank(G)\raisebox−0.43pt$\raisebox−1.075pt\scalebox.5def=$(rd,…,r0)∈Nd+1$ where $r_{i}$ is the number of transitions $t \in T$ such that the dimension of $\boldmathV\boldmathVG(t)$ is equal to $i$ . The rank of a KLM sequence $ξ$ defined as $(\boldmathx0G0\boldmathy0)\boldmatha1(\boldmathx1G1\boldmathy1)…\boldmatha\boldmathak(\boldmathxkGk\boldmathyk)$ is the vector $rank(ξ)\raisebox−0.43pt$\raisebox−1.075pt\scalebox.5def=$∑$

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

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The Reachability Problem for Petri Nets is Not Elementary

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

Improved lower and upper bounds on the tile complexity of uniquely self-assembling a thin rectangle non-cooperatively in 3D

Normalisers of primitive permutation groups in quasipolynomial time

1. Introduction

Vector addition systems with states

On a conceptual level,

On a computational level,

Contributions.

2. Background

Notations

Vector Addition Systems

Vector Addition Systems with States

Example 2.1.

Example 2.2.

Example 2.3.

Reachability

Problem: 1 (VASS reachability).

3. Decomposition Structures

3.1. KLM Sequences

Example 3.1.

3.2. Ranks and Sizes

Vector Spaces

Lemma 3.2.

Proof.

Ranks

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

Authors

Related Research

The Reachability Problem for Petri Nets is Not Primitive Recursive

Bisimulation Equivalence of First-Order Grammars is ACKERMANN-Complete

The Reachability Problem for Petri Nets is Not Elementary (Extended Abstract)

The Reachability Problem for Petri Nets is Not Elementary

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

Improved lower and upper bounds on the tile complexity of uniquely self-assembling a thin rectangle non-cooperatively in 3D

Normalisers of primitive permutation groups in quasipolynomial time

This week in AI

1. Introduction

Vector addition systems with states

On a conceptual level,

On a computational level,

Contributions.

2. Background

Notations

Vector Addition Systems

Vector Addition Systems with States

Example 2.1.

Example 2.2.

Example 2.3.

Reachability

Problem: 1 (VASS reachability).

3. Decomposition Structures

3.1. KLM Sequences

Example 3.1.

3.2. Ranks and Sizes

Vector Spaces

Lemma 3.2.

Proof.

Ranks