9 Search Results for "Guttenberg, Roland"

Document
DOI: 10.4230/LIPIcs.STACS.2026.60
The Asymptotic Size of Finite Irreducible Semigroups of Rational Matrices

Authors: Stefan Kiefer and Andrew Ryzhikov

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
We study finite semigroups of n × n matrices with rational entries. Such semigroups provide a rich generalization of transition monoids of unambiguous (and, in particular, deterministic) finite automata. In this paper we determine the maximum size of finite semigroups of rational n × n matrices, with the goal of shedding more light on the structure of such matrix semigroups. While in general such semigroups can be arbitrarily large in terms of n, a classical result of Schützenberger from 1962 implies an upper bound of 2^{𝒪(n² log n)} for irreducible semigroups, i.e., the only subspaces of ℚⁿ that are invariant for all matrices in the semigroup are ℚⁿ and the subspace consisting only of the zero vector. Irreducible matrix semigroups can be viewed as the building blocks of general matrix semigroups, and as such play an important role in mathematics and computer science. From the point of view of automata theory, they generalize strongly connected automata. Using a very different technique from that of Schützenberger, we improve the upper bound on the cardinality to 3^{n²}. This is the main result of the paper. The bound is in some sense tight, as we show that there exists, for every n, a finite irreducible semigroup with 3^{⌊ n²/4 ⌋} rational matrices. Our main result also leads to an improvement of a bound, due to Almeida and Steinberg, on the mortality threshold. The mortality threshold is a number 𝓁 such that if the zero matrix is in the semigroup, then the zero matrix can be written as a product of at most 𝓁 matrices from any subset that generates the semigroup.
Cite as

Stefan Kiefer and Andrew Ryzhikov. The Asymptotic Size of Finite Irreducible Semigroups of Rational Matrices. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 60:1-60:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kiefer_et_al:LIPIcs.STACS.2026.60,
  author =	{Kiefer, Stefan and Ryzhikov, Andrew},
  title =	{{The Asymptotic Size of Finite Irreducible Semigroups of Rational Matrices}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{60:1--60:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.60},
  URN =		{urn:nbn:de:0030-drops-255496},
  doi =		{10.4230/LIPIcs.STACS.2026.60},
  annote =	{Keywords: finite matrix semigroups, irreducible matrix semigroups, matrix mortality, aperiodic semigroups, unambiguous automata, transition monoids}
}
Document
DOI: 10.4230/LIPIcs.SAND.2025.1
The Expressive Power of Uniform Population Protocols with Logarithmic Space

Authors: Philipp Czerner, Vincent Fischer, and Roland Guttenberg

Published in: LIPIcs, Volume 330, 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)

Abstract
Population protocols are a model of computation in which indistinguishable mobile agents interact in pairs to decide a property of their initial configuration. Originally introduced by Angluin et. al. in 2004 with a constant number of states, research nowadays focuses on protocols where the space usage depends on the number of agents. The expressive power of population protocols has so far however only been determined for protocols using o(log n) states, which compute only semilinear predicates, and for Ω(n) states. This leaves a significant gap, particularly concerning protocols with Θ(log n) or Θ(polylog n) states, which are the most common constructions in the literature. In this paper we close the gap and prove that for any ε > 0 and f ∈ Ω(log n)∩𝒪(n^{1-ε}), both uniform and non-uniform population protocols with Θ(f(n)) states can decide exactly those predicates, whose unary encoding lies in NSPACE(f(n) log n).
Cite as

Philipp Czerner, Vincent Fischer, and Roland Guttenberg. The Expressive Power of Uniform Population Protocols with Logarithmic Space. In 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 330, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{czerner_et_al:LIPIcs.SAND.2025.1,
  author =	{Czerner, Philipp and Fischer, Vincent and Guttenberg, Roland},
  title =	{{The Expressive Power of Uniform Population Protocols with Logarithmic Space}},
  booktitle =	{4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)},
  pages =	{1:1--1:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-368-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{330},
  editor =	{Meeks, Kitty and Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2025.1},
  URN =		{urn:nbn:de:0030-drops-230540},
  doi =		{10.4230/LIPIcs.SAND.2025.1},
  annote =	{Keywords: Population Protocols, Uniform, Expressive Power}
}
Document
DOI: 10.4230/LIPIcs.SAND.2025.5
Undecidability of the Emptiness Problem for Weak Models of Distributed Computing

Authors: Flavio T. Principato, Javier Esparza, and Philipp Czerner

Published in: LIPIcs, Volume 330, 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)

Abstract
Esparza and Reiter have recently conducted a systematic comparative study of weak asynchronous models of distributed computing, in which a network of identical finite-state machines acts cooperatively to decide properties of the network’s graph. They introduced a distributed automata framework encompassing many different models, and proved that w.r.t. their expressive power (the graph properties they can decide) distributed automata collapse into seven equivalence classes. In this contribution, we turn our attention to the formal verification problem: Given a distributed automaton, does it decide a given graph property? We consider a fundamental instance of this question - the emptiness problem: Given a distributed automaton, does it accept any graph at all? Our main result is negative: the emptiness problem is undecidable for six of the seven equivalence classes, and trivially decidable for the remaining class.
Cite as

Flavio T. Principato, Javier Esparza, and Philipp Czerner. Undecidability of the Emptiness Problem for Weak Models of Distributed Computing. In 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 330, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{principato_et_al:LIPIcs.SAND.2025.5,
  author =	{Principato, Flavio T. and Esparza, Javier and Czerner, Philipp},
  title =	{{Undecidability of the Emptiness Problem for Weak Models of Distributed Computing}},
  booktitle =	{4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)},
  pages =	{5:1--5:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-368-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{330},
  editor =	{Meeks, Kitty and Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2025.5},
  URN =		{urn:nbn:de:0030-drops-230582},
  doi =		{10.4230/LIPIcs.SAND.2025.5},
  annote =	{Keywords: Undecidability, Emptiness Problem, distributed Automata}
}
Document
Brief Announcement
DOI: 10.4230/LIPIcs.DISC.2024.44
Brief Announcement: The Expressive Power of Uniform Population Protocols with Logarithmic Space

Authors: Philipp Czerner, Vincent Fischer, and Roland Guttenberg

Published in: LIPIcs, Volume 319, 38th International Symposium on Distributed Computing (DISC 2024)

Abstract
Population protocols are a model of computation in which indistinguishable mobile agents interact in pairs to decide a property of their initial configuration. Originally introduced by Angluin et. al. in 2004 with a constant number of states, research nowadays focuses on protocols where the space usage depends on the number of agents. The expressive power of population protocols has so far however only been determined for protocols using o(log n) states, which compute only semilinear predicates, and for Ω(n) states. This leaves a significant gap, particularly concerning protocols with Θ(log n) or Θ(polylog n) states, which are the most common constructions in the literature. In this paper we close the gap and prove that for any ε > 0 and f ∈ Ω(log n) ∩ 𝒪(n^{1-ε}), both uniform and non-uniform population protocols with Θ(f(n)) states can decide exactly NSPACE(f(n) log n).
Cite as

Philipp Czerner, Vincent Fischer, and Roland Guttenberg. Brief Announcement: The Expressive Power of Uniform Population Protocols with Logarithmic Space. In 38th International Symposium on Distributed Computing (DISC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 319, pp. 44:1-44:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{czerner_et_al:LIPIcs.DISC.2024.44,
  author =	{Czerner, Philipp and Fischer, Vincent and Guttenberg, Roland},
  title =	{{Brief Announcement: The Expressive Power of Uniform Population Protocols with Logarithmic Space}},
  booktitle =	{38th International Symposium on Distributed Computing (DISC 2024)},
  pages =	{44:1--44:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-352-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{319},
  editor =	{Alistarh, Dan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2024.44},
  URN =		{urn:nbn:de:0030-drops-212726},
  doi =		{10.4230/LIPIcs.DISC.2024.44},
  annote =	{Keywords: Population Protocols, Uniform, Expressive Power}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.141
Flattability of Priority Vector Addition Systems

Authors: Roland Guttenberg

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

Abstract
Vector addition systems (VAS), also known as Petri nets, are a popular model of concurrent systems. Many problems from many areas reduce to the reachability problem for VAS, which consists of deciding whether a target configuration of a VAS is reachable from a given initial configuration. One of the main approaches to solve the problem on practical instances is called flattening, intuitively removing nested loops. This technique is known to terminate for semilinear VAS due to [Jérôme Leroux, 2013]. In this paper, we prove that also for VAS with nested zero tests, called Priority VAS, flattening does in fact terminate for all semilinear reachability relations. Furthermore, we prove that Priority VAS admit semilinear inductive invariants. Both of these results are obtained by defining a well-quasi-order on runs of Priority VAS which has good pumping properties.
Cite as

Roland Guttenberg. Flattability of Priority Vector Addition Systems. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 141:1-141:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{guttenberg:LIPIcs.ICALP.2024.141,
  author =	{Guttenberg, Roland},
  title =	{{Flattability of Priority Vector Addition Systems}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{141:1--141:20},
  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.2024.141},
  URN =		{urn:nbn:de:0030-drops-202848},
  doi =		{10.4230/LIPIcs.ICALP.2024.141},
  annote =	{Keywords: Priority Vector Addition Systems, Semilinear, Inductive Invariants, Geometry, Flattability, Almost Semilinear, Transformer Relation}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.156
Verification of Population Protocols with Unordered Data

Authors: Steffen van Bergerem, Roland Guttenberg, Sandra Kiefer, Corto Mascle, Nicolas Waldburger, and Chana Weil-Kennedy

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

Abstract
Population protocols are a well-studied model of distributed computation in which a group of anonymous finite-state agents communicates via pairwise interactions. Together they decide whether their initial configuration, i. e., the initial distribution of agents in the states, satisfies a property. As an extension in order to express properties of multisets over an infinite data domain, Blondin and Ladouceur (ICALP'23) introduced population protocols with unordered data (PPUD). In PPUD, each agent carries a fixed data value, and the interactions between agents depend on whether their data are equal or not. Blondin and Ladouceur also identified the interesting subclass of immediate observation PPUD (IOPPUD), where in every transition one of the two agents remains passive and does not move, and they characterised its expressive power. We study the decidability and complexity of formally verifying these protocols. The main verification problem for population protocols is well-specification, that is, checking whether the given PPUD computes some function. We show that well-specification is undecidable in general. By contrast, for IOPPUD, we exhibit a large yet natural class of problems, which includes well-specification among other classic problems, and establish that these problems are in ExpSpace. We also provide a lower complexity bound, namely coNExpTime-hardness.
Cite as

Steffen van Bergerem, Roland Guttenberg, Sandra Kiefer, Corto Mascle, Nicolas Waldburger, and Chana Weil-Kennedy. Verification of Population Protocols with Unordered Data. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 156:1-156:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vanbergerem_et_al:LIPIcs.ICALP.2024.156,
  author =	{van Bergerem, Steffen and Guttenberg, Roland and Kiefer, Sandra and Mascle, Corto and Waldburger, Nicolas and Weil-Kennedy, Chana},
  title =	{{Verification of Population Protocols with Unordered Data}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{156:1--156:20},
  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.2024.156},
  URN =		{urn:nbn:de:0030-drops-202993},
  doi =		{10.4230/LIPIcs.ICALP.2024.156},
  annote =	{Keywords: Population protocols, Parameterized verification, Distributed computing, Well-specification}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2023.6
Geometry of Reachability Sets of Vector Addition Systems

Authors: Roland Guttenberg, Mikhail Raskin, and Javier Esparza

Published in: LIPIcs, Volume 279, 34th International Conference on Concurrency Theory (CONCUR 2023)

Abstract
Vector Addition Systems (VAS), aka Petri nets, are a popular model of concurrency. The reachability set of a VAS is the set of configurations reachable from the initial configuration. Leroux has studied the geometric properties of VAS reachability sets, and used them to derive decision procedures for important analysis problems. In this paper we continue the geometric study of reachability sets. We show that every reachability set admits a finite decomposition into disjoint almost hybridlinear sets enjoying nice geometric properties. Further, we prove that the decomposition of the reachability set of a given VAS is effectively computable. As a corollary, we derive a new proof of Hauschildt’s 1990 result showing the decidability of the question whether the reachability set of a given VAS is semilinear. As a second corollary, we prove that the complement of a reachability set, if it is infinite, always contains an infinite linear set.
Cite as

Roland Guttenberg, Mikhail Raskin, and Javier Esparza. Geometry of Reachability Sets of Vector Addition Systems. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{guttenberg_et_al:LIPIcs.CONCUR.2023.6,
  author =	{Guttenberg, Roland and Raskin, Mikhail and Esparza, Javier},
  title =	{{Geometry of Reachability Sets of Vector Addition Systems}},
  booktitle =	{34th International Conference on Concurrency Theory (CONCUR 2023)},
  pages =	{6:1--6:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-299-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{279},
  editor =	{P\'{e}rez, Guillermo A. and Raskin, Jean-Fran\c{c}ois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2023.6},
  URN =		{urn:nbn:de:0030-drops-190005},
  doi =		{10.4230/LIPIcs.CONCUR.2023.6},
  annote =	{Keywords: Vector Addition System, Petri net, Reachability Set, Almost hybridlinear, Partition, Geometry}
}
Document
DOI: 10.4230/LIPIcs.SAND.2022.11
Fast and Succinct Population Protocols for Presburger Arithmetic

Authors: Philipp Czerner, Roland Guttenberg, Martin Helfrich, and Javier Esparza

Published in: LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

Abstract
In their 2006 seminal paper in Distributed Computing, Angluin et al. present a construction that, given any Presburger predicate as input, outputs a leaderless population protocol that decides the predicate. The protocol for a predicate of size m (when expressed as a Boolean combination of threshold and remainder predicates with coefficients in binary) runs in 𝒪(m ⋅ n² log n) expected number of interactions, which is almost optimal in n, the number of interacting agents. However, the number of states of the protocol is exponential in m. This is a problem for natural computing applications, where a state corresponds to a chemical species and it is difficult to implement protocols with many states. Blondin et al. described in STACS 2020 another construction that produces protocols with a polynomial number of states, but exponential expected number of interactions. We present a construction that produces protocols with 𝒪(m) states that run in expected 𝒪(m⁷ ⋅ n²) interactions, optimal in n, for all inputs of size Ω(m). For this, we introduce population computers, a carefully crafted generalization of population protocols easier to program, and show that our computers for Presburger predicates can be translated into fast and succinct population protocols.
Cite as

Philipp Czerner, Roland Guttenberg, Martin Helfrich, and Javier Esparza. Fast and Succinct Population Protocols for Presburger Arithmetic. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{czerner_et_al:LIPIcs.SAND.2022.11,
  author =	{Czerner, Philipp and Guttenberg, Roland and Helfrich, Martin and Esparza, Javier},
  title =	{{Fast and Succinct Population Protocols for Presburger Arithmetic}},
  booktitle =	{1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)},
  pages =	{11:1--11:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-224-2},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{221},
  editor =	{Aspnes, James and Michail, Othon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.11},
  URN =		{urn:nbn:de:0030-drops-159535},
  doi =		{10.4230/LIPIcs.SAND.2022.11},
  annote =	{Keywords: population protocols, fast, succinct, population computers}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.FSTTCS.2021.2
State Complexity of Population Protocols (Invited Talk)

Authors: Javier Esparza

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract
Population protocols were introduced by Angluin et al. in 2004 to study the theoretical properties of networks of mobile sensors with very limited computational resources. They have also been proposed as a natural computing model, with molecules, cells, or microorganisms playing the role of sensors. In a population protocol an arbitrary number of indistinguishable, finite-state agents interact randomly in pairs to collectively decide if their initial global configuration satisfies a given property. The property is formalized as a predicate that maps each initial configuration to an output, 0 or 1. Starting from an initial configuration, the agents eventually agree to the correct output almost surely, and continue producing it forever. The protocol is said to stabilize to the correct output. It is well known that population protocols can decide exactly the semilinear predicates, or, equivalently, the predicates expressible in Presburger arithmetic. Current research concentrates on investigating the amount of resources needed to decide a given predicate. The standard resources, time and memory, translate for population protocols into expected time to stabilization, usually called parallel runtime, and number of states of each agent. In this talk we concentrate on the latter. A variant of population protocols allows for a leader, a distinguished finite-state agent that is added to the initial configuration and, intuitively, helps the other agents to organize the computation. In the last years my collaborators and I have obtained upper and lower bounds for the state complexity of population protocols with and without a leader. Define the state complexity of a predicate as the minimal number of states of a protocol that decides the predicate, and STATE(η) as the maximum state complexity of the predicates of size at most η, where predicates are encoded as quantifier-free formulas of Presburger arithmetic with coefficients written in binary. Using techniques from the theory of Petri nets and Vector Addition Systems, we have shown that STATE(η) is polynomially bounded, even for leaderless protocols; this improves on the exponential bound given in 2004 by Angluin and collaborators. We have also proved that STATE(η) ∈ Ω(log log η) for leaderless protocols, even for those deciding very simple predicates of the form x ≥ c for some constant c. In the talk I report on these results, and on two very recent, still unpublished results. Modulo the pending peer-review confirmation, the first result shows the existence of leaderless protocols with a polynomial number of states and linear parallel runtime, and the second, due to Leroux, gives a Ω((log log η)^{1/3}) lower bound for protocols with a leader.
Cite as

Javier Esparza. State Complexity of Population Protocols (Invited Talk). In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{esparza:LIPIcs.FSTTCS.2021.2,
  author =	{Esparza, Javier},
  title =	{{State Complexity of Population Protocols}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{2:1--2:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.2},
  URN =		{urn:nbn:de:0030-drops-155139},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.2},
  annote =	{Keywords: Population protocols, state complexity, Petri nets}
}
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