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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Recently, weighted ω-pushdown automata have been introduced by Droste, Ésik, Kuich. This new type of automaton has access to a stack and models quantitative aspects of infinite words. Here, we consider a simple version of those automata. The simple ω-pushdown automata do not use ε-transitions and have a very restricted stack access. In previous work, we could show this automaton model to be expressively equivalent to context-free ω-languages in the unweighted case. Furthermore, semiring-weighted simple ω-pushdown automata recognize all ω-algebraic series.
Here, we consider ω-valuation monoids as weight structures. As a first result, we prove that for this weight structure and for simple ω-pushdown automata, Büchi-acceptance and Muller-acceptance are expressively equivalent. In our second result, we derive a Nivat theorem for these automata stating that the behaviors of weighted ω-pushdown automata are precisely the projections of very simple ω-series restricted to ω-context-free languages. The third result is a weighted logic with the same expressive power as the new automaton model. To prove the equivalence, we use a similar result for weighted nested ω-word automata and apply our present result of expressive equivalence of Muller and Büchi acceptance.

Manfred Droste, Sven Dziadek, and Werner Kuich. Nivat-Theorem and Logic for Weighted Pushdown Automata on Infinite Words. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{droste_et_al:LIPIcs.FSTTCS.2020.44, author = {Droste, Manfred and Dziadek, Sven and Kuich, Werner}, title = {{Nivat-Theorem and Logic for Weighted Pushdown Automata on Infinite Words}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {44:1--44:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.44}, URN = {urn:nbn:de:0030-drops-132850}, doi = {10.4230/LIPIcs.FSTTCS.2020.44}, annote = {Keywords: Weighted automata, Pushdown automata, Infinite words, Weighted logic} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

In weighted automata theory, many classical results on formal languages have been extended into a quantitative setting. Here, we investigate weighted context-free languages of infinite words, a generalization of omega-context-free languages (Cohen, Gold 1977) and an extension of weighted context-free languages of finite words (Chomsky, Schützenberger 1963). As in the theory of formal grammars, these weighted languages, or omega-algebraic series, can be represented as solutions of mixed omega-algebraic systems of equations and by weighted omega-pushdown automata.
In our first main result, we show that mixed omega-algebraic systems can be transformed into Greibach normal form. Our second main result proves that simple omega-reset pushdown automata recognize all omega-algebraic series that are a solution of an omega-algebraic system in Greibach normal form. Simple reset automata do not use epsilon-transitions and can change the stack only by at most one symbol. These results generalize fundamental properties of context-free languages to weighted languages.

Manfred Droste, Sven Dziadek, and Werner Kuich. Greibach Normal Form for omega-Algebraic Systems and Weighted Simple omega-Pushdown Automata. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{droste_et_al:LIPIcs.FSTTCS.2019.38, author = {Droste, Manfred and Dziadek, Sven and Kuich, Werner}, title = {{Greibach Normal Form for omega-Algebraic Systems and Weighted Simple omega-Pushdown Automata}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {38:1--38:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.38}, URN = {urn:nbn:de:0030-drops-116003}, doi = {10.4230/LIPIcs.FSTTCS.2019.38}, annote = {Keywords: Weighted omega-Context-Free Grammars, Algebraic Systems, Greibach Normal Form, Weighted Automata, omega-Pushdown Automata} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

By fundamental results of Schützenberger, McNaughton and Papert from the 1970s, the classes of first-order definable and aperiodic languages coincide. Here, we extend this equivalence to a quantitative setting. For this, weighted automata form a general and widely studied model. We define a suitable notion of a weighted first-order logic. Then we show that this weighted first-order logic and aperiodic polynomially ambiguous weighted automata have the same expressive power. Moreover, we obtain such equivalence results for suitable weighted sublogics and finitely ambiguous or unambiguous aperiodic weighted automata. Our results hold for general weight structures, including all semirings, average computations of costs, bounded lattices, and others.

Manfred Droste and Paul Gastin. Aperiodic Weighted Automata and Weighted First-Order Logic. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 76:1-76:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{droste_et_al:LIPIcs.MFCS.2019.76, author = {Droste, Manfred and Gastin, Paul}, title = {{Aperiodic Weighted Automata and Weighted First-Order Logic}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {76:1--76:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.76}, URN = {urn:nbn:de:0030-drops-110203}, doi = {10.4230/LIPIcs.MFCS.2019.76}, annote = {Keywords: Weighted automata, weighted logic, aperiodic automata, first-order logic, unambiguous, finitely ambiguous, polynomially ambiguous} }

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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)

In the last years renewed investigation of operator precedence languages (OPL) led to discover important properties thereof: OPL are closed with respect to all major operations, are characterized, besides the original grammar family, in terms of an automata family (OPA) and an MSO logic; furthermore they significantly generalize the well-known visibly pushdown languages (VPL). In another area of research, quantitative models of systems are also greatly in demand. In this paper, we lay the foundation to marry these two research fields. We introduce weighted operator precedence automata and show how they are both strict extensions of OPA and weighted visibly pushdown automata. We prove a Nivat-like result which shows that quantitative OPL can be described by unweighted OPA and very particular weighted OPA. In a Büchi-like theorem, we show that weighted OPA are expressively equivalent to a weighted MSO-logic for OPL.

Manfred Droste, Stefan Dück, Dino Mandrioli, and Matteo Pradella. Weighted Operator Precedence Languages. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{droste_et_al:LIPIcs.MFCS.2017.31, author = {Droste, Manfred and D\"{u}ck, Stefan and Mandrioli, Dino and Pradella, Matteo}, title = {{Weighted Operator Precedence Languages}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {31:1--31:15}, 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.2017.31}, URN = {urn:nbn:de:0030-drops-81150}, doi = {10.4230/LIPIcs.MFCS.2017.31}, annote = {Keywords: Quantitative automata, operator precedence languages, input-driven languages, visibly pushdown languages, quantitative logic} }

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**Published in:** Dagstuhl Reports, Volume 4, Issue 1 (2014)

From Jan. 19 to Jan. 24, 2014, "Quantitative Models: Expressiveness, Analysis, and New Applications" was held in Schloss Dagstuhl-Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.

Manfred Droste, Paul Gastin, Kim Gulstrand Larsen, and Axel Legay. Quantitative Models: Expressiveness, Analysis, and New Applications (Dagstuhl Seminar 14041). In Dagstuhl Reports, Volume 4, Issue 1, pp. 104-124, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@Article{droste_et_al:DagRep.4.1.104, author = {Droste, Manfred and Gastin, Paul and Larsen, Kim Gulstrand and Legay, Axel}, title = {{Quantitative Models: Expressiveness, Analysis, and New Applications (Dagstuhl Seminar 14041)}}, pages = {104--124}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2014}, volume = {4}, number = {1}, editor = {Droste, Manfred and Gastin, Paul and Larsen, Kim Gulstrand and Legay, Axel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.4.1.104}, URN = {urn:nbn:de:0030-drops-45374}, doi = {10.4230/DagRep.4.1.104}, annote = {Keywords: quantitative models, quantitative analysis, timed and hybrid systems, probabilistic systems, weighted automata, systems biology, smart grid} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10031, Quantitative Models: Expressiveness and Analysis (2010)

From Jan 18 to Jan 22, 2010, the Dagstuhl Seminar 10031 ``Quantitative Models: Expressiveness and Analysis '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Christel Baier, Manfred Droste, Paul Gastin, and Kim Guldstrand Larsen. 10031 Abstracts Collection – Quantitative Models: Expressiveness and Analysis. In Quantitative Models: Expressiveness and Analysis. Dagstuhl Seminar Proceedings, Volume 10031, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{baier_et_al:DagSemProc.10031.1, author = {Baier, Christel and Droste, Manfred and Gastin, Paul and Larsen, Kim Guldstrand}, title = {{10031 Abstracts Collection – Quantitative Models: Expressiveness and Analysis}}, booktitle = {Quantitative Models: Expressiveness and Analysis}, pages = {1--15}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {10031}, editor = {Christel Baier and Manfred Droste and Paul Gastin and Kim Guldstrand Larsen}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10031.1}, URN = {urn:nbn:de:0030-drops-26839}, doi = {10.4230/DagSemProc.10031.1}, annote = {Keywords: Quantitative models, quantitative analysis, timed and hybrid systems, probabilistic systems, weighted automata} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10031, Quantitative Models: Expressiveness and Analysis (2010)

Quantitative models and quantitative analysis in Computer Science
are currently intensively studied, resulting in a revision of the
foundation of Computer Science where classical yes/no answers are
replaced by quantitative analyses. The potential application areas
are huge, e.g., performance analysis, operational research or
embedded systems.
The aim of the seminar was to address three fundamental topics
which are closely related: quantitative analysis of real-time and hybrid
systems; probabilistic analysis and stochastic automata; weighted
automata. These three areas of research have mainly evolved
independently so far and the relationship between them has emerged
only recently. The seminar brought together leading researchers
of the three areas, with the goal of future highly productive
cross-fertilizations.

Christel Baier, Manfred Droste, Paul Gastin, and Kim Guldstrand Larsen. 10031 Executive Summary – Quantitative Models: Expressiveness and Analysis. In Quantitative Models: Expressiveness and Analysis. Dagstuhl Seminar Proceedings, Volume 10031, pp. 1-2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{baier_et_al:DagSemProc.10031.2, author = {Baier, Christel and Droste, Manfred and Gastin, Paul and Larsen, Kim Guldstrand}, title = {{10031 Executive Summary – Quantitative Models: Expressiveness and Analysis}}, booktitle = {Quantitative Models: Expressiveness and Analysis}, pages = {1--2}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {10031}, editor = {Christel Baier and Manfred Droste and Paul Gastin and Kim Guldstrand Larsen}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10031.2}, URN = {urn:nbn:de:0030-drops-26824}, doi = {10.4230/DagSemProc.10031.2}, annote = {Keywords: Quantitative models, quantitative analysis, timed and hybrid systems, probabilistic systems, weighted automata} }

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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Volker Diekert, Manfred Droste, Anca Muscholl, and Doron Peled. Logic, Algebra, and Formal Verification of Concurrent Systems (Dagstuhl Seminar 00481). Dagstuhl Seminar Report 292, pp. 1-26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2001)

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@TechReport{diekert_et_al:DagSemRep.292, author = {Diekert, Volker and Droste, Manfred and Muscholl, Anca and Peled, Doron}, title = {{Logic, Algebra, and Formal Verification of Concurrent Systems (Dagstuhl Seminar 00481)}}, pages = {1--26}, ISSN = {1619-0203}, year = {2001}, type = {Dagstuhl Seminar Report}, number = {292}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.292}, URN = {urn:nbn:de:0030-drops-151760}, doi = {10.4230/DagSemRep.292}, }

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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Stephen Brookes, Manfred Droste, and Michael Mislove. Domain Theory and Its Applications (Dagstuhl Seminar 98181). Dagstuhl Seminar Report 209, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (1999)

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@TechReport{brookes_et_al:DagSemRep.209, author = {Brookes, Stephen and Droste, Manfred and Mislove, Michael}, title = {{Domain Theory and Its Applications (Dagstuhl Seminar 98181)}}, pages = {1--18}, ISSN = {1619-0203}, year = {1999}, type = {Dagstuhl Seminar Report}, number = {209}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.209}, URN = {urn:nbn:de:0030-drops-150951}, doi = {10.4230/DagSemRep.209}, }

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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Manfred Droste, Ernst-Rüdiger Olderog, Bernhard Steffen, and Glynn Winskel. Semantics of Concurrent Systems - Foundations and Applications (Dagstuhl Seminar 9619). Dagstuhl Seminar Report 144, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (1996)

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@TechReport{droste_et_al:DagSemRep.144, author = {Droste, Manfred and Olderog, Ernst-R\"{u}diger and Steffen, Bernhard and Winskel, Glynn}, title = {{Semantics of Concurrent Systems - Foundations and Applications (Dagstuhl Seminar 9619)}}, pages = {1--18}, ISSN = {1619-0203}, year = {1996}, type = {Dagstuhl Seminar Report}, number = {144}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.144}, URN = {urn:nbn:de:0030-drops-150310}, doi = {10.4230/DagSemRep.144}, }

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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Manfred Droste and Yuri Gurevich. Semantics of Programming Languages and Algebra (Dagstuhl Seminar 9323). Dagstuhl Seminar Report 65, pp. 1-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (1993)

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@TechReport{droste_et_al:DagSemRep.65, author = {Droste, Manfred and Gurevich, Yuri}, title = {{Semantics of Programming Languages and Algebra (Dagstuhl Seminar 9323)}}, pages = {1--24}, ISSN = {1619-0203}, year = {1993}, type = {Dagstuhl Seminar Report}, number = {65}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.65}, URN = {urn:nbn:de:0030-drops-149534}, doi = {10.4230/DagSemRep.65}, }

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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Manfred Droste and Yuri Gurevich. Semantics of Programming Languages and Model Theory (Dagstuhl Seminar 9126). Dagstuhl Seminar Report 16, pp. 1-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (1991)

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@TechReport{droste_et_al:DagSemRep.16, author = {Droste, Manfred and Gurevich, Yuri}, title = {{Semantics of Programming Languages and Model Theory (Dagstuhl Seminar 9126)}}, pages = {1--24}, ISSN = {1619-0203}, year = {1991}, type = {Dagstuhl Seminar Report}, number = {16}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.16}, URN = {urn:nbn:de:0030-drops-149042}, doi = {10.4230/DagSemRep.16}, }

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