25 research outputs found
The Reachability Problem for Petri Nets is Not Elementary
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.Comment: Final version of STOC'1
The reachability problem for Petri nets is not elementary
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 reachability problem for petri nets is not elementary
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
On Functions Weakly Computable by Pushdown Petri Nets and Related Systems
We consider numerical functions weakly computable by grammar-controlled
vector addition systems (GVASes, a variant of pushdown Petri nets). GVASes can
weakly compute all fast growing functions for
, hence they are computationally more powerful than
standard vector addition systems. On the other hand they cannot weakly compute
the inverses or indeed any sublinear function. The proof relies
on a pumping lemma for runs of GVASes that is of independent interest
Reachability in Fixed VASS: Expressiveness and Lower Bounds
The recent years have seen remarkable progress in establishing the complexity
of the reachability problem for vector addition systems with states (VASS),
equivalently known as Petri nets. Existing work primarily considers the case in
which both the VASS as well as the initial and target configurations are part
of the input. In this paper, we investigate the reachability problem in the
setting where the VASS is fixed and only the initial configuration is variable.
We show that fixed VASS fully express arithmetic on initial segments of the
natural numbers. It follows that there is a very weak reduction from any fixed
such number-theoretic predicate (e.g. primality or square-freeness) to
reachability in fixed VASS where configurations are presented in unary. If
configurations are given in binary, we show that there is a fixed VASS with
five counters whose reachability problem is PSPACE-hard
Geometry of VAS reachability sets
Vector Addition Systems (VAS) or equivalently petri-nets are a popular model
for representing concurrent systems. Many important decidability results about
VAS were obtained by considering geometric properties of their reachability
sets, i.e. the set of configurations reachable from some initial configuration
. For example, in 2012 Jerome Leroux proved that if a configuration
is not reachable, then there exists a semilinear inductive invariant separating
the reachability set from . This gave an alternative proof of decidability
of the reachability problem. The paper introduced the class of petri-sets,
proved that reachability sets are petri-sets, and that petri-sets have this
property. In a follow-up paper in 2013, Jerome Leroux again used the class of
petri-sets to prove that if a reachability set is semilinear, then a
representation of it can be computed. In this paper, we utilize the class of
petri-sets to answer the opposite type of question: Even if the reachability
set is non-semilinear, what form can it have? We give another proof that
semilinearity of the reachability set is decidable, which was first shown by
Hauschildt in 1990. We prove that reachability sets can be partitioned into
nicely shaped sets we call almost-hybridlinear, and how to utilize this to
decide semilinearity.Comment: 21 pages, 5 figure
Continuous Pushdown VASS in One Dimension are Easy
A pushdown vector addition system with states (PVASS) extends the model of
vector addition systems with a pushdown stack. The algorithmic analysis of
PVASS has applications such as static analysis of recursive programs
manipulating integer variables. Unfortunately, reachability analysis, even for
one-dimensional PVASS is not known to be decidable. We relax the model of
one-dimensional PVASS to make the counter updates continuous and show that in
this case reachability, coverability, and boundedness are decidable in
polynomial time. In addition, for the extension of the model with lower-bound
guards on the states, we show that coverability and reachability are in NP, and
boundedness is in coNP.Comment: 2 tables, 6 figures, 12 page