The Polynomial Complexity of Vector Addition Systems with States

Abstract

Vector addition systems are an important model in theoretical computer science and have been used in a variety of areas. In this paper, we consider vector addition systems with states over a parameterized initial configuration. For these systems, we are interested in the standard notion of computational complexity, i.e., we want to understand the length of the longest trace for a fixed vector addition system with states depending on the size of the initial configuration. We show that the asymptotic complexity of a given vector addition system with states is either Θ(N k ) for some computable integer k, where N is the size of the initial configuration, or at least exponential. We further show that k can be computed in polynomial time in the size of the considered vector addition system. Finally, we show that 1 ≤ k ≤ 2 n , where n is the dimension of the considered vector addition system.

Type-based analysis of logarithmic amortised complexity

“…We also mention the quest for abstract program models whose resource bound analysis problem is decidable, and for which the obtainable resource bounds can be precisely characterised. We list here the size-change abstraction, whose worst-case complexity has been completely characterised as polynomial (with rational coefficients) (Colcombet et al, 2014;Zuleger, 2015), vector-addition systems (Brázdil et al, 2018;Zuleger, 2020), for which polynomial complexity can be decided, and LOOP programs (Ben-Amram and Hamilton, 2019), for which multivariate polynomial bounds can be computed. We are not aware of similar results for program models that induce logarithmic bounds.…”

Type-Based Analysis of Logarithmic Amortised Complexity

“…We also mention the quest for abstract program models whose resource bound analysis problem is decidable, and for which the obtainable resource bounds can be precisely characterised. We list here the size-change abstraction, whose worst-case complexity has been completely characterised as polynomial (with rational coefficients) [14,56], vector-addition systems [12,57], for which polynomial complexity can be decided, and LOOP programs [10], for which multivariate polynomial bounds can be computed. We are not aware of similar results for programs models that induce logarithmic bounds.…”

Efficient Analysis of VASS Termination Complexity

“…□ A natural question is whether Theorem 4.2 can be extended to other complexity classes such as O (𝑛 𝑘 ). Recall that the problem whether L A ∈ O (𝑛 𝑘 ) for a given demonic VASS A is decidable in polynomial time, and ifL A ∉ O (𝑛 𝑘 ), then L A ∈ Ω(𝑛 𝑘+1 )[20]. However, our proof of Theorem 4.2 requires G 𝑘 being closed under function composition, which does not hold for O (𝑛 𝑘 ) and 𝑘 ≥ 2.…”

“…In the same paper, it is also shown that if L is not polynomial, then it is at least exponential, i.e., L ∈ 2 Ω (𝑛) . A recent result of [20] shows that if the termination complexity of a given demonic VASS is polynomial, then there is 𝑘 ∈ N computable in polynomial time such that L is Θ(𝑛 𝑘 ). A polynomial-time algorithm deciding the linearity of L for probabilistic VASS , i.e., VASS with demonic non-determinism and probabilistic choice 2 , is given in [2].…”