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 modeling 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, currently, the best published upper bound is non-primitive recursive Ackermannian of Leroux and Schmitz from Symposium on Logic in Computer Science 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, which are known to admit reductions from the Petri nets reachability problem, are also not elementary. Thirdly, it makes obsolete the current best lower bounds for the reachability problems for two key extensions of Petri nets: with branching and with a pushdown stack. We develop a construction that uses arbitrarily large pairs of values with ratio R to provide zero testable counters that are bounded by R . At the heart of our proof is then a novel gadget, the so-called factorial amplifier that, assuming availability of counters that are zero testable and bounded by k , guarantees to produce arbitrarily large pairs of values whose ratio is exactly the factorial of k . Repeatedly composing the factorial amplifier with itself by means of the former construction enables us to compute, in linear time, Petri nets that simulate Minsky machines whose counters are bounded by a tower of exponentials, which yields the non-elementary lower bound. By refining this scheme further, we, in fact, already establish hardness for h -exponential space for Petri nets with h + 13 counters.

Presburger arithmetic with stars, rational subsets of graph groups, and nested zero tests

“…Leaving aside decidability issues, one major obstacle in the automated analysis of VASS is the high computational complexity of decidable decision problems: control-state reachability is EXPSPACE-complete [39,49], and a non-elementary lower bound for the configuration-reachability problem has recently been established [11]. To the best of the authors' knowledge, no dedicated complexity results have been given for VASS with nested zero tests.…”

Forecasting the impact of environmental stresses on the frequent waves of COVID19

“…However, some areas have been hit harder than others in terms of infection and mortality rates. Maybe the differences have posed due to the impact of air pollution on the global spread of COVID-19 infections and mortality rates [11].…”

A lower bound for the coverability problem in acyclic pushdown VAS

“…In arbitrary dimension reachability reduces (and is thus equivalent) to the coverability problem, due to a simple logarithmic-space reduction that increases dimension by 1. Hence the lower bound of [3] is subsumed by a recent non-elementary lower bound for the reachability problem in VASS [6].…”