Efficient Algorithms for Asymptotic Bounds on Termination Time in VASS

Brázdil, Tomǎš; Chatterjee, Krishnendu; Kučera, Antonín; Novotný, Petr; Velan, Dominik; Zuleger, Florian

doi:10.1145/3209108.3209191

Cited by 12 publications

(25 citation statements)

References 78 publications

(128 reference statements)

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“…The results build on analogous results for non-probabilistic VASS established in [10], combining them with a novel probabilistic analysis.…”

supporting

confidence: 58%

“…A polynomial-time algorithm deciding asymptotic linearity of termination time for purely non-deterministic VASS (where the set Q p is empty) was given in [10]. Theorem 1 generalizes this result to VASS MDPs.…”

Section: Outline Of Techniquesmentioning

confidence: 83%

“…Theorem 1 generalizes this result to VASS MDPs. We start by recalling the results of [10] and sketching the main ideas behind the proof of Theorem 1. These ideas are then elaborated in subsequent sections.…”

Section: Outline Of Techniquesmentioning

confidence: 99%

“…The "only if" part is more elaborate. In [10], it was shown that if the increments are not contained in an open half-space with positive normal, then for all sufficiently large n, there is a non-terminating computation initiated in pn whose length is at least n 2 /c for some constant c. This computation consists of simple cycles and auxiliary short paths used to "switch" from one control state to another. Now let A be a VASS MDP with d counters.…”

Section: Outline Of Techniquesmentioning

confidence: 99%

“…The study on asymptotic termination complexity of non-probabilistic VASS, initiated in [10] was continued in [36], where the existence of some k such that L ∈ O(n k ) was also shown decidable in polynomial time.…”

mentioning

confidence: 99%

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Deciding Fast Termination for Probabilistic VASS with Nondeterminism

Brázdil

Chatterjee

Kučera

et al. 2019

Automated Technology for Verification and Analysis

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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

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“…The results build on analogous results for non-probabilistic VASS established in [10], combining them with a novel probabilistic analysis.…”

supporting

confidence: 58%

Section: Outline Of Techniquesmentioning

confidence: 83%

Section: Outline Of Techniquesmentioning

confidence: 99%

Section: Outline Of Techniquesmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Deciding Fast Termination for Probabilistic VASS with Nondeterminism

Brázdil

Chatterjee

Kučera

et al. 2019

Automated Technology for Verification and Analysis

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The Polynomial Complexity of Vector Addition Systems with States

Zuleger

2020

Lecture Notes in Computer Science

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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.

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ATLAS: Automated Amortised Complexity Analysis of Self-adjusting Data Structures

Leutgeb

Moser

Zuleger

2021

Computer Aided Verification

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Being able to argue about the performance of self-adjusting data structures such as splay trees has been a main objective, when Sleator and Tarjan introduced the notion of amortised complexity.Analysing these data structures requires sophisticated potential functions, which typically contain logarithmic expressions. Possibly for these reasons, and despite the recent progress in automated resource analysis, they have so far eluded automation. In this paper, we report on the first fully-automated amortised complexity analysis of self-adjusting data structures. Following earlier work, our analysis is based on potential function templates with unknown coefficients.We make the following contributions: 1) We encode the search for concrete potential function coefficients as an optimisation problem over a suitable constraint system. Our target function steers the search towards coefficients that minimise the inferred amortised complexity. 2) Automation is achieved by using a linear constraint system in conjunction with suitable lemmata schemes that encapsulate the required non-linear facts about the logarithm. We discuss our choices that achieve a scalable analysis. 3) We present our tool $$\mathsf {ATLAS}$$ ATLAS and report on experimental results for splay trees, splay heaps and pairing heaps. We completely automatically infer complexity estimates that match previous results (obtained by sophisticated pen-and-paper proofs), and in some cases even infer better complexity estimates than previously published.

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Efficient Algorithms for Asymptotic Bounds on Termination Time in VASS

Cited by 12 publications

References 78 publications

Deciding Fast Termination for Probabilistic VASS with Nondeterminism

Deciding Fast Termination for Probabilistic VASS with Nondeterminism

The Polynomial Complexity of Vector Addition Systems with States

ATLAS: Automated Amortised Complexity Analysis of Self-adjusting Data Structures

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