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.

A probabilistic vector addition system with states (pVASS) is a finite state Markov process augmented with non-negative integer counters that can be incremented or decremented during each state transition, blocking any behaviour that would cause a counter to decrease below zero. The pVASS can be used as abstractions of probabilistic programs with many decidable properties. The use of pVASS as abstractions requires the presence of nondeterminism in the model. In this paper, we develop techniques for checking fast termination of pVASS with nondeterminism. That is, for every initial configuration of size n, we consider the worst expected number of transitions needed to reach a configuration with some counter negative (the expected termination time). We show that the problem whether the asymptotic expected termination time is linear is decidable in polynomial time for a certain natural class of pVASS with nondeterminism. Furthermore, we show the following dichotomy: if the asymptotic expected termination time is not linear, then it is at least quadratic, i.e., in Ω(n 2 ).Keywords: angelic and demonic nondeterminism · termination time · probabilistic VASS IntroductionProbabilistic Programs & VASS Probabilistic systems play an important role in various areas of computing such as machine learning [26], network protocol design [25], robotics [45], privacy and security [5], and many others. For this reason, verification of probabilistic systems receives a considerable attention of the verification community. As in the classical (non-probabilistic) setting, in probabilistic verification one typically constructs a suitable abstract model overapproximating the real behaviour of the system. In the past, the verification research was focused mostly on finite-state probabilistic models [4] as well as some ⋆ Tomáš Brázdil

The termination complexity of a given VASS is a function L assigning to every 𝑛 the length of the longest nonterminating computation initiated in a configuration with all counters bounded by 𝑛. We show that for every VASS with demonic nondeterminism and every fixed 𝑘, the problem whether L ∈ G 𝑘 , where G 𝑘 is the 𝑘-th level in the Grzegorczyk hierarchy, is decidable in polynomial time. Furthermore, we show that if L ∉ G 𝑘 , then L grows at least as fast as the generator 𝐹 𝑘+1 of G 𝑘+1 . Hence, for every terminating VASS, the growth of L can be reasonably characterized by the leastFurthermore, we consider VASS with both angelic and demonic nondeterminism, i.e., VASS games where the players aim at lowering/raising the termination time. We prove that for every fixed 𝑘, the problem whether L ∈ G 𝑘 for a given VASS game is NP-complete. Furthermore, if L ∉ G 𝑘 , then L grows at least as fast as 𝐹 𝑘+1 .CCS Concepts: • Theory of computation → Abstract machines.

No abstract

No abstract