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**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

In this paper we propose two new subclasses of Petri nets with resets, for which the reachability and coverability problems become tractable. Namely, we add an acyclicity condition that only applies to the consumptions and productions, not the resets. The first class is acyclic Petri nets with resets, and we show that coverability is PSPACE-complete for them. This contrasts the known Ackermann-hardness for coverability in (not necessarily acyclic) Petri nets with resets. We prove that the reachability problem remains undecidable for acyclic Petri nets with resets. The second class concerns workflow nets, a practically motivated and natural subclass of Petri nets. Here, we show that both coverability and reachability in acyclic workflow nets with resets are PSPACE-complete. Without the acyclicity condition, reachability and coverability in workflow nets with resets are known to be equally hard as for Petri nets with resets, that being Ackermann-hard and undecidable, respectively.

Dmitry Chistikov, Wojciech Czerwiński, Piotr Hofman, Filip Mazowiecki, and Henry Sinclair-Banks. Acyclic Petri and Workflow Nets with Resets. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chistikov_et_al:LIPIcs.FSTTCS.2023.16, author = {Chistikov, Dmitry and Czerwi\'{n}ski, Wojciech and Hofman, Piotr and Mazowiecki, Filip and Sinclair-Banks, Henry}, title = {{Acyclic Petri and Workflow Nets with Resets}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {16:1--16:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.16}, URN = {urn:nbn:de:0030-drops-193892}, doi = {10.4230/LIPIcs.FSTTCS.2023.16}, annote = {Keywords: Petri nets, Workflow Nets, Resets, Acyclic, Reachability, Coverability} }

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**Published in:** LIPIcs, Volume 243, 33rd International Conference on Concurrency Theory (CONCUR 2022)

We consider the problems of language inclusion and language equivalence for Vector Addition Systems with States (VASSes) with the acceptance condition defined by the set of accepting states (and more generally by some upward-closed conditions). In general the problem of language equivalence is undecidable even for one-dimensional VASSes, thus to get decidability we investigate restricted subclasses. On one hand we show that the problem of language inclusion of a VASS in k-ambiguous VASS (for any natural k) is decidable and even in Ackermann. On the other hand we prove that the language equivalence problem is Ackermann-hard already for deterministic VASSes. These two results imply Ackermann-completeness for language inclusion and equivalence in several possible restrictions. Some of our techniques can be also applied in much broader generality in infinite-state systems, namely for some subclass of well-structured transition systems.

Wojciech Czerwiński and Piotr Hofman. Language Inclusion for Boundedly-Ambiguous Vector Addition Systems Is Decidable. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 16:1-16:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{czerwinski_et_al:LIPIcs.CONCUR.2022.16, author = {Czerwi\'{n}ski, Wojciech and Hofman, Piotr}, title = {{Language Inclusion for Boundedly-Ambiguous Vector Addition Systems Is Decidable}}, booktitle = {33rd International Conference on Concurrency Theory (CONCUR 2022)}, pages = {16:1--16:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-246-4}, ISSN = {1868-8969}, year = {2022}, volume = {243}, editor = {Klin, Bartek and Lasota, S{\l}awomir and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2022.16}, URN = {urn:nbn:de:0030-drops-170796}, doi = {10.4230/LIPIcs.CONCUR.2022.16}, annote = {Keywords: vector addition systems, language inclusion, language equivalence, determinism, unambiguity, bounded ambiguity, Petri nets, well-structured transition systems} }

Document

**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

We study languages of unambiguous VASS, that is, Vector Addition Systems with States, whose transitions read letters from a finite alphabet, and whose acceptance condition is defined by a set of final states (i.e., the coverability language). We show that the problem of universality for unambiguous VASS is ExpSpace-complete, in sheer contrast to Ackermann-completeness for arbitrary VASS, even in dimension 1. When the dimension d ∈ ℕ is fixed, the universality problem is PSpace-complete if d ≥ 2, and coNP-hard for 1-dimensional VASSes (also known as One Counter Nets).

Wojciech Czerwiński, Diego Figueira, and Piotr Hofman. Universality Problem for Unambiguous VASS. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{czerwinski_et_al:LIPIcs.CONCUR.2020.36, author = {Czerwi\'{n}ski, Wojciech and Figueira, Diego and Hofman, Piotr}, title = {{Universality Problem for Unambiguous VASS}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {36:1--36:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.36}, URN = {urn:nbn:de:0030-drops-128486}, doi = {10.4230/LIPIcs.CONCUR.2020.36}, annote = {Keywords: unambiguity, vector addition systems, universality problems} }

Document

**Published in:** LIPIcs, Volume 171, 31st International Conference on Concurrency Theory (CONCUR 2020)

We study the language universality problem for One-Counter Nets, also known as 1-dimensional Vector Addition Systems with States (1-VASS), parameterized either with an initial counter value, or with an upper bound on the allowed counter value during runs. The language accepted by an OCN (defined by reaching a final control state) is monotone in both parameters. This yields two natural questions: 1) does there exist an initial counter value that makes the language universal? 2) does there exist a sufficiently high ceiling so that the bounded language is universal?
Although the ordinary universality problem is decidable (and Ackermann-complete) and these parameterized variants seem to reduce to checking basic structural properties of the underlying automaton, we show that in fact both problems are undecidable. We also look into the complexities of the problems for several decidable subclasses, namely for unambiguous, and deterministic systems, and for those over a single-letter alphabet.

Shaull Almagor, Udi Boker, Piotr Hofman, and Patrick Totzke. Parametrized Universality Problems for One-Counter Nets. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 47:1-47:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{almagor_et_al:LIPIcs.CONCUR.2020.47, author = {Almagor, Shaull and Boker, Udi and Hofman, Piotr and Totzke, Patrick}, title = {{Parametrized Universality Problems for One-Counter Nets}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {47:1--47:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.47}, URN = {urn:nbn:de:0030-drops-128592}, doi = {10.4230/LIPIcs.CONCUR.2020.47}, annote = {Keywords: Counter net, VASS, Unambiguous Automata, Universality} }

Document

**Published in:** LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

Timed basic parallel processes (TBPP) extend communication-free Petri nets (aka. BPP or commutative context-free grammars) by a global notion of time. TBPP can be seen as an extension of timed automata (TA) with context-free branching rules, and as such may be used to model networks of independent timed automata with process creation. We show that the coverability and reachability problems (with unary encoded target multiplicities) are PSPACE-complete and EXPTIME-complete, respectively. For the special case of 1-clock TBPP, both are NP-complete and hence not more complex than for untimed BPP. This contrasts with known super-Ackermannian-completeness and undecidability results for general timed Petri nets. As a result of independent interest, and basis for our NP upper bounds, we show that the reachability relation of 1-clock TA can be expressed by a formula of polynomial size in the existential fragment of linear arithmetic, which improves on recent results from the literature.

Lorenzo Clemente, Piotr Hofman, and Patrick Totzke. Timed Basic Parallel Processes. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{clemente_et_al:LIPIcs.CONCUR.2019.15, author = {Clemente, Lorenzo and Hofman, Piotr and Totzke, Patrick}, title = {{Timed Basic Parallel Processes}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {15:1--15:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-121-4}, ISSN = {1868-8969}, year = {2019}, volume = {140}, editor = {Fokkink, Wan and van Glabbeek, Rob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.15}, URN = {urn:nbn:de:0030-drops-109171}, doi = {10.4230/LIPIcs.CONCUR.2019.15}, annote = {Keywords: Timed Automata, Petri Nets} }

Document

**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

Following a recently considered generalization of linear equations to unordered data vectors, we perform a further generalization to ordered data vectors. These generalized equations naturally appear in the analysis of vector addition systems (or Petri nets) extended with ordered data. We show that nonnegative-integer solvability of linear equations is computationally equivalent (up to an exponential blowup) to the reachability problem for (plain) vector addition systems. This high complexity is surprising, and contrasts with NP-completeness for unordered data vectors. This also contrasts with our second result, namely polynomial time complexity of the solvability problem when the nonnegative-integer restriction on solutions is relaxed.

Piotr Hofman and Slawomir Lasota. Linear Equations with Ordered Data. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hofman_et_al:LIPIcs.CONCUR.2018.24, author = {Hofman, Piotr and Lasota, Slawomir}, title = {{Linear Equations with Ordered Data}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {24:1--24:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.24}, URN = {urn:nbn:de:0030-drops-95624}, doi = {10.4230/LIPIcs.CONCUR.2018.24}, annote = {Keywords: Linear equations, Petri nets, Petri nets with data, vector addition systems, sets with atoms, orbit-finite sets} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

A vector addition system (VAS) with an initial and a final marking and transition labels induces a language. In part because the reachability problem in VAS remains far from being well-understood, it is difficult to devise decision procedures for such languages. This is especially true for checking properties that state the existence of infinitely many words of a particular shape. Informally, we call these unboundedness properties.
We present a simple set of axioms for predicates that can express unboundedness properties. Our main result is that such a predicate is decidable for VAS languages as soon as it is decidable for regular languages. Among other results, this allows us to show decidability of (i) separability by bounded regular languages, (ii) unboundedness of occurring factors from a language K with mild conditions on K, and (iii) universality of the set of factors.

Wojciech Czerwinski, Piotr Hofman, and Georg Zetzsche. Unboundedness Problems for Languages of Vector Addition Systems. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 119:1-119:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{czerwinski_et_al:LIPIcs.ICALP.2018.119, author = {Czerwinski, Wojciech and Hofman, Piotr and Zetzsche, Georg}, title = {{Unboundedness Problems for Languages of Vector Addition Systems}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {119:1--119:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.119}, URN = {urn:nbn:de:0030-drops-91235}, doi = {10.4230/LIPIcs.ICALP.2018.119}, annote = {Keywords: vector addition systems, decision problems, unboundedness, separability} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Equivalence of deterministic pushdown automata is a famous problem in theoretical computer science whose decidability has been shown by Sénizergues. Our first result shows that decidability no longer holds when moving from finite words to infinite words. This solves an open problem that has recently been raised by Löding. In fact, we show that already the equivalence problem for deterministic Büchi one-counter automata is undecidable. Hence, the decidability border is rather tight when taking into account a recent result by Löding and Repke that equivalence of deterministic weak parity pushdown automata (a subclass of deterministic Büchi pushdown automata) is decidable.
Another known result on finite words is that the universality problem for vector addition systems is decidable. We show undecidability when moving to infinite words. In fact, we prove that already the universality problem for nondeterministic Büchi one-counter nets (or equivalently vector addition systems with one unbounded dimension) is undecidable.

Stanislav Böhm, Stefan Göller, Simon Halfon, and Piotr Hofman. On Büchi One-Counter Automata. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bohm_et_al:LIPIcs.STACS.2017.14, author = {B\"{o}hm, Stanislav and G\"{o}ller, Stefan and Halfon, Simon and Hofman, Piotr}, title = {{On B\"{u}chi One-Counter Automata}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {14:1--14:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.14}, URN = {urn:nbn:de:0030-drops-70194}, doi = {10.4230/LIPIcs.STACS.2017.14}, annote = {Keywords: infinite words, deterministic pushdown automata} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Given two finite-state automata, are the Parikh images of the languages they generate equivalent? This problem was shown decidable in coNEXP by Huynh in 1985 within the more general setting of context-free commutative grammars. Huynh conjectured that a Pi_2^P upper bound might be possible, and Kopczynski and To established in 2010 such an upper bound when the size of the alphabet is fixed. The contribution of this paper is to show that the language equivalence problem for regular and context-free commutative grammars is actually coNEXP-complete. In addition, our lower bound immediately yields further coNEXP-completeness results for equivalence problems for regular commutative expressions, reversal-bounded counter automata and communication-free Petri nets. Finally, we improve both lower and upper bounds for language equivalence for exponent-sensitive commutative grammars.

Christoph Haase and Piotr Hofman. Tightening the Complexity of Equivalence Problems for Commutative Grammars. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{haase_et_al:LIPIcs.STACS.2016.41, author = {Haase, Christoph and Hofman, Piotr}, title = {{Tightening the Complexity of Equivalence Problems for Commutative Grammars}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {41:1--41:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.41}, URN = {urn:nbn:de:0030-drops-57422}, doi = {10.4230/LIPIcs.STACS.2016.41}, annote = {Keywords: language equivalence, commutative grammars, presburger arithmetic, semi-linear sets, petri nets} }

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**Published in:** LIPIcs, Volume 31, 18th International Conference on Database Theory (ICDT 2015)

The separability problem for regular languages asks, given two regular languages I and E, whether there exists a language S that separates the two, that is, includes I but contains nothing from E. Typically, S comes from a simple, less expressive class of languages than I and E. In general, a simple separator $S$ can be seen as an approximation of I or as an explanation of how I and E are different. In a database context, separators can be used for explaining the result of regular path queries or for finding explanations for the difference between paths in a graph database, that is, how paths from given nodes u_1 to v_1 are different from those from u_2 to v_2. We study the complexity of separability of regular languages by combinations of subsequences or subwords of a given length k. The rationale is that the parameter k can be used to influence the size and simplicity of the separator. The emphasis of our study is on tracing the tractability of the problem.

Piotr Hofman and Wim Martens. Separability by Short Subsequences and Subwords. In 18th International Conference on Database Theory (ICDT 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 31, pp. 230-246, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hofman_et_al:LIPIcs.ICDT.2015.230, author = {Hofman, Piotr and Martens, Wim}, title = {{Separability by Short Subsequences and Subwords}}, booktitle = {18th International Conference on Database Theory (ICDT 2015)}, pages = {230--246}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-79-8}, ISSN = {1868-8969}, year = {2015}, volume = {31}, editor = {Arenas, Marcelo and Ugarte, Mart{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2015.230}, URN = {urn:nbn:de:0030-drops-49878}, doi = {10.4230/LIPIcs.ICDT.2015.230}, annote = {Keywords: separability, complexity, graph data, debugging} }

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**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

One-counter nets (OCN) are Petri nets with exactly one unbounded place.
They are equivalent to a subclass of one-counter automata with just a weak test for zero. Unlike many other semantic equivalences, strong and weak simulation preorder are decidable for OCN, but the computational complexity was an open problem. We show that both strong and weak simulation preorder on OCN are Pspace-complete.

Piotr Hofman, Slawomir Lasota, Richard Mayr, and Patrick Totzke. Simulation Over One-counter Nets is PSPACE-Complete. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 515-526, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{hofman_et_al:LIPIcs.FSTTCS.2013.515, author = {Hofman, Piotr and Lasota, Slawomir and Mayr, Richard and Totzke, Patrick}, title = {{Simulation Over One-counter Nets is PSPACE-Complete}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {515--526}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.515}, URN = {urn:nbn:de:0030-drops-43970}, doi = {10.4230/LIPIcs.FSTTCS.2013.515}, annote = {Keywords: Simulation preorder; one-counter nets; complexity} }

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