The Reachability Problem for Petri Nets is Not Elementary

Abstract: Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modelling and analysis of hardware, software and database systems, as well as chemical, biological and business processes. The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of verification. Decidability was proved by Mayr in his seminal STOC 1981 work, and the currently best published upper bound is non-primitive recursive Ackermannian of Leroux and Schmitz from LICS 2019. We establish a non-elementary lower bound, i.e. that the reachability problem needs a tower of exponentials of time and space. Until this work, the best lower bound has been exponential space, due to Lipton in 1976. The new lower bound is a major breakthrough for several reasons. Firstly, it shows that the reachability problem is much harder than the coverability (i.e., state reachability) problem, which is also ubiquitous but has been known to be complete for exponential space since the late 1970s. Secondly, it implies that a plethora of problems from formal languages, logic, concurrent systems, process calculi and other areas, that are known to admit reductions from the Petri nets reachability problem, are also not elementary. Thirdly, it makes obsolete the currently best lower bounds for the reachability problems for two key extensions of Petri nets: with branching and with a pushdown stack.

• The paper establishes a non-elementary lower bound, proving that Petri net reachability exceeds any finite tower of exponentials.
• The paper introduces a novel factorial amplifier gadget that simulates Minsky machines with exponentially large counter bounds.
• The paper’s findings imply that verification problems reducible from Petri nets require exponential resources, challenging efficient algorithm design.

The Reachability Problem for Petri Nets is Not Elementary

Introduction

"The Reachability Problem for Petri Nets is Not Elementary" (1809.07115) addresses a longstanding question concerning the complexity of the reachability problem in Petri nets. Petri nets are a well-known model for representing concurrent systems, having extensive applications in various domains, including hardware, software, and process modeling. The core problem, called the reachability problem, is to determine if a Petri net can transition from a given initial state to a specified final state. Although decidability was established decades ago, the precise complexity class was elusive. This paper establishes a non-elementary lower bound, significantly advancing our understanding of this critical problem in verification theory.

Core Contributions

The authors demonstrate that the reachability problem requires more than elementary time and space. Specifically, they show that the problem's complexity exceeds any finite tower of exponentials, refuting the conjecture that positioned the reachability problem as -complete. This extends beyond prior work that only suggested exponential space lower bounds [Lipton, 1976].

1. Non-Elementary Lower Bound: The paper demonstrates that the reachability problem cannot be resolved within an elementary function of the input size. This result implies that the complexity grows faster than any feasible combination of exponentials and iterations of the Ackermann function.
2. Implications for Related Problems: The implications extend to various problems reducible from Petri net reachability, indicating that problems in areas such as formal languages, logic, and process calculi are similarly non-elementary.
3. Novel Gadget - Factorial Amplifier: Central to their proof is a novel construction called the "factorial amplifier," which uses counters limited by $k$ to produce large integer pairs with a precise ratio derived from factorials. This concept facilitates constructing Petri nets that simulate complex computations.

Implementation Details

1. Petri Net Construction: The factorial amplifier plays a crucial role in creating net configurations that simulate Minsky machines with counters bounded by exponentially large values. This involves constructing a sequence of operations and gadgets that ensure the reachability complexity escalates in non-elementary fashion.
2. Simulating Minsky Machines: The authors leverage a sequence of amplifiers composed with themselves to simulate Minsky machines with rapidly growing counter bounds. This provides the basis for establishing non-elementary complexity.
3. Counterexamples and Proof Strategies: Schmitz’s survey and other theoretical frameworks serve as a foundation for establishing these measures. The application of ratio and counter-programming demonstrate meticulous construction by deploying zero tests and maintaining invariants throughout execution trails.

Implications

The non-elementary lower bound suggests that even with advancements in algorithmic theory, reachability for Petri nets resists efficient solutions. This positions Petri nets and associated computations in a category necessitating exponential resources, challenging efficiency in future computational frameworks. The findings further obsolesce prior bounds for structures including branching and Pushdown Vector Addition Systems (VASS).

Conclusion

The paper marks a significant milestone in verification theory by resolving a central open question concerning the complexity of Petri nets' reachability. It provides not just a deeper theoretical insight but also sets a direction for future explorations into related system complexity challenges. As computational models evolve, understanding these theoretical limit bounds will critically align with practical advancements in complexity-driven system design and analysis.