Coverability in Vector Addition Systems

• The coverability problem in VAS is defined as determining if a starting configuration can eventually reach or exceed a target configuration coordinate-wise.
• This topic examines both classical and parameterized complexity, highlighting results like XNL-completeness for unary encoding and para-PSPACE-hardness for binary cases.
• It explores proof techniques, algorithmic bounds, and open research questions that influence verification methods and automated reasoning in concurrent systems.

Vector Addition Systems (VAS) are a fundamental model for the analysis of concurrent computations, parameterized counter systems, and Petri nets. The coverability problem is a core computational question in VAS theory, central to verification, static analysis, and decidability theory in computer science. This article details the formal statement of the coverability problem, its parameterized and classical complexity, bounds, reductions, and leading open questions, referencing precise results established in recent research.

1. Vector Addition Systems and the Coverability Problem

A $d$-dimensional Vector Addition System (VAS) is defined as a finite set $V \subseteq \mathbb{Z}^d$ of vectors. A configuration is a vector $s \in \mathbb{N}^d$. The systems operate by iteratively applying transition vectors from $V$:

• The one-step transition relation is defined as $s \rightarrow_V s+v$ if $v \in V$ and $s+v \in \mathbb{N}^d$, i.e., all counters remain non-negative.
• The reflexive-transitive closure is denoted as $s \rightarrow^*_V u$ if a finite sequence of transitions yields configuration $u$ from $s$.

Given $s, t \in \mathbb{N}^d$, $s$ covers $t$ (notation: $s \Rightarrow^*_V t'$ with $t' \succeq t$) if there exists $u \in \mathbb{N}^d$ such that $u \succeq t$ and $s \rightarrow^*_V u$ coordinate-wise.

The coverability problem is formulated as: given $V \subseteq \mathbb{Z}^d$, and configurations $s, t \in \mathbb{N}^d$, decide if $s$ covers $t$ in $V$ (Pilipczuk et al., 24 Nov 2025).

2. Natural Parameterizations and Main Complexity Results

There are two principal parameterizations of the VAS coverability problem:

• Parameterization by dimension $d$: Denoted $p$-dim-COVERABILITY(VAS), where $d$ is the parameter.
• Parameterization by the size of $V$: Denoted $p$-size-COVERABILITY(VAS), where the parameter is $\|V\|$, the total bit-length of the vectors in $V$.

Both unary (each integer in $V$, $s$, $t$ written in unary) and binary encodings are considered.

Main Complexity Results

Parameterization	Input Encoding	Complexity Class	Notes
Dimension ($d$)	Unary	XNL-complete	Under PL-reductions
Size ($\|V\|$)	Unary	XNL-complete	Under PL-reductions
Dimension ($d$), Size ($\|V\|$)	Binary	para-PSPACE-complete	Under FPT-reductions

These results are established in (Pilipczuk et al., 24 Nov 2025), with XNL-completeness achieved by reducing from the XNL-complete $p$-INTERSECTION-NONEMPTINESS(DFA) problem and para-PSPACE-completeness leveraging known PSPACE-hardness of fixed-dimension binary VAS coverability.

In the unary encoding, there exists a deterministic $n^{2^{O(d)}}$-time algorithm, placing $p$-dim-COVERABILITY(VAS) in the class XP, but the sharper XNL bound prevails.

3. Proof Techniques and Bounds

XNL Membership and Hardness

• Membership: Upper bounds build on Rackoff-type arguments: the length of any covering run in a $d$-dimensional unary VAS does not exceed $n^{2^{O(d)}}$, allowing stepwise nondeterministic simulation within $O(2^d \cdot \log n)$ space.
• Hardness: Hardness is by PL-space reductions from XNL-complete problems. In particular, for parameterization by $\|V\|$, XNL-hardness follows via reductions to DFA intersection.

para-PSPACE-hardness (Binary Encoding)

• For binary encodings, Draghici, Haase & Ryzhikov (2024) show the existence of a fixed VAS $V_0$ where coverability is PSPACE-hard, even with constant dimension, yielding para-PSPACE-hardness under FPT-reductions (Pilipczuk et al., 24 Nov 2025).

Reduction Equivalences

Structural equivalences allow translation between VAS and VASS (with possible increase by 3 in dimension), and Petri nets are computationally equivalent. Unary p-size and p-dim parameterizations reduce to one another within polynomial logarithmic space.

Algorithmic and Witness Length Bounds

A deterministic $n^{2^{O(d)}}$-time algorithm exists for $d$-dimensional unary encoded VAS (Pilipczuk et al., 24 Nov 2025). Recent work improves upper bounds on witness length to $n^{2^{O(d)}}$, matching lower bounds up to the exact exponent and establishing ETH-based optimality (Künnemann et al., 2023).

4. Position among Related Models and Restrictions

• Fixed-Dimension and Unary: For fixed $d$ and unary encodings, coverability is in XNL, and lower bounds show no $n^{o(2^d)}$-time algorithm exists unless ETH fails (Künnemann et al., 2023).
• 2-VASS and Mixed Encoding: For 2-VASS, coverability is PSPACE-complete (binary-binary encoding) and in NP for systems where one counter is unary and the other is binary, leveraging compressed linear form schemes (Mazowiecki et al., 2023, Blondin et al., 2014).
• 1-VASS and Disequality Tests: In 1-VASS, coverability is in NC$^2$ without tests, and remains in P with disequality tests (Almagor et al., 2019).
• Pushdown VAS and Extensions: In 1-dimensional pushdown VAS, coverability is decidable and EXPSPACE upper bounds are known (Leroux et al., 2015). However, with resets, coverability becomes undecidable even in dimension one (Schmitz et al., 2019).
• VAS with Zero-Tests: Coverability is decidable for VAS with a single zero-test, using a Karp-Miller-style construction enhanced via filtered covers (Bonnet et al., 2012).

5. Open Problems and Future Directions

Key open problems highlighted in (Pilipczuk et al., 24 Nov 2025) include:

• Fixed-Parameter Tractability by Size: Is COVERABILITY parameterized by the size of $V$ (in unary encoding) fixed-parameter tractable? That is, does an algorithm exist with runtime $f(\|V\|) \cdot \mathrm{poly}(n)$? This remains open.
• Shortest Witness Length: For a fixed VAS, does every shortest witness for coverability have length $O(n)$? If so, p-size COVERABILITY could admit para-NP, para-L, or even FPT algorithms.
• Structural Parameterizations: Whether additional structural parameters (e.g., largest entry, dependency graph treewidth) yield FPT algorithms for coverability/reachability remains open.
• Reachability vs. Coverability: Reachability parameterized by $|V|$ is unresolved, with the best known bounds non-elementary, and its inclusion in FPT is unexplored.
• High-Parameter Restrictions: On the "high-parameter" side, identification of restrictions yielding tractability is suggested but not resolved.

6. Connections to Verification and Theory

The coverability problem for VAS abstracts vital questions in verification—specifically, the checking of safety properties in parameterized distributed systems. The sharp complexity landscape in fixed-dimension and fixed-size VAS, and the crucial role of encoding (unary vs. binary), now provide a refined understanding of which verification tasks are algorithmically feasible for restricted systems (Pilipczuk et al., 24 Nov 2025).

Coverability in VAS directly influences the design and limits of automated reasoning tools in model checking, especially with extensions such as zero-tests and pushdown stacks, where decidability and complexity boundaries display sensitivity to even minor model enrichments (Bonnet et al., 2012, Schmitz et al., 2019, Leroux et al., 2015).

• Parameterized Complexity: "A Note on the Parameterised Complexity of Coverability in Vector Addition Systems" (Pilipczuk et al., 24 Nov 2025)
• Classical Bounds and Lower Bounds: "Coverability in VASS Revisited: Improving Rackoff's Bound to Obtain Conditional Optimality" (Künnemann et al., 2023)
• Decidability for Pushdown VAS: "On the Coverability Problem for Pushdown Vector Addition Systems in One Dimension" (Leroux et al., 2015)
• Decidability with Zero-Tests: "Model Checking Vector Addition Systems with one zero-test" (Bonnet et al., 2012)
• Undecidability with Resets: "Coverability is Undecidable in One-dimensional Pushdown Vector Addition Systems with Resets" (Schmitz et al., 2019)
• 1-VASS, Disequality: "Coverability in 1-VASS with Disequality Tests" (Almagor et al., 2019)
• 2-VASS, Mixed Encoding: "Coverability in 2-VASS with One Unary Counter is in NP" (Mazowiecki et al., 2023)
• 2-VASS, PSPACE: "Reachability in Two-Dimensional Vector Addition Systems with States is PSPACE-complete" (Blondin et al., 2014)

These works collectively establish a precise and nuanced picture of VAS coverability, with sharp demarcations between polynomial, super-polynomial, and undecidable regimes according to VAS dimension, encoding, and added features.