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Deciding Polynomial Termination Complexity for VASS Programs [article]

Michal Ajdarów, Antonín Kučera
<span title="2021-12-05">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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We show that for every fixed k≥ 3, the problem whether the termination/counter complexity of a given demonic VASS is 𝒪(n^k), Ω(n^k), and Θ(n^k) is coNP-complete, NP-complete, and DP-complete, respectively  ...  This shows that the polynomial-time algorithm designed for strongly connected demonic VASS in previous works cannot be extended to the general case.  ...  In this paper, we give a full classification of the computational complexity of deciding polynomial termination/counter complexity for demonic VASS and VASS games, and solve open problems formulated in  ... 
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Michal Ajdarów, Antonín Kučera. "Deciding Polynomial Termination Complexity for VASS Programs." arXiv (2021)

Efficient Algorithms for Checking Fast Termination in VASS [article]

Tomáš Brázdil, Krishnendu Chatterjee, Antonín Kučera, Petr Novotný, Dominik Velan
<span title="2017-08-29">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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Hence we focus on polynomial asymptotic bounds for VASS.  ...  VASS provide a fundamental model for analysis of concurrent processes, parametrized systems, and they are also used as abstract models for programs for bounds analysis.  ...  In the course of the proof we describe a polynomial time-algorithm for deciding whether given VASS has linear termination complexity.  ... 
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Tomáš Brázdil, Krishnendu Chatterjee, Antonín Kučera, Petr Novotný, Dominik Velan. "Efficient Algorithms for Checking Fast Termination in VASS." arXiv (2017)

Efficient Algorithms for Asymptotic Bounds on Termination Time in VASS [article]

Tomáš Brázdil, Krishnendu Chatterjee, Antonín Kučera, Petr Novotný, Dominik Velan, Florian Zuleger
<span title="2018-04-29">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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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 show that certain singularities in these properties are the key reason for non-polynomial asymptotic complexity of VASS.  ...  Polynomial termination time We now concentrate on VASS with polynomial termination complexity. For simplicity, we restrict ourselves to strongly connected VASS.  ... 
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Citation

Tomáš Brázdil, Krishnendu Chatterjee, Antonín Kučera, Petr Novotný, Dominik Velan, Florian Zuleger. "Efficient Algorithms for Asymptotic Bounds on Termination Time in VASS." arXiv (2018)

The Polynomial Complexity of Vector Addition Systems with States [chapter]

Florian Zuleger
<span title="">2020</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;

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For these systems, we are interested in the standard notion of computational time complexity, i.e., we want to understand the length of the longest trace for a fixed vector addition system with states  ...  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  ...  [15] presents a PTIME procedure for deciding whether a VASS is polynomial or at least exponential, but does not give a precise analysis in case of polynomial complexity.  ... 
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Florian Zuleger. "The Polynomial Complexity of Vector Addition Systems with States." Lecture Notes in Computer Science (2020) 622-641

Reachability in Register Machines with Polynomial Updates [chapter]

Alain Finkel, Stefan Göller, Christoph Haase
<span title="">2013</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;

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We moreover give a classification of the complexity of reachability according to the type of polynomials allowed and the geometry induced by the range-constraining formula.  ...  This paper introduces a class of register machines whose registers can be updated by polynomial functions when a transition is taken, and the domain of the registers can be constrained by linear constraints  ...  However for polynomials of higher degree and loops with a richer control structure, deciding termination becomes non-obvious.  ... 
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Alain Finkel, Stefan Göller, Christoph Haase. "Reachability in Register Machines with Polynomial Updates." Lecture Notes in Computer Science (2013) 409-420

Resource control graphs

Jean-Yves Moyen
<span title="2009-08-01">2009</span> <i title="Association for Computing Machinery (ACM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3kjjbv4edjbezl2w37ipy5lo5a" style="color: black;">ACM Transactions on Computational Logic</a> </i> &nbsp;

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This allows to build termination proofs similar to the ones in Size Change Termination. But the size of states can also be used to represent the space used by the program at each point.  ...  This new tool is able to encompass several existing analyses and similarities with other studies hint that even more might be expressable in this framework thus giving hopes for a generic tool for studying  ...  Hofmann for pointing out the name "homogeneous coordinates" and its use in computer graphics.  ... 
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Citation

Jean-Yves Moyen. "Resource control graphs." ACM Transactions on Computational Logic 10.4 (2009) 1-44

Exhibition of a Structural Bug with Wings [chapter]

Florent Avellaneda, Rémi Morin
<span title="">2014</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;

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The algorithm searches for a counterexample in the form of a multiset of arcs computed by means of linear programming.  ...  We show how to solve this problem in polynomial time by a reduction to a linear program using a separation algorithm.  ...  Recall that checking termination of a Petri net requires exponential space [17] whereas we can check structural termination of a VASS in polynomial time.  ... 
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Florent Avellaneda, Rémi Morin. "Exhibition of a Structural Bug with Wings." Lecture Notes in Computer Science (2014) 253-272

Long-Run Average Behavior of Vector Addition Systems with States

Krishnendu Chatterjee, Thomas A. Henzinger, Jan Otop, Michael Wagner
<span title="2019-08-26">2019</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/iv4yo5vao5ctfjfushi4akt5xi" style="color: black;">International Conference on Concurrency Theory</a> </i> &nbsp;

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For uniform cost functions, we show that the regular long-run emptiness problem is (a) decidable in polynomial time for integer-valued VASS, and (b) decidable but nonelementarily hard for natural-valued  ...  For general cost functions, we show that the problem is (c) NP-complete for integer-valued VASS, and (d) undecidable for natural-valued VASS.  ...  Acknowledgements We thank Petr Novotný for helpful discussion on the literature.  ... 
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Krishnendu Chatterjee, Thomas A. Henzinger, Jan Otop, Michael Wagner. "Long-Run Average Behavior of Vector Addition Systems with States." International Conference on Concurrency Theory (2019) 27:1-27:16

Long-Run Average Behavior of Vector Addition Systems with States [article]

Krishnendu Chatterjee, Thomas A. Henzinger, Jan Otop
<span title="2019-05-14">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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For uniform cost functions, we show that the regular long-run emptiness problem is (a)~decidable in polynomial time for integer-valued VASS, and (b)~decidable but nonelementarily hard for natural-valued  ...  For general cost functions, we show that the problem is (c)~NP-complete for integer-valued VASS, and (d)~undecidable for natural-valued VASS.  ...  Acknowledgment We would like to thank Petr Novotn for discussion on literature on VASS and the long-run average property.  ... 
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Citation

Krishnendu Chatterjee, Thomas A. Henzinger, Jan Otop. "Long-Run Average Behavior of Vector Addition Systems with States." arXiv (2019)

Multi-dimensional Long-Run Average Problems for Vector Addition Systems with States [article]

Krishnendu Chatterjee, Thomas A. Henzinger, Jan Otop
<span title="2020-07-17">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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); and (c) can be solved in polynomial time for probabilistic integer-valued VASS, and probabilistic natural-valued VASS when all computations are non-terminating.  ...  In this work we consider multi-dimensional long-run average objectives for VASS and probabilistic VASS.  ...  For natural-valued probabilistic VASS, the presence of terminating runs makes even defining the probability measure complex.  ... 
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Krishnendu Chatterjee, Thomas A. Henzinger, Jan Otop. "Multi-dimensional Long-Run Average Problems for Vector Addition Systems with States." arXiv (2020)

Multi-Dimensional Long-Run Average Problems for Vector Addition Systems with States

Krishnendu Chatterjee, Thomas A. Henzinger, Jan Otop, Laura Kovács, Igor Konnov
<span title="2020-08-26">2020</span> <i title="Schloss Dagstuhl - Leibniz-Zentrum für Informatik"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/iv4yo5vao5ctfjfushi4akt5xi" style="color: black;">International Conference on Concurrency Theory</a> </i> &nbsp;

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); and (c) can be solved in polynomial time for probabilistic integer-valued VASS, and probabilistic natural-valued VASS when all computations are non-terminating.  ...  In this work we consider multi-dimensional long-run average objectives for VASS and probabilistic VASS.  ...  Acknowledgements We want to thank the anonymous reviewers for their helpful comments, which contributed to the final version of this paper.  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200828192535/https://drops.dagstuhl.de/opus/volltexte/2020/12835/pdf/LIPIcs-CONCUR-2020-23.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/5f/b3/5fb32aa30793ea58dedebd29baca63901a814147.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.concur.2020.23"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>
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Krishnendu Chatterjee, Thomas A. Henzinger, Jan Otop, Laura Kovács, Igor Konnov. "Multi-Dimensional Long-Run Average Problems for Vector Addition Systems with States." International Conference on Concurrency Theory (2020) 23:1-23:22

Ranking Functions for Vector Addition Systems [article]

Florian Zuleger
<span title="2017-10-23">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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Moreover, we show how ranking functions can be used for the computational complexity analysis of vector addition systems (here complexity refers to the number of steps the vector addition system under  ...  addition system, which can be reported to the user of a verification tool, and can be used as certificates for termination.  ...  Witness for Non-Termination Next we state a criterion for non-termination of a VASS. Definition 2 (Witness for non-termination). Let V be a VASS.  ... 
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Florian Zuleger. "Ranking Functions for Vector Addition Systems." arXiv (2017)

Reachability Analysis of Concurrent Boolean Programs with Symbolic Counter Abstraction

Junyan Qian, Min Li, Lingzhong Zhao
<span title="">2012</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cx3f4s3qmfe6bg4qvuy2cxezyu" style="color: black;">Procedia Computer Science</a> </i> &nbsp;

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We present a model checking algorithm for thread-state reachability analysis of concurrent Boolean programs, constructing Karp-Miller coverability tree directly on a Boolean program.  ...  A basic problem in software model checking is the choice of a model for software. Boolean program is the most popular representation and is amenable to model checking.  ...  As a result, the time complexity of thread-state reachability problem is equivalent to that of the coverability problem for VASS that are polynomial. Corollary 2.  ... 
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Junyan Qian, Min Li, Lingzhong Zhao. "Reachability Analysis of Concurrent Boolean Programs with Symbolic Counter Abstraction." Procedia Computer Science (2012) 1363-1370

On the complexity of the linear-time μ-calculus for Petri Nets [chapter]

Peter Habermehl
<span title="">1997</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;

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We also show that for the subclass of BPPs the problem has already the same complexity as for arbitrary nets.  ...  In this paper we show that the complexity in the size of the formula can be reduced to polynomial space. We also prove that this is the best one can do.  ...  Acknowledgement: The author would like to thank Javier Esparza for very helpful discussions on this topic.  ... 
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Peter Habermehl. "On the complexity of the linear-time μ-calculus for Petri Nets." Lecture Notes in Computer Science (1997) 102-116

The Polynomial Complexity of Vector Addition Systems with States [article]

Florian Zuleger
<span title="2020-03-14">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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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  ...  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  ...  [15] presents a PTIME procedure for deciding whether a VASS is polynomial or at least exponential, but does not give a precise analysis in case of polynomial complexity.  ... 
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Florian Zuleger. "The Polynomial Complexity of Vector Addition Systems with States." arXiv (2020)

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