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**Published in:** LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Fagin’s seminal result characterizing NP in terms of existential second-order logic started the fruitful field of descriptive complexity theory. In recent years, there has been much interest in the investigation of quantitative (weighted) models of computations. In this paper, we start the study of descriptive complexity based on weighted Turing machines over arbitrary semirings. We provide machine-independent characterizations (over ordered structures) of the weighted complexity classes NP[𝒮], FP[𝒮], FPLOG[𝒮], FPSPACE[𝒮], and FPSPACE_poly[𝒮] in terms of definability in suitable weighted logics for an arbitrary semiring 𝒮. In particular, we prove weighted versions of Fagin’s theorem (even for arbitrary structures, not necessarily ordered, provided that the semiring is idempotent and commutative), the Immerman-Vardi’s theorem (originally for 𝖯) and the Abiteboul-Vianu-Vardi’s theorem (originally for PSPACE). We also discuss a recent open problem proposed by Eiter and Kiesel.
Recently, the above mentioned weighted complexity classes have been investigated in connection to classical counting complexity classes. Furthermore, several classical counting complexity classes have been characterized in terms of particular weighted logics over the semiring ℕ of natural numbers. In this work, we cover several of these classes and obtain new results for others such as NPMV, ⊕𝖯, or the collection of real-valued languages realized by polynomial-time real-valued nondeterministic Turing machines. Furthermore, our results apply to classes based on many other important semirings, such as the max-plus and the min-plus semirings over the natural numbers which correspond to the classical classes MaxP[O(log n)] and MinP[O(log n)], respectively.

Guillermo Badia, Manfred Droste, Carles Noguera, and Erik Paul. Logical Characterizations of Weighted Complexity Classes. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{badia_et_al:LIPIcs.MFCS.2024.14, author = {Badia, Guillermo and Droste, Manfred and Noguera, Carles and Paul, Erik}, title = {{Logical Characterizations of Weighted Complexity Classes}}, booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)}, pages = {14:1--14:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-335-5}, ISSN = {1868-8969}, year = {2024}, volume = {306}, editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.14}, URN = {urn:nbn:de:0030-drops-205707}, doi = {10.4230/LIPIcs.MFCS.2024.14}, annote = {Keywords: Descriptive complexity, Weighted Turing machines, Weighted logics, Semirings} }

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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

The HOM-problem asks whether the image of a regular tree language under a given tree homomorphism is again regular. It was recently shown to be decidable by Godoy, Giménez, Ramos, and Àlvarez. In this paper, the ℕ-weighted version of this problem is considered and its decidability is proved. More precisely, it is decidable in polynomial time whether the image of a regular ℕ-weighted tree language under a nondeleting, nonerasing tree homomorphism is regular.

Andreas Maletti, Andreea-Teodora Nász, and Erik Paul. Weighted HOM-Problem for Nonnegative Integers. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 51:1-51:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{maletti_et_al:LIPIcs.STACS.2024.51, author = {Maletti, Andreas and N\'{a}sz, Andreea-Teodora and Paul, Erik}, title = {{Weighted HOM-Problem for Nonnegative Integers}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {51:1--51:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.51}, URN = {urn:nbn:de:0030-drops-197614}, doi = {10.4230/LIPIcs.STACS.2024.51}, annote = {Keywords: Weighted Tree Automaton, Decision Problem, Subtree Equality Constraint, Tree Homomorphism, HOM-Problem, Weighted Tree Grammar, Weighted HOM-Problem} }

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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)

We show that the finite sequentiality problem is decidable for finitely ambiguous max-plus tree automata. A max-plus tree automaton is a weighted tree automaton over the max-plus semiring. A max-plus tree automaton is called finitely ambiguous if the number of accepting runs on every tree is bounded by a global constant. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton.

Erik Paul. Finite Sequentiality of Finitely Ambiguous Max-Plus Tree Automata. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 137:1-137:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{paul:LIPIcs.ICALP.2020.137, author = {Paul, Erik}, title = {{Finite Sequentiality of Finitely Ambiguous Max-Plus Tree Automata}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {137:1--137:15}, 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.2020.137}, URN = {urn:nbn:de:0030-drops-125447}, doi = {10.4230/LIPIcs.ICALP.2020.137}, annote = {Keywords: Weighted Tree Automata, Max-Plus Tree Automata, Finite Sequentiality, Decidability, Finite Ambiguity} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

We show the decidability of the finite sequentiality problem for unambiguous max-plus tree automata. A max-plus tree automaton is called unambiguous if there is at most one accepting run on every tree. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton.

Erik Paul. Finite Sequentiality of Unambiguous Max-Plus Tree Automata. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 55:1-55:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{paul:LIPIcs.STACS.2019.55, author = {Paul, Erik}, title = {{Finite Sequentiality of Unambiguous Max-Plus Tree Automata}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {55:1--55:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.55}, URN = {urn:nbn:de:0030-drops-102946}, doi = {10.4230/LIPIcs.STACS.2019.55}, annote = {Keywords: Weighted Tree Automata, Max-Plus Tree Automata, Finite Sequentiality, Decidability, Ambiguity} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

We prove a weighted Feferman-Vaught decomposition theorem for disjoint unions and products of finite structures. The classical Feferman-Vaught Theorem describes how the evaluation of a first order sentence in a generalized product of relational structures can be reduced to the evaluation of sentences in the contributing structures and the index structure. The logic we employ for our weighted extension is based on the weighted MSO logic introduced by Droste and Gastin to obtain a Büchi-type result for weighted automata. We show that for disjoint unions and products of structures, the evaluation of formulas from two respective fragments of the logic can be reduced to the evaluation of formulas in the contributing structures. We also prove that the respective restrictions are necessary. Surprisingly, for the case of disjoint unions, the fragment is the same as the one used in the Büchi-type result of weighted automata. In fact, even the formulas used to show that the respective restrictions are necessary are the same in both cases. However, here proving that they do not allow for a Feferman-Vaught-like decomposition is more complex and employs Ramsey's Theorem. We also show how translation schemes can be applied to go beyond disjoint unions and products.

Manfred Droste and Erik Paul. A Feferman-Vaught Decomposition Theorem for Weighted MSO Logic. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 76:1-76:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{droste_et_al:LIPIcs.MFCS.2018.76, author = {Droste, Manfred and Paul, Erik}, title = {{A Feferman-Vaught Decomposition Theorem for Weighted MSO Logic}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {76:1--76:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.76}, URN = {urn:nbn:de:0030-drops-96581}, doi = {10.4230/LIPIcs.MFCS.2018.76}, annote = {Keywords: Quantitative Logic, Quantitative Model Theory, Feferman-Vaught Theorem, Translation Scheme, Transduction} }

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

We introduce a new logic called Monitor Logic and show that it is expressively equivalent to Quantitative Monitor Automata.

Erik Paul. Monitor Logics for Quantitative Monitor Automata. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{paul:LIPIcs.MFCS.2017.14, author = {Paul, Erik}, title = {{Monitor Logics for Quantitative Monitor Automata}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {14:1--14:13}, 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.2017.14}, URN = {urn:nbn:de:0030-drops-81133}, doi = {10.4230/LIPIcs.MFCS.2017.14}, annote = {Keywords: Quantitative Monitor Automata, Nested Weighted Automata, Monitor Logics, Weighted Logics} }

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

We show that the equivalence, unambiguity and sequentiality problems are decidable for finitely ambiguous max-plus tree automata.

Erik Paul. The Equivalence, Unambiguity and Sequentiality Problems of Finitely Ambiguous Max-Plus Tree Automata are Decidable. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 53:1-53:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{paul:LIPIcs.MFCS.2017.53, author = {Paul, Erik}, title = {{The Equivalence, Unambiguity and Sequentiality Problems of Finitely Ambiguous Max-Plus Tree Automata are Decidable}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {53:1--53:13}, 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.2017.53}, URN = {urn:nbn:de:0030-drops-81147}, doi = {10.4230/LIPIcs.MFCS.2017.53}, annote = {Keywords: Tree Automata, Max-Plus Automata, Equivalence, Unambiguity, Sequentiality, Decidability} }

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