Efficient algorithms for asymptotic bounds on termination time in VASS Conference Paper


Author(s): Brázdil, Tomáš; Chatterjee, Krishnendu; Kučera, Antonín; Novotný, Petr; Velan, Dominik; Zuleger, Florian
Title: Efficient algorithms for asymptotic bounds on termination time in VASS
Title Series: ACM/IEEE Symposium on Logic in Computer Science
Affiliation IST Austria
Abstract: 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(nk), 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 (nk). Our results are based on new insights into the geometry of VASS dynamics, which hold the potential for further applicability to VASS analysis.
Keywords: Asymptotic complexity; Polynomial approximation; Polynomial-time algorithms; Vectors; Vector addition systems; Computer circuits; Asymptotic bounds; Concurrent process; Fundamental models; Parameterized system; Resource bound analysis
Conference Title: LICS: Logic in Computer Science
Volume: Part F138033
Conference Dates: 9 - 12 July, 2018
Conference Location: Oxford, UK
ISBN: 10436871 (ISSN); 9781450355834 (ISBN); 9781450355834 (ISBN)
Publisher: IEEE  
Date Published: 2018-07-09
Start Page: 185
End Page: 194
DOI: 10.1145/3209108.3209191
Open access: no
IST Austria Authors
  1. Petr Novotny
    11 Novotny
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