On Functions Weakly Computable by Petri Nets and Vector Addition Systems

Leroux, Jérôme; Schnoebelen, Philippe

doi:10.1007/978-3-319-11439-2_15

Cited by 7 publications

(7 citation statements)

References 32 publications

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“…For example, we cannot hope to implement a macro whose unique complete execution performs some given counter operations exactly a triply exponential number of times. (2) It has been known for many years how Petri nets can compute various functions weakly, in the sense that the result may be nondeterministically either correct or smaller [34,41] 2 . Most notably, for all natural numbers n, Grzegorczyk's function [38] F n is computable weakly by a Petri net of size O(n).…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 99%

The Reachability Problem for Petri Nets Is Not Elementary

Czerwiński

Lasota

Lazić

et al. 2020

J. ACM

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“…The reachability problem is decidable [22,24,28], and EXPSPACE-hard [27], and the current bestknown upper bound is cubic Ackermannian [25], a complexity class belonging to the third level of a fast-growing complexity hierarchy introduced in [31]. Functions (non)computable by VASS are studied in [26]. Our algorithm for computing polynomial bounds can be seen as the dual (in the sense of linear programming) of the algorithm of [23]; this connection is the basis for the completeness of our ranking function construction (we further comment on the connection to [23] in Section 4).…”

Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for Asymptotic Bounds on Termination Time in VASS

Brázdil

Chatterjee

Kučera

et al. 2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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Vector Addition Systems with States (VASS) provide a well-known and fundamental model for the analysis of concurrent processes, parameterized systems, and are also used as abstract models of programs in resource bound analysis. In this paper we study the problem of obtaining asymptotic bounds on the termination time of a given VASS. In particular, we focus on the practically important case of obtaining polynomial bounds on termination time. Our main contributions are as follows: First, we present a polynomial-time algorithm for deciding whether a given VASS has a linear asymptotic complexity. We also show that if the complexity of a VASS is not linear, it is at least quadratic. Second, we classify VASS according to quantitative properties of their cycles. We show that certain singularities in these properties are the key reason for non-polynomial asymptotic complexity of VASS. In absence of singularities, we show that the asymptotic complexity is always polynomial and of the form Θ(n k ), for some integer k ≤ d, where d is the dimension of the VASS. We present a polynomial-time algorithm computing the optimal k. For general VASS, the same algorithm, which is based on a complete technique for the construction of ranking functions in VASS, produces a valid lower bound, i.e., a k such that the termination complexity is Ω(n k ). Our results are based on new insights into the geometry of VASS dynamics, which hold the potential for further applicability to VASS analysis.

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“…Since this definition does not accommodate nondeterminism nicely (which is natural for multiset rewriting), we say a multiset rewriting system weakly computes a numerical function f : N → N if there exists a run with M f (O) = f (n) and for any other run holds that M f (O) ≤ f (n). This definition is an adoption of weak computability for Petri nets [14].…”

Section: Expressive Powermentioning

confidence: 99%