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Documents authored by Schütze, Lia

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Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.153
On the Length of Strongly Monotone Descending Chains over ℕ^d

Authors: Sylvain Schmitz and Lia Schütze

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
A recent breakthrough by Künnemann, Mazowiecki, Schütze, Sinclair-Banks, and Węgrzycki (ICALP 2023) bounds the running time for the coverability problem in d-dimensional vector addition systems under unary encoding to n^{2^{O(d)}}, improving on Rackoff’s n^{2^{O(dlg d)}} upper bound (Theor. Comput. Sci. 1978), and provides conditional matching lower bounds. In this paper, we revisit Lazić and Schmitz' "ideal view" of the backward coverability algorithm (Inform. Comput. 2021) in the light of this breakthrough. We show that the controlled strongly monotone descending chains of downwards-closed sets over ℕ^d that arise from the dual backward coverability algorithm of Lazić and Schmitz on d-dimensional unary vector addition systems also enjoy this tight n^{2^{O(d)}} upper bound on their length, and that this also translates into the same bound on the running time of the backward coverability algorithm. Furthermore, our analysis takes place in a more general setting than that of Lazić and Schmitz, which allows to show the same results and improve on the 2EXPSPACE upper bound derived by Benedikt, Duff, Sharad, and Worrell (LICS 2017) for the coverability problem in invertible affine nets.
Cite as

Sylvain Schmitz and Lia Schütze. On the Length of Strongly Monotone Descending Chains over ℕ^d. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 153:1-153:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{schmitz_et_al:LIPIcs.ICALP.2024.153,
  author =	{Schmitz, Sylvain and Sch\"{u}tze, Lia},
  title =	{{On the Length of Strongly Monotone Descending Chains over \mathbb{N}^d}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{153:1--153:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.153},
  URN =		{urn:nbn:de:0030-drops-202964},
  doi =		{10.4230/LIPIcs.ICALP.2024.153},
  annote =	{Keywords: Vector addition system, coverability, well-quasi-order, order ideal, affine net}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2023.131
Coverability in VASS Revisited: Improving Rackoff’s Bound to Obtain Conditional Optimality

Authors: Marvin Künnemann, Filip Mazowiecki, Lia Schütze, Henry Sinclair-Banks, and Karol Węgrzycki

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
Seminal results establish that the coverability problem for Vector Addition Systems with States (VASS) is in EXPSPACE (Rackoff, '78) and is EXPSPACE-hard already under unary encodings (Lipton, '76). More precisely, Rosier and Yen later utilise Rackoff’s bounding technique to show that if coverability holds then there is a run of length at most n^{2^𝒪(d log d)}, where d is the dimension and n is the size of the given unary VASS. Earlier, Lipton showed that there exist instances of coverability in d-dimensional unary VASS that are only witnessed by runs of length at least n^{2^Ω(d)}. Our first result closes this gap. We improve the upper bound by removing the twice-exponentiated log(d) factor, thus matching Lipton’s lower bound. This closes the corresponding gap for the exact space required to decide coverability. This also yields a deterministic n^{2^𝒪(d)}-time algorithm for coverability. Our second result is a matching lower bound, that there does not exist a deterministic n^{2^o(d)}-time algorithm, conditioned upon the Exponential Time Hypothesis. When analysing coverability, a standard proof technique is to consider VASS with bounded counters. Bounded VASS make for an interesting and popular model due to strong connections with timed automata. Withal, we study a natural setting where the counter bound is linear in the size of the VASS. Here the trivial exhaustive search algorithm runs in 𝒪(n^{d+1})-time. We give evidence to this being near-optimal. We prove that in dimension one this trivial algorithm is conditionally optimal, by showing that n^{2-o(1)}-time is required under the k-cycle hypothesis. In general fixed dimension d, we show that n^{d-2-o(1)}-time is required under the 3-uniform hyperclique hypothesis.
Cite as

Marvin Künnemann, Filip Mazowiecki, Lia Schütze, Henry Sinclair-Banks, and Karol Węgrzycki. Coverability in VASS Revisited: Improving Rackoff’s Bound to Obtain Conditional Optimality. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 131:1-131:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kunnemann_et_al:LIPIcs.ICALP.2023.131,
  author =	{K\"{u}nnemann, Marvin and Mazowiecki, Filip and Sch\"{u}tze, Lia and Sinclair-Banks, Henry and W\k{e}grzycki, Karol},
  title =	{{Coverability in VASS Revisited: Improving Rackoff’s Bound to Obtain Conditional Optimality}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{131:1--131:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.131},
  URN =		{urn:nbn:de:0030-drops-181834},
  doi =		{10.4230/LIPIcs.ICALP.2023.131},
  annote =	{Keywords: Vector Addition System, Coverability, Reachability, Fine-Grained Complexity, Exponential Time Hypothesis, k-Cycle Hypothesis, Hyperclique Hypothesis}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2022.124
Reachability in Bidirected Pushdown VASS

Authors: Moses Ganardi, Rupak Majumdar, Andreas Pavlogiannis, Lia Schütze, and Georg Zetzsche

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
A pushdown vector addition system with states (PVASS) extends the model of vector addition systems with a pushdown store. A PVASS is said to be bidirected if every transition (pushing/popping a symbol or modifying a counter) has an accompanying opposite transition that reverses the effect. Bidirectedness arises naturally in many models; it can also be seen as a overapproximation of reachability. We show that the reachability problem for bidirected PVASS is decidable in Ackermann time and primitive recursive for any fixed dimension. For the special case of one-dimensional bidirected PVASS, we show reachability is in PSPACE, and in fact in polynomial time if the stack is polynomially bounded. Our results are in contrast to the directed setting, where decidability of reachability is a long-standing open problem already for one dimensional PVASS, and there is a PSPACE-lower bound already for one-dimensional PVASS with bounded stack. The reachability relation in the bidirected (stateless) case is a congruence over ℕ^d. Our upper bounds exploit saturation techniques over congruences. In particular, we show novel elementary-time constructions of semilinear representations of congruences generated by finitely many vector pairs. In the case of one-dimensional PVASS, we employ a saturation procedure over bounded-size counters. We complement our upper bound with a TOWER-hardness result for arbitrary dimension and k-EXPSPACE hardness in dimension 2k+6 using a technique by Lazić and Totzke to implement iterative exponentiations.
Cite as

Moses Ganardi, Rupak Majumdar, Andreas Pavlogiannis, Lia Schütze, and Georg Zetzsche. Reachability in Bidirected Pushdown VASS. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 124:1-124:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ganardi_et_al:LIPIcs.ICALP.2022.124,
  author =	{Ganardi, Moses and Majumdar, Rupak and Pavlogiannis, Andreas and Sch\"{u}tze, Lia and Zetzsche, Georg},
  title =	{{Reachability in Bidirected Pushdown VASS}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{124:1--124:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.124},
  URN =		{urn:nbn:de:0030-drops-164651},
  doi =		{10.4230/LIPIcs.ICALP.2022.124},
  annote =	{Keywords: Vector addition systems, Pushdown, Reachability, Decidability, Complexity}
}
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