{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T03:10:58Z","timestamp":1780629058750,"version":"3.54.1"},"publisher-location":"Cham","reference-count":21,"publisher":"Springer International Publishing","isbn-type":[{"value":"9783030452308","type":"print"},{"value":"9783030452315","type":"electronic"}],"license":[{"start":{"date-parts":[[2020,1,1]],"date-time":"2020-01-01T00:00:00Z","timestamp":1577836800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>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 time 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<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varTheta (N^k)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>\u0398<\/mml:mi><mml:mo>(<\/mml:mo><mml:msup><mml:mi>N<\/mml:mi><mml:mi>k<\/mml:mi><\/mml:msup><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>for some computable integer<jats:italic>k<\/jats:italic>, where<jats:italic>N<\/jats:italic>is the size of the initial configuration, or at least exponential. We further show that<jats:italic>k<\/jats:italic>can be computed in polynomial time in the size of the considered vector addition system. Finally, we show that<jats:inline-formula><jats:alternatives><jats:tex-math>$$1 \\le k \\le 2^n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>\u2264<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>\u2264<\/mml:mo><mml:msup><mml:mn>2<\/mml:mn><mml:mi>n<\/mml:mi><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where<jats:italic>n<\/jats:italic>is the dimension of the considered vector addition system.<\/jats:p>","DOI":"10.1007\/978-3-030-45231-5_32","type":"book-chapter","created":{"date-parts":[[2020,4,17]],"date-time":"2020-04-17T10:02:53Z","timestamp":1587117773000},"page":"622-641","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["The Polynomial Complexity of Vector Addition Systems with States"],"prefix":"10.1007","author":[{"given":"Florian","family":"Zuleger","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,4,17]]},"reference":[{"key":"32_CR1","doi-asserted-by":"crossref","unstructured":"Parosh Aziz Abdulla, Giorgio Delzanno, and Laurent van Begin. A language-based comparison of extensions of Petri nets with and without whole-place operations. In LATA, pages 71\u201382, 2009.","DOI":"10.1007\/978-3-642-00982-2_6"},{"key":"32_CR2","doi-asserted-by":"crossref","unstructured":"Benjamin Aminof, Sasha Rubin, and Florian Zuleger. On the expressive power of communication primitives in parameterised systems. In LPAR, pages 313\u2013328, 2015.","DOI":"10.1007\/978-3-662-48899-7_22"},{"key":"32_CR3","doi-asserted-by":"crossref","unstructured":"Benjamin Aminof, Sasha Rubin, Florian Zuleger, and Francesco Spegni. Liveness of parameterized timed networks. In ICALP, pages 375\u2013387, 2015.","DOI":"10.1007\/978-3-662-47666-6_30"},{"key":"32_CR4","doi-asserted-by":"crossref","unstructured":"Roderick Bloem, Swen Jacobs, Ayrat Khalimov, Igor Konnov, Sasha Rubin, Helmut Veith, and Josef Widder. Decidability in parameterized verification. SIGACT News, 47(2):53\u201364, 2016.","DOI":"10.1145\/2951860.2951873"},{"key":"32_CR5","doi-asserted-by":"crossref","unstructured":"Tom\u00e1s Br\u00e1zdil, Krishnendu Chatterjee, Anton\u00edn Kucera, Petr Novotn\u00fd, Dominik Velan, and Florian Zuleger. Efficient algorithms for asymptotic bounds on termination time in VASS. In LICS, pages 185\u2013194, 2018.","DOI":"10.1145\/3209108.3209191"},{"key":"32_CR6","doi-asserted-by":"crossref","unstructured":"Leonard Dickson. Finiteness of the odd perfect and primitive abundant numbers with $$n$$ distinct prime factors. Am. J. Math, 35:413\u2013422, 1913.","DOI":"10.2307\/2370405"},{"key":"32_CR7","unstructured":"Javier Esparza and Mogens Nielsen. Decidability issues for Petri nets - a survey. Elektronische Informationsverarbeitung und Kybernetik, 30(3):143\u2013160, 1994."},{"key":"32_CR8","doi-asserted-by":"crossref","unstructured":"Alain Finkel, Gilles Geeraerts, Jean-Fran\u00e7ois Raskin, and Laurent van Begin. On the omega-language expressive power of extended Petri nets. TCS, 356(3):374\u2013386, 2006.","DOI":"10.1016\/j.tcs.2006.02.008"},{"key":"32_CR9","doi-asserted-by":"crossref","unstructured":"Steven M. German and A. Prasad Sistla. Reasoning about systems with many processes. J. ACM, 39(3), 675\u2013735, 1992.","DOI":"10.1145\/146637.146681"},{"key":"32_CR10","doi-asserted-by":"crossref","unstructured":"John E. Hopcroft and Jean-Jacques Pansiot. On the reachability problem for 5-dimensional vector addition systems. TCS, 8:135\u2013159, 1979.","DOI":"10.1016\/0304-3975(79)90041-0"},{"key":"32_CR11","doi-asserted-by":"crossref","unstructured":"Annu John, Igor Konnov, Ulrich Schmid, Helmut Veith, and Josef Widder. Parameterized model checking of fault-tolerant distributed algorithms by abstraction. In FMCAD, pages 201\u2013209, 2013.","DOI":"10.1007\/978-3-642-39176-7_14"},{"key":"32_CR12","doi-asserted-by":"crossref","unstructured":"Alexander Kaiser, Daniel Kroening, and Thomas Wahl. A widening approach to multithreaded program verification. TOPLAS, 36(4):14:1\u201314:29, 2014.","DOI":"10.1145\/2629608"},{"key":"32_CR13","doi-asserted-by":"crossref","unstructured":"Richard M. Karp and Raymond E. Miller. Parallel program schemata. J. Comput. Syst. Sci., 3(2):147\u2013195, 1969.","DOI":"10.1016\/S0022-0000(69)80011-5"},{"key":"32_CR14","doi-asserted-by":"crossref","unstructured":"S. Rao Kosaraju and Gregory F. Sullivan. Detecting cycles in dynamic graphs in polynomial time (preliminary version). In STOC, pages 398\u2013406, 1988.","DOI":"10.1145\/62212.62251"},{"key":"32_CR15","unstructured":"J\u00e9r\u00f4me Leroux. Polynomial vector addition systems with states. In ICALP, pages 134:1\u2013134:13, 2018."},{"key":"32_CR16","unstructured":"Richard J. Lipton. The Reachability Problem Requires Exponential space. Research report 62. Department of Computer Science, Yale University, 1976."},{"key":"32_CR17","doi-asserted-by":"crossref","unstructured":"Charles Rackoff. The covering and boundedness problems for vector addition systems. TCS, 6:223\u2013231, 1978.","DOI":"10.1016\/0304-3975(78)90036-1"},{"key":"32_CR18","doi-asserted-by":"crossref","unstructured":"Moritz Sinn, Florian Zuleger, and Helmut Veith. A simple and scalable static analysis for bound analysis and amortized complexity analysis. In CAV, pages 745\u2013761, 2014.","DOI":"10.1007\/978-3-319-08867-9_50"},{"key":"32_CR19","doi-asserted-by":"crossref","unstructured":"Moritz Sinn, Florian Zuleger, and Helmut Veith. Difference constraints: An adequate abstraction for complexity analysis of imperative programs. In FMCAD, pages 144\u2013151, 2015.","DOI":"10.1109\/FMCAD.2015.7542264"},{"key":"32_CR20","doi-asserted-by":"crossref","unstructured":"Moritz Sinn, Florian Zuleger, and Helmut Veith. Complexity and resource bound analysis of imperative programs using difference constraints. JAR, 59:3\u201345, 2017.","DOI":"10.1007\/s10817-016-9402-4"},{"key":"32_CR21","unstructured":"Florian Zuleger. The polynomial complexity of vector addition systems with states. CoRR, abs\/1907.01076, 2019."}],"container-title":["Lecture Notes in Computer Science","Foundations of Software Science and Computation Structures"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-030-45231-5_32","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,21]],"date-time":"2022-10-21T16:03:50Z","timestamp":1666368230000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-030-45231-5_32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020]]},"ISBN":["9783030452308","9783030452315"],"references-count":21,"URL":"https:\/\/doi.org\/10.1007\/978-3-030-45231-5_32","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020]]},"assertion":[{"value":"17 April 2020","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"FoSSaCS","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference on Foundations of Software Science and Computation Structures","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Dublin","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Ireland","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2020","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"25 April 2020","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"30 April 2020","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"23","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"fossacs2020","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/www.etaps.org\/2020\/fossacs","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Single-blind","order":1,"name":"type","label":"Type","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"EasyChair","order":2,"name":"conference_management_system","label":"Conference Management System","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"98","order":3,"name":"number_of_submissions_sent_for_review","label":"Number of Submissions Sent for Review","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"31","order":4,"name":"number_of_full_papers_accepted","label":"Number of Full Papers Accepted","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"0","order":5,"name":"number_of_short_papers_accepted","label":"Number of Short Papers Accepted","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"32% - The value is computed by the equation \"Number of Full Papers Accepted \/ Number of Submissions Sent for Review * 100\" and then rounded to a whole number.","order":6,"name":"acceptance_rate_of_full_papers","label":"Acceptance Rate of Full Papers","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"3","order":7,"name":"average_number_of_reviews_per_paper","label":"Average Number of Reviews per Paper","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"12","order":8,"name":"average_number_of_papers_per_reviewer","label":"Average Number of Papers per Reviewer","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"Yes","order":9,"name":"external_reviewers_involved","label":"External Reviewers Involved","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"The conference could not take place due to the COVID-19 pandemic. There was an online event on July 2, 2020.","order":10,"name":"additional_info_on_review_process","label":"Additional Info on Review Process","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}}]}}