Reachability in Two-Dimensional Unary Vector Addition Systems with States is NL-Complete

Englert, Matthias; Lazić, Ranko; Totzke, Patrick

doi:10.1145/2933575.2933577

Cited by 27 publications

(29 citation statements)

References 8 publications

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“…The latter include investigating implications for the reachability problem for fixed-dimension flat vector addition systems with states (cf. [3,12,35]).…”

Section: Discussionmentioning

confidence: 99%

The Reachability Problem for Petri Nets Is Not Elementary

Czerwiński

Lasota

Lazić

et al. 2020

J. ACM

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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.

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“…The latter include investigating implications for the reachability problem for fixed-dimension flat vector addition systems with states (cf. [3,12,35]).…”

Section: Discussionmentioning

confidence: 99%

The Reachability Problem for Petri Nets Is Not Elementary

Czerwiński

Lasota

Lazić

et al. 2020

J. ACM

Self Cite

View full text Add to dashboard Cite

show abstract

“…The latter include investigating implications for the reachability problem for fixed-dimension flat vector addition systems with states (cf. [35,3,12]).…”

Section: Discussionmentioning

confidence: 99%

The reachability problem for Petri nets is not elementary

Czerwiński

Lasota

Lazić

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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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.At the heart of our proof is a novel gadget so called the 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. We also develop a novel construction that uses arbitrarily large pairs of values with ratio R to provide zero testable counters that are bounded by R. Repeatedly composing the factorial amplifier with itself by means of the construction then 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 establish hardness for h-exponential space already for Petri nets with h + 13 counters. *

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“…This article has been produced by completing, integrating and streamlining the conference papers [3] and [13]. Ever since their publication, the main results as well as technical results that we present in this paper have found applications for a variety of problems.…”

Section: Contributionsmentioning

confidence: 98%

The Reachability Problem for Two-Dimensional Vector Addition Systems with States

Blondin

EnglertMatthias

FinkelAlain

et al. 2021

J. ACM

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We prove that the reachability problem for two-dimensional vector addition systems with states is NL-complete or PSPACE-complete, depending on whether the numbers in the input are encoded in unary or binary. As a key underlying technical result, we show that, if a configuration is reachable, then there exists a witnessing path whose sequence of transitions is contained in a bounded language defined by a regular expression of pseudo-polynomially bounded length. This, in turn, enables us to prove that the lengths of minimal reachability witnesses are pseudo-polynomially bounded.

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Reachability in Two-Dimensional Unary Vector Addition Systems with States is NL-Complete

Cited by 27 publications

References 8 publications

The Reachability Problem for Petri Nets Is Not Elementary

The Reachability Problem for Petri Nets Is Not Elementary

The reachability problem for Petri nets is not elementary

The Reachability Problem for Two-Dimensional Vector Addition Systems with States

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