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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

The universality problem asks whether a given finite state automaton accepts all the input words. For quantitative models of automata, where input words are mapped to real values, this is naturally extended to ask whether all the words are mapped to values above (or below) a given threshold. This is known to be undecidable for commonly studied examples such as weighted automata over the positive rational (plus-times) or the integer tropical (min-plus) semirings, or equivalently cost register automata (CRAs) over these semirings. In this paper, we prove that when restricted to CRAs with only three registers, the universality problem is still undecidable, even with additional restrictions for the CRAs to be copyless linear with resets.
In contrast, we show that, assuming the unary encoding of updates, the ∀-exact problem (does the CRA output zero on all the words?) for integer min-plus linear CRAs can be decided in polynomial time if the number of registers is constant. Without the restriction on the number of registers this problem is known to be PSPACE-complete.

Laure Daviaud and Andrew Ryzhikov. Universality and Forall-Exactness of Cost Register Automata with Few Registers. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 40:1-40:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{daviaud_et_al:LIPIcs.MFCS.2023.40, author = {Daviaud, Laure and Ryzhikov, Andrew}, title = {{Universality and Forall-Exactness of Cost Register Automata with Few Registers}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {40:1--40:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.40}, URN = {urn:nbn:de:0030-drops-185744}, doi = {10.4230/LIPIcs.MFCS.2023.40}, annote = {Keywords: cost register automata, universality, forall-exact problem, decidability} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

The Strahler number of a rooted tree is the largest height of a perfect binary tree that is its minor. The Strahler number of a parity game is proposed to be defined as the smallest Strahler number of the tree of any of its attractor decompositions. It is proved that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices n and linear in (d/(2k))^k, where d is the number of priorities and k is the Strahler number. This complexity is quasi-polynomial because the Strahler number is at most logarithmic in the number of vertices. The proof is based on a new construction of small Strahler-universal trees.
It is shown that the Strahler number of a parity game is a robust, and hence arguably natural, parameter: it coincides with its alternative version based on trees of progress measures and - remarkably - with the register number defined by Lehtinen (2018). It follows that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices and linear in (d/(2k))^k, where k is the register number. This significantly improves the running times and space achieved for parity games of bounded register number by Lehtinen (2018) and by Parys (2020).
The running time of the algorithm based on small Strahler-universal trees yields a novel trade-off k ⋅ lg(d/k) = O(log n) between the two natural parameters that measure the structural complexity of a parity game, which allows solving parity games in polynomial time. This includes as special cases the asymptotic settings of those parameters covered by the results of Calude, Jain Khoussainov, Li, and Stephan (2017), of Jurdziński and Lazić (2017), and of Lehtinen (2018), and it significantly extends the range of such settings, for example to d = 2^O(√{lg n}) and k = O(√{lg n}).

Laure Daviaud, Marcin Jurdziński, and K. S. Thejaswini. The Strahler Number of a Parity Game. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 123:1-123:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{daviaud_et_al:LIPIcs.ICALP.2020.123, author = {Daviaud, Laure and Jurdzi\'{n}ski, Marcin and Thejaswini, K. S.}, title = {{The Strahler Number of a Parity Game}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {123:1--123:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.123}, URN = {urn:nbn:de:0030-drops-125304}, doi = {10.4230/LIPIcs.ICALP.2020.123}, annote = {Keywords: parity game, attractor decomposition, progress measure, universal tree, Strahler number} }

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

An improved translation from alternating parity automata on infinite words to alternating weak automata is given. The blow-up of the number of states is related to the size of the smallest universal ordered trees and hence it is quasi-polynomial, and it is polynomial if the asymptotic number of priorities is at most logarithmic in the number of states. This is an exponential improvement on the translation of Kupferman and Vardi (2001) and a quasi-polynomial improvement on the translation of Boker and Lehtinen (2018). Any slightly better such translation would (if - like all presently known such translations - it is efficiently constructive) lead to algorithms for solving parity games that are asymptotically faster in the worst case than the current state of the art (Calude, Jain, Khoussainov, Li, and Stephan, 2017; Jurdziński and Lazić, 2017; and Fearnley, Jain, Schewe, Stephan, and Wojtczak, 2017), and hence it would yield a significant breakthrough.

Laure Daviaud, Marcin Jurdziński, and Karoliina Lehtinen. Alternating Weak Automata from Universal Trees. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{daviaud_et_al:LIPIcs.CONCUR.2019.18, author = {Daviaud, Laure and Jurdzi\'{n}ski, Marcin and Lehtinen, Karoliina}, title = {{Alternating Weak Automata from Universal Trees}}, booktitle = {30th International Conference on Concurrency Theory (CONCUR 2019)}, pages = {18:1--18:14}, 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.2019.18}, URN = {urn:nbn:de:0030-drops-109208}, doi = {10.4230/LIPIcs.CONCUR.2019.18}, annote = {Keywords: alternating automata, weak automata, B\"{u}chi automata, parity automata, parity games, universal trees} }

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

The containment problem for quantitative automata is the natural quantitative generalisation of the classical language inclusion problem for Boolean automata. We study it for probabilistic automata, where it is known to be undecidable in general. We restrict our study to the class of probabilistic automata with bounded ambiguity. There, we show decidability (subject to Schanuel's conjecture) when one of the automata is assumed to be unambiguous while the other one is allowed to be finitely ambiguous. Furthermore, we show that this is close to the most general decidable fragment of this problem by proving that it is already undecidable if one of the automata is allowed to be linearly ambiguous.

Laure Daviaud, Marcin Jurdzinski, Ranko Lazic, Filip Mazowiecki, Guillermo A. Pérez, and James Worrell. When is Containment Decidable for Probabilistic Automata?. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 121:1-121:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{daviaud_et_al:LIPIcs.ICALP.2018.121, author = {Daviaud, Laure and Jurdzinski, Marcin and Lazic, Ranko and Mazowiecki, Filip and P\'{e}rez, Guillermo A. and Worrell, James}, title = {{When is Containment Decidable for Probabilistic Automata?}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {121:1--121:14}, 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.2018.121}, URN = {urn:nbn:de:0030-drops-91251}, doi = {10.4230/LIPIcs.ICALP.2018.121}, annote = {Keywords: Probabilistic automata, Containment, Emptiness, Ambiguity} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Weighted automata over the tropical semiring Zmax are closely related to finitely generated semigroups of matrices over Zmax. In this paper, we use results in automata theory to study two quantities associated with sets of matrices: the joint spectral radius and the ultimate rank. We prove that these two quantities are not computable over the tropical semiring, i.e. there is no algorithm that takes as input a finite set of matrices S and provides as output the joint spectral radius (resp. the ultimate rank) of S. On the other hand, we prove that the joint spectral radius is nevertheless approximable and we exhibit restricted cases in which the joint spectral radius and the ultimate rank are computable. To reach this aim, we study the problem of comparing functions computed by weighted automata over the tropical semiring. This problem is known to be undecidable, and we prove that it remains undecidable in some specific subclasses of automata.

Laure Daviaud, Pierre Guillon, and Glenn Merlet. Comparison of Max-Plus Automata and Joint Spectral Radius of Tropical Matrices. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 19:1-19:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{daviaud_et_al:LIPIcs.MFCS.2017.19, author = {Daviaud, Laure and Guillon, Pierre and Merlet, Glenn}, title = {{Comparison of Max-Plus Automata and Joint Spectral Radius of Tropical Matrices}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.19}, URN = {urn:nbn:de:0030-drops-81052}, doi = {10.4230/LIPIcs.MFCS.2017.19}, annote = {Keywords: max-plus automata, max-plus matrices, weighted automata, tropical semiring, joint spectral radius, ultimate rank} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Max-plus automata are quantitative extensions of automata designed to associate an integer with every non-empty word. A pair of distinct words is said to be an identity for a class of max-plus automata if each of the automata in the class computes the same value on the two words. We give the shortest identities holding for the class of max-plus automata with two states. For this, we exhibit an interesting list of necessary conditions for an identity to hold. Moreover, this result provides a counter-example of a conjecture of Izhakian, concerning the minimality of certain identities.

Laure Daviaud and Marianne Johnson. The Shortest Identities for Max-Plus Automata with Two States. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 48:1-48:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{daviaud_et_al:LIPIcs.MFCS.2017.48, author = {Daviaud, Laure and Johnson, Marianne}, title = {{The Shortest Identities for Max-Plus Automata with Two States}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {48:1--48:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.48}, URN = {urn:nbn:de:0030-drops-81048}, doi = {10.4230/LIPIcs.MFCS.2017.48}, annote = {Keywords: Max-plus automata, Weighted automata, Identities, Tropical matrices} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We study various models of transducers equipped with origin information. We consider the semantics of these models as particular graphs, called origin graphs, and we characterise the families of such graphs recognised by streaming string transducers.

Mikolaj Bojanczyk, Laure Daviaud, Bruno Guillon, and Vincent Penelle. Which Classes of Origin Graphs Are Generated by Transducers. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 114:1-114:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bojanczyk_et_al:LIPIcs.ICALP.2017.114, author = {Bojanczyk, Mikolaj and Daviaud, Laure and Guillon, Bruno and Penelle, Vincent}, title = {{Which Classes of Origin Graphs Are Generated by Transducers}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {114:1--114:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian 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.ICALP.2017.114}, URN = {urn:nbn:de:0030-drops-73984}, doi = {10.4230/LIPIcs.ICALP.2017.114}, annote = {Keywords: Streaming String Transducers, Origin Semantics, String-to-String Transductions, MSO Definability} }

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

Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalising two results on regular languages to cost functions: Eilenberg's varieties theorem and profinite equational characterisations of lattices of regular languages. This opens interesting new perspectives, but the specificities of cost functions introduce difficulties that prevent these generalisations to be straightforward. In contrast, although syntactic algebras can be defined for formal power series over a commutative ring, no such notion is known for series over semirings and in particular over the tropical semiring.

Laure Daviaud, Denis Kuperberg, and Jean-Éric Pin. Varieties of Cost Functions. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 30:1-30:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{daviaud_et_al:LIPIcs.STACS.2016.30, author = {Daviaud, Laure and Kuperberg, Denis and Pin, Jean-\'{E}ric}, title = {{Varieties of Cost Functions}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {30:1--30:14}, 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.2016.30}, URN = {urn:nbn:de:0030-drops-57319}, doi = {10.4230/LIPIcs.STACS.2016.30}, annote = {Keywords: Cost functions, regular language, varieties, syntactic algebra} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Distance automata are automata weighted over the semiring (\mathbb{N} \cup \infty,\min,+) (the tropical semiring). Such automata compute functions from words to \mathbb{N} \cup \infty such as the number of occurrences of a given letter. It is known that testing f <= g is an undecidable problem for f,g computed by distance automata. The main contribution of this paper is to show that an approximation of this problem becomes decidable.
We present an algorithm which, given epsilon > 0 and two functions f,g computed by distance automata, answers "yes" if f <= (1-epsilon) g, "no" if $f \not\leq g$, and may answer "yes" or "no" in all other cases. This result highly refines previously known decidability results of the same type.
The core argument behind this quasi-decision procedure is an algorithm which is able to provide an approximated finite presentation to the closure under products of sets of matrices over the tropical semiring.
We also provide another theorem, of affine domination, which shows that previously known decision procedures for cost-automata have an improved precision when used over distance automata.

Thomas Colcombet and Laure Daviaud. Approximate comparison of distance automata. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 574-585, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{colcombet_et_al:LIPIcs.STACS.2013.574, author = {Colcombet, Thomas and Daviaud, Laure}, title = {{Approximate comparison of distance automata}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {574--585}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.574}, URN = {urn:nbn:de:0030-drops-39667}, doi = {10.4230/LIPIcs.STACS.2013.574}, annote = {Keywords: Distance automata, tropical semiring, decidability, cost functions} }

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